This leads to the introduction of an additional quantum number called isospin. We can define a single-nucleon state function in terms of the quantum numbers \( n \), \( j \), \( m_j \), \( l \), \( s \), \( \tau \) and \( \tau_z \). Using our definitions in terms of an uncoupled basis, we can define a single-particle state with the state function $$ \psi_{njm_j;ls}=\sum_{m_lm_s}\langle lm_lsm_s|jm_j\rangle\phi_{nlm_lsm_s}, $$ which we can now extend to $$ \psi_{njm_j;ls}\xi_{\tau\tau_z}=\sum_{m_lm_s}\langle lm_lsm_s|jm_j\rangle\phi_{nlm_lsm_s}\xi_{\tau\tau_z}, $$ with the isospin spinors defined as $$ \xi_{\tau=1/2\tau_z=+1/2}=\left(\begin{array}{c} 1 \\ 0\end{array}\right), $$ and $$ \xi_{\tau=1/2\tau_z=-1/2}=\left(\begin{array}{c} 0 \\ 1\end{array}\right). $$ We can then define the proton state function as $$ \psi^p(\mathbf{r}) =\psi_{njm_j;ls}(\mathbf{r})\left(\begin{array}{c} 0 \\ 1\end{array}\right), $$ and similarly for neutrons as $$ \psi^n(\mathbf{r}) =\psi_{njm_j;ls}(\mathbf{r})\left(\begin{array}{c} 1 \\ 0\end{array}\right). $$
We can in turn define the isospin Pauli matrices (in the same as we define the spin matrices) as $$ \hat{\tau}_x =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), $$ $$ \hat{\tau}_y =\left(\begin{array}{cc} 0 & -\imath \\ \imath & 0 \end{array}\right), $$ and $$ \hat{\tau}_z =\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), $$ and operating with \( \hat{\tau}_z \) on the proton state function we have $$ \hat{\tau}_z\psi^p(\mathbf{r})=-\frac{1}{2}\psi^p(\mathbf{r}), $$ and for neutrons we have $$ \hat{\tau}\psi^n(\mathbf{r})=\frac{1}{2}\psi^n(\mathbf{r}). $$
We can now define the so-called charge operator as $$ \frac{\hat{Q}}{e} = \frac{1}{2}\left(1-\hat{\tau}_z\right)=\begin{Bmatrix} 0 & 0 \\ 0 & 1 \end{Bmatrix}, $$ which results in $$ \frac{\hat{Q}}{e}\psi^p(\mathbf{r})=\psi^p(\mathbf{r}), $$ and $$ \frac{\hat{Q}}{e}\psi^n(\mathbf{r})=0, $$ as it should be.
The total isospin is defined as $$ \hat{T}=\sum_{i=1}^A\hat{\tau}_i, $$ and its corresponding isospin projection as $$ \hat{T}_z=\sum_{i=1}^A\hat{\tau}_{z_i}, $$ with eigenvalues \( T(T+1) \) for \( \hat{T} \) and \( 1/2(N-Z) \) for \( \hat{T}_z \), where \( N \) is the number of neutrons and \( Z \) the number of protons.
If charge is conserved, the Hamiltonian \( \hat{H} \) commutes with \( \hat{T}_z \) and all members of a given isospin multiplet (that is the same value of \( T \)) have the same energy and there is no \( T_z \) dependence and we say that \( \hat{H} \) is a scalar in isospin space.
The aim is to give you an overview over central features of the nucleon-nucleon interaction and how it is constructed, with both technical and theoretical approaches.
For nuclei with \( A>4 \), the energy saturates: Volume and binding energies of nuclei are proportional to the mass number \( A \).
Nuclei are also bound. The average distance between nucleons in nuclei is about \( 2 \) fm which must roughly correspond to the range of the attractive part.
Charge-symmetry breaking (CSB), after electromagnetic effects have been removed:
To derive the above famous form of the nuclear force using field theoretical concepts, we will need some elements from relativistic quantum mechanics. These derivations will be given below. The material here gives some background to this.
| Baryons | Mass (MeV) | Mesons | Mass (MeV) |
|---|---|---|---|
| \( p,n \) | 938.926 | \( \pi \) | 138.03 |
| \( \Lambda \) | 1116.0 | \( \eta \) | 548.8 |
| \( \Sigma \) | 1197.3 | \( \sigma \) | \( \approx 550.0 \) |
| \( \Delta \) | 1232.0 | \( \rho \) | 770 |
| \( \omega \) | 782.6 | ||
| \( \delta \) | 983.0 | ||
| \( K \) | 495.8 | ||
| \( K^{\star} \) | 895.0 |
But before we proceed, we will look into specific quantum numbers of the relative system and study expectation vaues of the various terms of $$ V(\mathbf{r})= \left\{ C_c + C_\mathbf{\sigma} \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 + C_T \left( 1 + {3\over m_\alpha r} + {3\over \left(m_\alpha r\right)^2}\right) S_{12} (\hat r)\right. $$ $$ \left. + C_{SL} \left( {1\over m_\alpha r} + {1\over \left( m_\alpha r\right)^2} \right) \mathbf{L}\cdot \mathbf{S} \right\} \frac{e^{-m_\alpha r}}{m_\alpha r} $$
When solving the scattering equation or solving the two-nucleon problem, it is convenient to rewrite the Schroedinger equation, due to the spherical symmetry of the Hamiltonian, in relative and center-of-mass coordinates. This will also define the quantum numbers of the relative and center-of-mass system and will aid us later in solving the so-called Lippman-Schwinger equation for the scattering problem.
We define the center-of-mass (CoM) momentum as $$ \mathbf{K}=\sum_{i=1}^A\mathbf{k}_i, $$ with \( \hbar=c=1 \) the wave number \( k_i=p_i \), with \( p_i \) the pertinent momentum of a single-particle state. We have also the relative momentum $$ \mathbf{k}_{ij}=\frac{1}{2}(\mathbf{k}_i-\mathbf{k}_j). $$ We will below skip the indices \( ij \) and simply write \( \mathbf{k} \)
In a similar fashion we can define the CoM coordinate $$ \mathbf{R}=\frac{1}{A}\sum_{i=1}^{A}\mathbf{r}_i, $$ and the relative distance $$ \mathbf{r}_{ij}=(\mathbf{r}_i-\mathbf{r}_j). $$
With the definitions $$ \mathbf{K}=\sum_{i=1}^A\mathbf{k}_i, $$ and $$ \mathbf{k}_{ij}=\frac{1}{2}(\mathbf{k}_i-\mathbf{k}_j). $$ we can rewrite the two-particle kinetic energy (note that we use \( \hbar=c=1 \) as $$ \frac{\mathbf{k}_1^2}{2m_n}+\frac{\mathbf{k}_2^2}{2m_n}=\frac{\mathbf{k}^2}{m_n}+\frac{\mathbf{K}^2}{4m_n}, $$ where \( m_n \) is the average of the proton and the neutron masses.
Since the two-nucleon interaction depends only on the relative distance, this means that we can separate Schroedinger's equation in an equation for the center-of-mass motion and one for the relative motion.
With an equation for the relative motion only and a separate one for the center-of-mass motion we need to redefine the two-body quantum numbers.
Previously we had a two-body state vector defined as \( |(j_1j_2)JM_J\rangle \) in a coupled basis. We will now define the quantum numbers for the relative motion. Here we need to define new orbital momenta (since these are the quantum numbers which change). We define $$ \hat{l}_1+\hat{l}_2=\hat{\lambda}=\hat{l}+\hat{L}, $$ where \( \hat{l} \) is the orbital momentum associated with the relative motion and \( \hat{L} \) the corresponding one linked with the CoM. The total spin \( S \) is unchanged since it acts in a different space. We have thus that $$ \hat{J}=\hat{l}+\hat{L}+\hat{S}, $$ which allows us to define the angular momentum of the relative motion $$ { \cal J} = \hat{l}+\hat{S}, $$ where \( { \cal J} \) is the total angular momentum of the relative motion.
The total two-nucleon state function has to be anti-symmetric. The total function contains a spatial part, a spin part and an isospin part. If isospin is conserved, this leads to in case we have an \( s \)-wave with spin \( S=0 \) to an isospin two-body state with \( T=1 \) since the spatial part is symmetric and the spin part is anti-symmetric.
Since the projections for \( T \) are \( T_z=-1,0,1 \), we can have a \( pp \), an \( nn \) and a \( pn \) state.
For \( l=0 \) and \( S=1 \), a so-called triplet state, \( ^3S_1 \), we must have \( T=0 \), meaning that we have only one state, a \( pn \) state. For other partial waves, the following table lists states up to \( f \) waves. We can systemize this in a table as follows, recalling that \( |\mathbf{l}-\mathbf{S}| \le |\mathbf{J}| \le |\mathbf{l}+\mathbf{S}| \),
| \( ^{2S+1}l_J \) | \( J \) | \( l \) | \( S \) | \( T \) | \( \vert pp\rangle \) | \( \vert pn\rangle \) | \( \vert nn\rangle \) |
| \( ^{1}S_0 \) | 0 | 0 | 0 | 1 | yes | yes | yes |
| \( ^{3}S_1 \) | 1 | 0 | 1 | 0 | no | yes | no |
| \( ^{3}P_0 \) | 0 | 1 | 1 | 1 | yes | yes | yes |
| \( ^{1}P_1 \) | 1 | 1 | 0 | 0 | no | yes | no |
| \( ^{3}P_1 \) | 1 | 1 | 1 | 1 | yes | yes | yes |
| \( ^{3}P_2 \) | 2 | 1 | 1 | 1 | yes | yes | yes |
| \( ^{3}D_1 \) | 1 | 2 | 1 | 0 | no | yes | no |
| \( ^{3}F_2 \) | 2 | 3 | 1 | 1 | yes | yes | yes |
The tensor force is given by $$ S_{12} (\hat r) = \frac{3}{r^2}\left(\mathbf{\sigma}_1\cdot \mathbf{r}\right) \left(\mathbf{\sigma}_2\cdot \mathbf{r}\right) -\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2$$ where the Pauli matrices are defined as $$ \sigma_x =\begin{Bmatrix} 0 & 1 \\ 1 & 0 \end{Bmatrix}, $$ $$ \sigma_y =\begin{Bmatrix} 0 & -\imath \\ \imath & 0 \end{Bmatrix}, $$ and $$ \sigma_z =\begin{Bmatrix} 1 & 0 \\ 0 & -1 \end{Bmatrix}, $$ with the properties \( \sigma = 2\mathbf{S} \) (the spin of the system, being \( 1/2 \) for nucleons), \( \sigma^2_x=\sigma^2_y=\sigma_z=\mathbf{1} \) and obeying the commutation and anti-commutation relations \( \{\sigma_x,\sigma_y\} =0 \) \( [\sigma_x,\sigma_y] =\imath\sigma_z \) etc.
When we look at the expectation value of \( \langle \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle \), we can rewrite this expression in terms of the spin \( \mathbf{S}=\mathbf{s}_1+\mathbf{s}_2 \), resulting in $$ \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=2(S^2-s_1^2-s_2^2)=2S(S+1)-3, $$ where we \( s_1=s_2=1/2 \) leading to $$ \left\{ \begin{array}{cc} \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=1 & \mathrm{if} \hspace{0.2cm} S=1\\ \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=-3 & \mathrm{if} \hspace{0.2cm} S=0\\\end{array}\right. $$
Similarly, the expectation value of the spin-orbit term is $$ \langle \mathbf{l}\mathbf{S} \rangle = \frac{1}{2}\left( J(J+1)-l(l+1)-S(S+1)\right), $$ which means that for \( s \)-waves with either \( S=0 \) and thereby \( J=0 \) or \( S=1 \) and \( J=1 \), the expectation value for the spin-orbit force is zero. With the above phenomenological model, the only contributions to the expectation value of the potential energy for \( s \)-waves stem from the central and the spin-spin components since the expectation value of the tensor force is also zero.
For \( s=1/2 \) spin values only for two nucleons, the expectation value of the tensor force operator is
| \( l' \) | |||
| \( l \) | \( J+1 \) | \( J \) | \( J-1 \) |
| \( J+1 \) | \( -\frac{2J(J+2)}{2J+1} \) | 0 | \( \frac{6\sqrt{J(J+1)}}{2J+1} \) |
| \( J \) | 0 | 2 | 0 |
| \( J-1 \) | \( \frac{6\sqrt{J(J+1)}}{2J+1} \) | 0 | \( -\frac{2(2J+1)}{2J+1} \) |
If we now add isospin to our simple \( V_4 \) interaction model, we end up with \( 8 \) operators, popularly dubbed \( V_8 \) interaction model. The explicit form reads $$ V(\mathbf{r})= \left\{ C_c + C_\mathbf{\sigma} \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 + C_T \left( 1 + {3\over m_\alpha r} + {3\over \left(m_\alpha r\right)^2}\right) S_{12} (\hat r)\right. $$ $$ \left. + C_{SL} \left( {1\over m_\alpha r} + {1\over \left( m_\alpha r\right)^2} \right) \mathbf{L}\cdot \mathbf{S} \right\} \frac{e^{-m_\alpha r}}{m_\alpha r} $$ $$ + \left\{ C_{c\tau} + C_{\sigma\tau}\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 + C_{T\tau} \left( 1 + {3\over m_\alpha r} + {3\over \left(m_\alpha r\right)^2}\right) S_{12} (\hat r)\right. $$ $$ \left. + C_{SL\tau} \left( {1\over m_\alpha r} + {1\over \left( m_\alpha r\right)^2} \right) \mathbf{L}\cdot \mathbf{S} \right\}\mathbf{\tau}_1\cdot\mathbf{\tau}_2 \frac{e^{-m_\alpha r}}{m_\alpha r} $$
From 1950 till approximately 2000: One-Boson-Exchange (OBE) models dominate. These are models which typically include several low-mass mesons, that is with masses below 1 GeV. Potentials which are based upon the standard non-relativistic operator structure are called "Phenomenological Potentials" Some historically important examples are
The total two-nucleon state function has to be anti-symmetric. The total function contains a spatial part, a spin part and an isospin part. If isospin is conserved, this leads to in case we have an \( s \)-wave with spin \( S=0 \) to an isospin two-body state with \( T=1 \) since the spatial part is symmetric and the spin part is anti-symmetric.
Since the projections for \( T \) are \( T_z=-1,0,1 \), we can have a \( pp \), an \( nn \) and a \( pn \) state.
For \( l=0 \) and \( S=1 \), a so-called triplet state, \( ^3S_1 \), we must have \( T=0 \), meaning that we have only one state, a \( pn \) state. For other partial waves, see exercises below.
The one-pion exchange contribution (see derivation below), can be written as $$ V_{\pi}(\mathbf{r})= -\frac{f_{\pi}^{2}}{4\pi m_{\pi}^{2}}\mathbf{ \tau}_1\cdot\mathbf{\tau}_2 \frac{1}{3}\left\{\mathbf{ \sigma}_1\cdot\mathbf{ \sigma}_2+\left( 1 + {3\over m_\pi r} + {3\over\left(m_\pi r\right)^2}\right) S_{12} (\hat r)\right\} \frac{e^{-m_\pi r}}{m_\pi r}. $$ Here the constant \( f_{\pi}^{2}/4\pi\approx 0.08 \) and the mass of the pion is \( m_\pi\approx 140 \) MeV/$\mbox{c}^2$.
Let us look closer at specific partial waves for which one-pion exchange is applicable. If we have \( S=0 \) and \( T=0 \), the orbital momentum has to be an odd number in order for the total anti-symmetry to be obeyed. For \( S=0 \), the tensor force component is zero, meaning that the only contribution is $$ V_{\pi}(\mathbf{r})=\frac{3f_{\pi}^{2}}{4\pi m_{\pi}^{2}}\frac{e^{-m_\pi r}}{m_\pi r}, $$ since \( \langle\mathbf{ \sigma}_1\cdot\mathbf{ \sigma}_2\rangle=-3 \), that is we obtain a repulsive contribution to partial waves like \( ^1P_0 \).
Since \( S=0 \) yields always a zero tensor force contribution, for the combination of \( T=1 \) and then even \( l \) values, we get an attractive contribution $$ V_{\pi}(\mathbf{r})=-\frac{f_{\pi}^{2}}{4\pi m_{\pi}^{2}}\frac{e^{-m_\pi r}}{m_\pi r}. $$ With \( S=1 \) and \( T=0 \), \( l \) can only take even values in order to obey the anti-symmetry requirements and we get $$ V_{\pi}(\mathbf{r})= -\frac{f_{\pi}^{2}}{4\pi m_{\pi}^{2}} \left(1+( 1 + {3\over m_\pi r} + {3\over\left(m_\pi r\right))^2}) S_{12} (\hat r)\right) \frac{e^{-m_\pi r}}{m_\pi r}, $$ while for \( S=1 \) and \( T=1 \), \( l \) can only take odd values, resulting in a repulsive contribution $$ V_{\pi}(\mathbf{r})= \frac{1}{3}\frac{f_{\pi}^{2}}{4\pi m_{\pi}^{2}}\left(1+( 1 + {3\over m_\pi r} + {3\over\left(m_\pi r\right)^2}) S_{12} (\hat r)\right) \frac{e^{-m_\pi r}}{m_\pi r}. $$
The central part of one-pion exchange interaction, arising from the spin-spin term, is thus attractive for \( s \)-waves and all even \( l \) values. For \( p \)-waves and all other odd values it is repulsive. However, its overall strength is weak. This is discussed further in one of exercises below.
For spin \( 1/2 \) baryons, the fields \( \Psi \) are expanded in terms of the Dirac spinors (positive energy solution shown here with \( \overline{u}u=1 \)) $$ u(k\sigma)=\sqrt{\frac{E(k)+m}{2m}} \left(\begin{array}{c} \chi\\ \\ \frac{\mathbf{\sigma}\mathbf{k}}{E(k)+m}\chi \end{array}\right), $$ with \( \chi \) the familiar Pauli spinor and \( E(k) =\sqrt{m^2 +|\mathbf{k}|^2} \). The positive energy part of the field \( \Psi \) reads $$ \Psi (x)={\displaystyle \frac{1}{(2\pi )^{3/2}} \sum_{\mathbf{k}\mathbf{\sigma}}u(k\mathbf{\sigma})\exp{-(ikx)}a_{\mathbf{k}\mathbf{\sigma}}}, $$ with \( a \) being a fermion annihilation operator.
Expanding the free Dirac spinors in terms of \( 1/m \) (\( m \) is here the mass of the relevant baryon) results, to lowest order, in the familiar non-relativistic expressions for baryon-baryon potentials. The configuration space version of the interaction can be approximated as $$ V(\mathbf{r})= \left\{ C^0_C + C^1_C + C_\sigma \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 + C_T \left( 1 + {3\over m_\alpha r} + {3\over \left(m_\alpha r\right)^2} \right) S_{12} (\hat r)\right. $$ $$ + C_{SL}\left. \left( {1\over m_\alpha r} + {1\over \left( m_\alpha r\right)^2} \right) \mathbf{L}\cdot \mathbf{S} \right\} \frac{\exp{-(m_\alpha r)}}{m_\alpha r}, $$ where \( m_{\alpha} \) is the mass of the relevant meson and \( S_{12} \) is the familiar tensor term.
We derive now the non-relativistic one-pion exchange interaction.
Here \( p_{1} \), \( p_{1}' \), \( p_{2} \), \( p_{2}' \) and \( k=p_{1}-p_{1}' \) denote four-momenta. The vertices are given by the pseudovector Lagrangian $$ {\cal L}_{pv}=\frac{f_{\pi}}{m_{\pi}}\overline{\psi}\gamma_{5}\gamma_{\mu} \psi\partial^{\mu}\phi_{\pi}. $$ From the Feynman diagram rules we can write the two-body interaction as $$ V^{pv}=\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\frac{\overline{u}(p_{1}')\gamma_{5} \gamma_{\mu}(p_{1}-p_{1}')^{\mu}u(p_{1})\overline{u}(p_{2}')\gamma_{5} \gamma_{\nu}(p_{2}'-p_{2})^{\nu}u(p_{2})}{(p_{1}-p_{1}')^{2}-m_{\pi}^{2}}. $$
The factors \( p_{1}-p_{1}'=p_{2}'-p_{2} \) are both the four-momentum of the exchanged meson and come from the derivative of the meson field in the interaction Lagrangian. The Dirac spinors obey $$ \begin{align*} \gamma_{\mu}p^{\mu}u(p)&=mu(p) \nonumber \\ \overline{u}(p)\gamma_{\mu}p^{\mu}&=m\overline{u}(p). \nonumber \end{align*} $$
Using these relations, together with \( \{\gamma_{5},\gamma_{\mu}\}=0 \), we find $$ \begin{align*} \overline{u}(p_{1}')\gamma_{5}\gamma_{\mu}(p_{1}-p_{1}')^{\mu}u(p_{1}) &=m\overline{u}(p_{1}')\gamma_{5}u(p_{1})+\overline{u}(p_{1}')\gamma_{\mu} p_{1}'^{\mu}\gamma_{5}u(p_{1}) \nonumber \\ & =2m\overline{u}(p_{1}')\gamma_{5}u(p_{1}) \nonumber \end{align*} $$ and $$ \overline{u}(p_{2}')\gamma_{5}\gamma_{\mu}(p_{2}'-p_{2})^{\mu}= -2m\overline{u}(p_{2}')\gamma_{5}u(p_{1}). $$
We get $$ V^{pv}=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}4m^{2}\frac{\overline{u}(p_{1}') \gamma_{5}u(p_{1})\overline{u}(p_{2}')\gamma_{5}u(p_{2})}{(p_{1}-p_{1}') ^{2}-m_{\pi}^{2}}. $$ By inserting expressions for the Dirac spinors, we find $$ \begin{align*} \overline{u}(p_{1}')\gamma_{5}u(p_{1})&=\sqrt{\frac{(E_{1}'+m)(E_{1}+m)} {4m^{2}}}\left(\begin{array}{cc}\chi^{\dagger}&-\frac{\sigma_{1}\cdot{ \bf p_{1}}}{E_{1}' +m}\chi^{\dagger}\end{array}\right)\left(\begin{array}{cc}0&1\\1&0\end{array} \right)\nonumber \\ &\times \left(\begin{array}{c}\chi\\ \frac{\sigma_{1}\cdot\mathbf{p_{1}}}{E_{1}+m}\chi \end{array}\right) \nonumber \\ &=\sqrt{\frac{(E_{1}'+m)(E_{1}+m)}{4m^{2}}}\left(\frac{\sigma_{1}\cdot \mathbf{p_{1}}}{E_{1}+m}-\frac{\sigma_{1}\cdot\mathbf{p_{1}'}}{E_{1}'+m}\right) \nonumber \end{align*} $$
Similarly $$ \overline{u}(p_{2}')\gamma_{5}u(p_{2})=\sqrt{\frac{(E_{2}'+m)(E_{2}+m)} {4m^{2}}}\left(\frac{\sigma_{2}\cdot \mathbf{p}_{2}}{E_{2}+m}- \frac{\sigma_{2}\cdot\mathbf{p'}_{2}}{E_{2}'+m}\right). $$ In the CM system we have \( \mathbf{p}_{2}=-\mathbf{p}_{1} \), \( \mathbf{p'}_{2}= -\mathbf{p'}_{1} \) and so \( E_{2}=E_{1} \), \( E_{2}'=E_{1}' \). We can then write down the relativistic contribution to the NN potential in the CM system: $$ \begin{align} V^{pv}&=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}4m^{2}\frac{1}{(p_{1}-p_{1}')^{2}- m_{\pi}^{2}}\frac{(E_{1}+m)(E_{1}'+m)}{4m^{2}} \nonumber \\ &\times\left(\frac{\sigma_{1}\cdot\mathbf{p}_{1}}{E_{1}+m}-\frac{\sigma_{1} \cdot\mathbf{p'}_{1}}{E_{1}'+m}\right)\left(\frac{\sigma_{2}\cdot\mathbf{p}_{1}} {E_{1}+m}-\frac{\sigma_{2}\cdot\mathbf{p'}_{1}}{E_{1}'+m}\right). \nonumber \end{align} $$
In the non-relativistic limit we have to lowest order $$ E_{1}=\sqrt{\mathbf{p}_{1}^{2}+m^{2}}\approx m \approx E_{1}' $$ and then \( (p_{1}-p_{1}')^{2}=-\mathbf{k}^{2} \), so we get for the contribution to the NN potential $$ \begin{align} V^{pv}&=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}4m^{2}\frac{1}{\mathbf{k}^{2}+m^{2}} \frac{2m\cdot 2m}{4m^{2}}\frac{\sigma_{1}}{2m}\cdot(\mathbf{p}_{1}-\mathbf{p'}_{1}) \frac{\sigma_{2}}{2m}\cdot (\mathbf{p}_{1}-\mathbf{p'}_{1}) \nonumber \\ &=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \frac{(\sigma_{1}\cdot\mathbf{k})(\sigma_{2}\cdot\mathbf{k})}{\mathbf{k}^{2}+m_{\pi}^{2}}. \nonumber \end{align} $$ We have omitted exchange terms and the isospin term \( \mathbf{\tau}_1\cdot\mathbf{\tau}_2 \).
We have $$ V^{pv}(k)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \frac{(\sigma_{1}\cdot\mathbf{k})(\sigma_{2}\cdot\mathbf{k})}{\mathbf{k}^{2}+m_{\pi}^{2}}. $$ In coordinate space we have $$ V^{pv}(r)=\int\frac{d^3k}{(2\pi)^3}e^{i\mathbf{kr}}V^{pv}(k) $$ resulting in $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla} \int\frac{d^3k}{(2\pi)^3}e^{i\mathbf{kr}}\frac{1}{\mathbf{k}^{2}+m_{\pi}^{2}}. $$
We obtain $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla}\frac{e^{-m_{\pi}r}}{r}. $$
Carrying out the differentation of $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla}\frac{e^{-m_{\pi}r}}{r}. $$ we arrive at the famous one-pion exchange potential with central and tensor parts $$ V(\mathbf{r})= -\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\left\{C_{\sigma}\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2+ C_T \left( 1 + \frac{3}{m_\alpha r} + \frac{3}{\left(m_\alpha r\right)^2}\right) S_{12}(\hat r)\right\}\frac{\exp{-m_\pi r}}{m_\pi r}. $$ For the full potential add the exchange part and the \( \mathbf{\tau}_1\cdot\mathbf{\tau}_2 \) term as well. (Subtle point: there is a divergence which gets cancelled by using cutoffs) This leads to coefficients \( C_{\sigma} \) and \( C_T \) which are fitted to data.
When we perform similar non-relativistic expansions for scalar and vector mesons we obtain for the \( \sigma \) meson $$ V^{\sigma}= g_{\sigma NN}^{2}\frac{1}{\mathbf{k}^{2}+m_{\sigma}^{2}}\left (-1+\frac{\mathbf{q}^{2}}{2M_N^2} -\frac{\mathbf{k}^{2}}{8M_N^2}-\frac{\mathbf{LS}}{2M_N^2}\right). $$ We note an attractive central force and spin-orbit force. This term has an intermediate range. We have defined \( 1/2(p_{1}+p_{1}')=\mathbf{q} \). For the full potential add the exchange part and the isospin dependence as well.
We obtain for the \( \omega \) meson $$ V^{\omega}= g_{\omega NN}^{2}\frac{1}{\mathbf{k}^{2}+m_{\omega}^{2}}\left (1-3\frac{\mathbf{LS}}{2M_N^2}\right). $$ We note a repulsive central force and an attractive spin-orbit force. This term has short range. For the full potential add the exchange part and the isospin dependence as well.
Finally for the \( \rho \) meson $$ V^{\rho}= g_{\rho NN}^{2}\frac{\mathbf{k}^{2}}{\mathbf{k}^{2}+m_{\rho}^{2}}\left ( -2\sigma_{1}\sigma_{2}+S_{12}(\hat{k})\right)\tau_{1}\tau_{2}. $$ We note a tensor force with sign opposite to that of the pion. This term has short range. For the full potential add the exchange part and the isospin dependence as well.
We have seen that:
What follows now is a more technical discussion on how we can solve the two-nucleon problem. This will lead us to the so-called Lippman-Schwinger equation for the scattering problem and a rewrite of Schroedinger's equation in relative and center-of-mass coordinates.
Before we break down the Schroedinger equation into a partial wave decomposition, we derive now the so-called Lippman-Schwinger equation. We will do this in an operator form first. Thereafter, we rewrite it in terms of various quantum numbers such as relative momenta, orbital momenta etc. The Schroedinger equation in abstract vector representation is $$ \left( \hat{H}_0 + \hat{V} \right) \vert \psi_n \rangle = E_n \vert\psi_n \rangle. $$ In our case for the two-body problem \( \hat{H}_0 \) is just the kinetic energy. We rewrite it as $$ \left( \hat{H}_0 -E_n \right)\vert\psi_n \rangle =-\hat{V}\vert \psi_n \rangle . $$ We assume that the invers of \( \left( \hat{H}_0 -E_n\right) \) exists and rewrite this equation as $$ \vert\psi_n \rangle =\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \psi_n \rangle . $$
The equation $$ \vert \psi_n \rangle =\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \psi_n \rangle, $$ is normally solved in an iterative fashion. We assume first that $$ \vert\psi_n \rangle = \vert\phi_n \rangle, $$ where \( \vert\phi_n \rangle \) are the eigenfunctions of $$ \hat{H}_0\vert \phi_n \rangle=\omega_n\vert \phi_n \rangle $$ the so-called unperturbed problem. In our case, these will simply be the kinetic energies of the relative motion.
Inserting \( \vert\phi_n \rangle \) on the right-hand side of $$ \vert \psi_n \rangle =\frac{1}{( E_n -\hat{H}_0)}\hat{V}\vert \psi_n \rangle, $$ yields $$ \vert \psi_n \rangle =\vert\phi_n \rangle+\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \phi_n \rangle, $$ as our first iteration. Reinserting again gives $$ \vert \psi_n \rangle =\vert\phi_n \rangle+\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \phi_n \rangle+\frac{1}{( E_n -\hat{H}_0)}\hat{V}\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \phi_n \rangle, $$ and continuing we obtain $$ \vert \psi_n \rangle =\sum_{i=0}^{\infty}\left[\frac{1}{( E_n -\hat{H}_0)}\hat{V}\right]^i\vert \phi_n \rangle. $$
It is easy to see that $$ \vert \psi_n \rangle =\sum_{i=0}^{\infty}\left[\frac{1}{(E_n -\hat{H}_0)}\hat{V}\right]^i\vert \phi_n \rangle, $$ can be rewritten as $$ \vert \psi_n \rangle =\vert\phi_n \rangle+\frac{1}{( E_n -\hat{H}_0)} \hat{V}\left(1+ \frac{1}{(E_n -\hat{H}_0)}\hat{V}+\frac{1}{(E_n -\hat{H}_0)}\hat{V}\frac{1}{(E_n -\hat{H}_0)}\hat{V}+\dots\right]\vert \phi_n \rangle, $$ which we rewrite as $$ \vert \psi_n \rangle =\vert\phi_n \rangle+\frac{1}{(E_n -\hat{H}_0)}\hat{V}\vert \psi_n \rangle. $$
In operator form we have thus $$ \vert \psi_n \rangle =\vert\phi_n \rangle+\frac{1}{(E_n -\hat{H}_0)}\hat{V}\vert \psi_n \rangle. $$ We multiply from the left with \( \hat{V} \) and \( \langle \phi_m \vert \) and obtain $$ \langle \phi_m \vert\hat{V}\vert \psi_n \rangle =\langle \phi_m \vert\hat{V}\vert\phi_n \rangle+\langle \phi_m \vert\hat{V}\frac{1}{(E_n -\hat{H}_0)}\hat{V}\vert \psi_n \rangle. $$ We define thereafter the so-called \( T \)-matrix as $$ \langle \phi_m \vert\hat{T}\vert \phi_n \rangle=\langle \phi_m \vert\hat{V}\vert \psi_n \rangle. $$ We can rewrite our equation as $$ \langle \phi_m \vert\hat{T}\vert \phi_n \rangle =\langle \phi_m \vert\hat{V}\vert\phi_n \rangle+\langle \phi_m \vert\hat{V}\frac{1}{(E_n -\hat{H}_0)}\hat{T}\vert \phi_n \rangle. $$
The equation $$ \langle \phi_m \vert\hat{T}\vert \phi_n \rangle =\langle \phi_m \vert\hat{V}\vert\phi_n \rangle+\langle \phi_m \vert\hat{V}\frac{1}{(E_n -\hat{H}_0)}\hat{T}\vert \phi_n \rangle, $$ is called the Lippman-Schwinger equation. Inserting the completeness relation $$ \mathbf{1} = \sum_n \vert \phi_n\rangle\langle \phi_n \vert, \:\: \langle \phi_n\vert \phi_{n'} \rangle = \delta_{n,n'} $$ we have $$ \langle \phi_m \vert\hat{T}\vert \phi_n \rangle =\langle \phi_m \vert\hat{V}\vert\phi_n \rangle+\sum_k \langle \phi_m \vert\hat{V}\vert \phi_k\rangle\frac{1}{(E_n -\omega_k)}\langle \phi_k \vert\hat{T}\vert \phi_n \rangle, $$ which is (when we specify the state \( \vert\phi_n \rangle \)) an integral equation that can actually be solved by matrix inversion easily! The unknown quantity is the \( T \)-matrix.
Now we wish to introduce a partial wave decomposition in order to solve the Lippman-Schwinger equation. With a partial wave decomposition we can reduce a three-dimensional integral equation to a one-dimensional one.
Let us continue with our Schroedinger equation in the abstract vector representation $$ \left(T + V\right)\vert\psi_n\rangle = E_n\vert\psi_n \rangle $$ Here \( T \) is the kinetic energy operator and \( V \) is the potential operator. The eigenstates form a complete orthonormal set according to $$ \mathbf{1}=\sum_n\vert\psi_n\rangle\langle\psi_n\vert, \:\: \langle\psi_n\vert\psi_{n'}\rangle =\delta_{n,n'} $$
The most commonly used representations are the coordinate and the momentum space representations. They define the completeness relations $$ \begin{align*} \mathbf{1}&= \int d\mathbf{r} \:\vert\mathbf{r} \rangle \langle \mathbf{r}\vert, \:\: \langle \mathbf{r}\vert \mathbf{r'} \rangle = \delta ( \mathbf{r}-\mathbf{r'}) \\ \mathbf{1} &= \int d\mathbf{k} \:\vert \mathbf{k}\rangle \langle \mathbf{k}\vert, \:\: \langle\mathbf{k}\vert \mathbf{k'} \rangle = \delta ( \mathbf{k}-\mathbf{k'}) \end{align*} $$ Here the basis states in both \( \mathbf{r} \)- and \( \mathbf{k} \)-space are dirac-delta function normalized. From this it follows that the plane-wave states are given by, $$ \langle\mathbf{r}\vert\mathbf{k} \rangle =\left(\frac{1}{2\pi}\right)^{3/2}\exp\left(i\mathbf{k\cdot r} \right) $$ which is a transformation function defining the mapping from the abstract \( \vert\mathbf{k}\rangle \) to the abstract \( \vert\mathbf{r}\rangle \) space.
That the \( \mathbf{r} \)-space basis states are delta-function normalized follows from $$ \delta ( \mathbf{r}-\mathbf{r'}) = \langle \mathbf{r} \vert \mathbf{r}'\rangle = \langle \mathbf{r} \vert \mathbf{1} \vert \mathbf{r}'\rangle = \int d\mathbf{k} \langle \mathbf{r}\vert \mathbf{k} \rangle \langle \mathbf{k}\vert \mathbf{r}' \rangle =\left( {1\over 2\pi}\right)^3 \int d\mathbf{k} e^{i \mathbf{k}(\mathbf{r} - \mathbf{r}')} $$ and the same for the momentum space basis states, $$ \delta ( \mathbf{k}-\mathbf{k'}) = \langle \mathbf{k} \vert \mathbf{k}'\rangle = \langle \mathbf{k} \vert \mathbf{1} \vert \mathbf{k}'\rangle =\int d\mathbf{r} \langle \mathbf{k}\vert \mathbf{r} \rangle \langle \mathbf{r}\vert \mathbf{k}' \rangle = \left( {1\over 2\pi}\right)^3 \int d\mathbf{r} e^{i \mathbf{r}(\mathbf{k} - \mathbf{k}')} $$
Projecting on momentum states, we obtain the momentum space Schroedinger equation as $$ \begin{equation} \frac{\hbar^2}{2\mu}k^2\psi_n(\mathbf{k})+\int d\mathbf{k'}V(\mathbf{k}, \mathbf{k'}) \psi_n(\mathbf{k'})=E_n \psi_n(\mathbf{k}) \tag{1} \end{equation} $$ Here the notation \( \psi_n(\mathbf{k}) =\langle\mathbf{k}\vert\psi_n\rangle \) and \( \langle\mathbf{k}\vert V\vert\mathbf{k}' \rangle =V(\mathbf{k}, \mathbf{k'}) \) has been introduced. The potential in momentum space is given by a double Fourier-transform of the potential in coordinate space, i.e. $$ V(\mathbf{k},\mathbf{k'}) = \left( \frac{1}{2\pi}\right)^3\int d\mathbf{r}\int d\mathbf{r}'\exp{-i\mathbf{kr}}V(\mathbf{r},\mathbf{r}')\exp{i\mathbf{k}'\mathbf{r}'} $$
Here it is assumed that the potential interaction does not contain any spin dependence. Instead of a differential equation in coordinate space, the Schroedinger equation becomes an integral equation in momentum space. This has many tractable features. Firstly, most realistic nucleon-nucleon interactions derived from field-theory are given explicitly in momentum space. Secondly, the boundary conditions imposed on the differential equation in coordinate space are automatically built into the integral equation. And last, but not least, integral equations are easy to numerically implement, and convergence is obtained by just increasing the number of integration points. Instead of solving the three-dimensional integral equation, an infinite set of 1-dimensional equations can be obtained via a partial wave expansion.
The wave function \( \psi_n(\mathbf{k}) \) can be expanded in a complete set of spherical harmonics, that is $$ \begin{equation} \psi_n(\mathbf{k}) = \sum _{lm} \psi_{nlm}(k)Y_{lm}(\hat{k}) \hspace{1cm} \psi_{nlm}(k) = \int d\hat{k} Y_{lm}^*(\hat{k})\psi_n(\mathbf{k}). , \tag{2} \end{equation} $$ By inserting equation (2) in equation (1), and projecting from the left \( Y_{lm}(\hat{k}) \), the three-dimensional Schroedinger Eq. (1) is reduced to an infinite set of 1-dimensional angular momentum coupled integral equations, $$ \begin{equation} \left( \frac{\hbar^2}{2\mu} k^2-E_{nlm}\right)\psi_{nlm}(k) = -\sum_{l'm'}\int_{0}^\infty dk' {k'}^2 V_{lm, l'm'}(k,k') \psi_{nl'm'}(k') \tag{3} \end{equation} $$ where the angular momentum projected potential takes the form, $$ \begin{equation} V_{lm, l'm'}(k,k') = \int d{\hat{k}} \int d{\hat{k}'}Y_{lm}^*(\hat{k})V(\mathbf{k}\mathbf{k'})Y_{l'm'}(\hat{k}') \tag{4} \end{equation} $$ here \( d\hat{k} = d\theta\sin(\theta)d\varphi \). Note that we discuss only the orbital momentum, we will include angular momentum and spin later.
The potential is often given in position space. It is then convenient to establish the connection between \( V_{lm, l'm'}(k,k') \) and \( V_{lm, l'm'}(r,r') \). Inserting the completeness relation for the position quantum numbers in equation (4) results in $$ \begin{equation} V =\int d\mathbf{r}\int d\mathbf{r}'\left\{\int d{\hat{k}}Y_{lm}^*(\hat{k})\langle \mathbf{k}\vert \mathbf{r}\rangle\right\}\langle\mathbf{r}\vert V\vert\mathbf{r}'\rangle\left\{\int d\hat{k}'Y_{lm}(\hat{k}')\langle\mathbf{r'}\vert\mathbf{k}'\rangle\right\} \tag{5} \end{equation} $$
Since the plane waves depend only on the absolute values of position and momentum, \( \vert\mathbf{k}\vert \) and \( \vert\mathbf{r}\vert \), and the angle between them, \( \theta_{kr} \), they may be expanded in terms of bipolar harmonics of zero rank, i.e. $$ \exp{(i \mathbf{k}\cdot \mathbf{r})} = 4\pi\sum_{l=0}^{\infty} i^l j_l(kr)\left( Y_l(\hat{k}) \cdot Y_l(\hat{r}) \right)= \sum_{l=0}^{\infty} (2l+1)i^l j_l(kr) P_l(\cos \theta_{kr}) $$ where the addition theorem for spherical harmonics has been used in order to write the expansion in terms of Legendre polynomials. The spherical Bessel functions, \( j_l(z) \), are given in terms of Bessel functions of the first kind with half integer orders, $$ j_l(z) = \sqrt{\pi \over 2 z} J_{l+1/2}(z). $$
Inserting the plane-wave expansion into the brackets of Eq. (5) yields, $$ \begin{align*} \nonumber \int d{\hat{k}} Y_{lm}^*(\hat{k})\langle \mathbf{k}\vert \mathbf{r} \rangle & = \left( {1\over 2\pi} \right) ^{3/2}4\pi i^{-l} j_l(kr) Y_{lm}^*(\hat{r}), \\ \nonumber \int d{\hat{k}'}\: Y_{lm}(\hat{k}') \langle \mathbf{r'}\vert \mathbf{k}' \rangle & = \left( {1\over 2\pi} \right) ^{3/2}4\pi i^{l'} j_{l'}(k'r') Y_{l'm'}(\hat{r}). \end{align*} $$
The connection between the momentum- and position space angular momentum projected potentials are then given, $$ V_{lm, l'm'}(k,k')=\frac{2}{\pi}i^{l'-l}\int_0^\infty drr^2 \int_0^\infty dr'{r'}^2j_l(kr) V_{lm,l'm'}(r,r') j_{l'}(k'r') \tag{6} $$ which is known as a double Fourier-Bessel transform. The position space angular momentum projected potential is given by $$ V_{lm, l'm'}(r,r') = \int d{\hat{r}} \int d{\hat{r}'}Y_{lm}^*(\hat{r})V(\mathbf{r}, \mathbf{r'})Y_{l'm'}(\hat{r}'). \tag{7} $$
No assumptions of locality/non-locality and deformation of the interaction has so far been made, and the result in Eq. (6) is general. In position space the Schroedinger equation takes form of an integro-differential equation in case of a non-local interaction, in momentum space the Schroedinger equation is an ordinary integral equation of the Fredholm type, see Eq. (3). This is a further advantage of the momentum space approach as compared to the standard position space approach. If we assume that the interaction is of local character, i.e. $$ \langle \mathbf{r}\vert V \vert \mathbf{r'}\rangle = V(\mathbf{r}) \delta( \mathbf{r}-\mathbf{r}' ) = V(\mathbf{r}) {\delta( { r}-{r}' ) \over r^2} \delta ( \cos \theta - \cos \theta' ) \delta (\varphi-\varphi'), $$ then Eq. (7) reduces to $$ \begin{equation} V_{lm, l'm'}(r,r') = \frac{\delta({r}-{r}')}{r^2}\int d{\hat{r}}\: Y_{lm}^*(\hat{r})V(\mathbf{r})Y_{l'm'}(\hat{r}), \tag{8} \end{equation} $$ and Eq. (6) reduces to $$ \begin{equation} V_{lm, l'm'}(k,k') = \frac{2}{\pi}i^{l' -l}\int_0^\infty drr^2j_l(kr) V_{lm,l'm'}(r) j_{l'}(k'r) \tag{9} \end{equation} $$ where $$ \begin{equation} V_{lm, l'm'}(r) = \int d{\hat{r}}Y_{lm}^*(\hat{r})V(\mathbf{r})Y_{l'm'}(\hat{r}), \tag{10} \end{equation} $$
In the case that the interaction is central, \( V(\mathbf{r}) = V(r) \), then $$ \begin{equation} V_{lm, l'm'}(r) = V(r) \int d{\hat{r}}Y_{lm}^*(\hat{r})Y_{l'm'}(\hat{r}) = V(r) \delta_{l,l'}\delta_{m,m'}, \tag{11} \end{equation} $$ and $$ \begin{equation} V_{lm, l'm'}(k,k') = \frac{2}{\pi} \int_0^\infty drr^2j_l(kr) V(r) j_{l'}(k'r)\delta_{l,l'}\delta_{m,m'} = V_l(k,k') \delta_{l,l'}\delta_{m,m'} \tag{12} \end{equation} $$ where the momentum space representation of the interaction finally reads, $$ \begin{equation} V_{l}(k,k') = {2 \over \pi} \int_0^\infty dr\: r^2 \: j_l(kr) V(r) j_{l}(k'r). \tag{13} \end{equation} $$
For a local and spherical symmetric potential, the coupled momentum space Schroedinger equations given in Eq. (3) decouples in angular momentum, giving $$ \begin{equation} \frac{\hbar^2}{2\mu} k^2 \psi_{n l}(k) +\int_{0}^\infty dk' {k'}^2 V_{l}(k,k') \psi_{n l }(k')=E_{n l} \psi_{n l}(k) \tag{14} \end{equation} $$ Where we have written \( \psi_{n l }(k)=\psi_{nlm}(k) \), since the equation becomes independent of the projection \( m \) for spherical symmetric interactions. The momentum space wave functions \( \psi_{n l}(k) \) defines a complete orthogonal set of functions, which spans the space of functions with a positive finite Euclidean norm (also called \( l^2 \)-norm), \( \sqrt{\langle\psi_n\vert\psi_n\rangle} \), which is a Hilbert space. The corresponding normalized wave function in coordinate space is given by the Fourier-Bessel transform $$ \phi_{n l}(r) = \sqrt{\frac{2}{\pi}}\int dk k^2 j_l(kr) \psi_{n l}(k) $$
We will thus assume that the interaction is spherically symmetric and use the partial wave expansion of the plane waves in terms of spherical harmonics. This means that we can separate the radial part of the wave function from its angular dependence. The wave function of the relative motion is described in terms of plane waves as $$ \exp{(\imath \mathbf{kr})}=\langle\mathbf{r}\vert\mathbf{k}\rangle=4\pi\sum_{lm}\imath^{l}j_{l}(kr)Y_{lm}^{*}(\mathbf{\hat{k}})Y_{lm}(\mathbf{\hat{r}}), $$ where \( j_l \) is a spherical Bessel function and \( Y_{lm} \) the spherical harmonics.
In terms of the relative and center-of-mass momenta \( \mathbf{k} \) and \( \mathbf{K} \), the potential in momentum space is related to the nonlocal operator \( V(\mathbf{r},\mathbf{r}') \) by $$ \langle\mathbf{k'K'}\vert V \vert \mathbf{kK}\rangle =\int d\mathbf{r}d \mathbf{r'} \exp{-(\imath \mathbf{k'r'})}V(\mathbf{r'},\mathbf{r})\exp{\imath \mathbf{kr}}\delta(\mathbf{K},\mathbf{K'}). $$ We will assume that the interaction is spherically symmetric. Can separate the radial part of the wave function from its angular dependence. The wave function of the relative motion is described in terms of plane waves as $$ \exp{(\imath \mathbf{kr})} =\langle\mathbf{r}\vert\mathbf{k}\rangle= 4\pi\sum_{lm}\imath^{l}j_{l}(kr)Y_{lm}^{*}(\mathbf{\hat{k}})Y_{lm}(\mathbf{\hat{r}}), $$ where \( j_l \) is a spherical Bessel function and \( Y_{lm} \) the spherical harmonic.
This partial wave basis is useful for defining the operator for the nucleon-nucleon interaction, which is symmetric with respect to rotations, parity and isospin transformations. These symmetries imply that the interaction is diagonal with respect to the quantum numbers of total relative angular momentum \( {\cal J} \), spin \( S \) and isospin \( T \) (we skip isospin for the moment). Using the above plane wave expansion, and coupling to final \( {\cal J} \) and \( S \) and \( T \) we get $$ \langle\mathbf{k'}\vert V \vert\mathbf{k}\rangle= (4\pi)^2 \sum_{STll'm_lm_{l'}{\cal J}}\imath^{l+l'} Y_{lm}^{*}(\mathbf{\hat{k}}) Y_{l'm'}(\mathbf{\hat{k}'}) $$ $$ \langle lm_lSm_S|{\cal J}M\rangle \langle l'm_{l'}Sm_S|{\cal J}M\rangle\langle k'l'S{\cal J}M\vert V \vert klS{\cal J}M\rangle, $$ where we have defined $$ \langle k'l'S{\cal J}M\vert V \vert klS{\cal J}M\rangle=\int j_{l'}(k'r')\langle l'S{\cal J}M\vert V(r',r)\vert lS{\cal J}M\rangle j_l(kr) {r'}^2 dr' r^2 dr. $$ We have omitted the momentum of the center-of-mass motion \( \mathbf{K} \) and the corresponding orbital momentum \( L \), since the interaction is diagonal in these variables.
We wrote the Lippman-Schwinger equation as $$ \langle \phi_m \vert\hat{T}\vert \phi_n \rangle =\langle \phi_m \vert\hat{V}\vert\phi_n \rangle+\sum_k \langle \phi_m \vert\hat{V}\vert \phi_k\rangle\frac{1}{(E_n -\omega_k)}\langle \phi_k \vert\hat{T}\vert \phi_n \rangle. $$ How do we rewrite it in a partial wave expansion with momenta \( k \)?
The general structure of the \( T \)-matrix in partial waves is $$ T_{ll'}^{\alpha}(kk'K\omega)=V_{ll'}^{\alpha}(kk') $$ $$ \begin{equation} +{\displaystyle \frac{2}{\pi}\sum_{l''m_{l''}M_S}\int_{0}^{\infty} d \mathbf{q} (\langle l''m_{l''}Sm_S|{\cal J}M\rangle)^2 \frac{Y_{l''m_{l''}}^*(\hat{\mathbf{q}}) Y_{l''m_{l''}}(\hat{\mathbf{q}}) V_{ll''}^{\alpha}(kq) T_{l''l'}^{\alpha}(qk'K\omega)} {\omega -H_0}}, \tag{15} \end{equation} $$
The shorthand notation $$ T_{ll'}^{\alpha}(kk'K\omega)= \langle kKlL{\cal J}S\vert T(\omega)\vert k'Kl'L{\cal J}S\rangle, $$ denotes the \( T \)-matrix with momenta \( k \) and \( k' \) and orbital momenta \( l \) and \( l' \) of the relative motion, and \( K \) is the corresponding momentum of the center-of-mass motion. Further, \( L \), \( {\cal J} \), \( S \) and \( T \) are the orbital momentum of the center-of-mass motion, the total angular momentum, spin and isospin, respectively. Due to the nuclear tensor force, the interaction is not diagonal in \( ll' \).
Using the orthogonality properties of the Clebsch-Gordan coefficients and the spherical harmonics, we obtain the well-known one-dimensional angle independent integral equation $$ T_{ll'}^{\alpha}(kk'K\omega)=V_{ll'}^{\alpha}(kk') +\frac{2}{\pi}\sum_{l''}\int_{0}^{\infty} dqq^2 \frac{V_{ll''}^{\alpha}(kq) T_{l''l'}^{\alpha}(qk'K\omega)} {\omega -H_0}. $$ Inserting the denominator we arrive at $$ \hat{T}_{ll'}^{\alpha}(kk'K)=\hat{V}_{ll'}^{\alpha}(kk') +\frac{2}{\pi}\sum_{l''}\int_{0}^{\infty} dqq^2 \hat{V}_{ll''}^{\alpha}(kq) \frac{1}{k^2-q^2 +i\epsilon} \hat{T}_{l''l'}^{\alpha}(qk'K). $$ To parameterize the nucleon-nucleon interaction we solve the Lippman-Scwhinger equation $$ T_{ll'}^{\alpha}(kk'K)=V_{ll'}^{\alpha}(kk') +\frac{2}{\pi}\sum_{l''}\int_{0}^{\infty} dqq^2 V_{ll''}^{\alpha}(kq) \frac{1}{k^2-q^2 +i\epsilon} T_{l''l'}^{\alpha}(qk'K). $$ The shorthand notation $$ T(\hat{V})_{ll'}^{\alpha}(kk'K\omega)=\langle kKlL{\cal J}S\vert T(\omega)\vert k'Kl'L{\cal J}S\rangle, $$ denotes the \( T(V) \)-matrix with momenta \( k \) and \( k' \) and orbital momenta \( l \) and \( l' \) of the relative motion, and \( K \) is the corresponding momentum of the center-of-mass motion. Further, \( L \), \( {\cal J} \), and \( S \) are the orbital momentum of the center-of-mass motion, the total angular momentum and spin, respectively. We skip for the moment isospin.
For scattering states, the energy is positive, \( E>0 \). The Lippman-Schwinger equation (a rewrite of the Schroedinger equation) is an integral equation where we have to deal with the amplitude \( R(k,k') \) (reaction matrix, which is the real part of the full complex \( T \)-matrix) defined through the integral equation for one partial wave (no coupled-channels) $$ \begin{equation} R_l(k,k') = V_l(k,k') +\frac{2}{\pi}{\cal P} \int_0^{\infty}dqq^2V_l(k,q)\frac{1}{E-q^2/m}R_l(q,k'). \tag{16} \end{equation} $$ For negative energies (bound states) and intermediate states scattering states blocked by occupied states below the Fermi level.
The symbol \( {\cal P} \) in the previous slide indicates that Cauchy's principal-value prescription is used in order to avoid the singularity arising from the zero of the denominator.
The total kinetic energy of the two incoming particles in the center-of-mass system is $$ E=\frac{k_0^2}{m_n}. $$
The matrix \( R_l(k,k') \) relates to the the phase shifts through its diagonal elements as $$ \begin{equation} R_l(k_0,k_0)=-\frac{tan\delta_l}{mk_0}. \tag{17} \end{equation} $$
From now on we will drop the subscript \( l \) in all equations. In order to solve the Lippman-Schwinger equation in momentum space, we need first to write a function which sets up the mesh points. We need to do that since we are going to approximate an integral through $$ \int_a^bf(x)dx\approx\sum_{i=1}^Nw_if(x_i), $$ where we have fixed \( N \) lattice points through the corresponding weights \( w_i \) and points \( x_i \). Typically obtained via methods like Gaussian quadrature.
If you use Gauss-Legendre the points are determined for the interval \( x_i\in [-1,1] \) You map these points over to the limits in your integral. You can then use the following mapping $$ k_i=const\times tan\left\{\frac{\pi}{4}(1+x_i)\right\}, $$ and $$ \omega_i= const\frac{\pi}{4}\frac{w_i}{cos^2\left(\frac{\pi}{4}(1+x_i)\right)}. $$ If you choose units fm$^{-1}$ for \( k \), set \( const=1 \). If you choose to work with MeV, set \( const\sim 200 \) (\( \hbar c=197 \) MeVfm).
The principal value integral is rather tricky to evaluate numerically, mainly since computers have limited precision. We will here use a subtraction trick often used when dealing with singular integrals in numerical calculations. We introduce first the calculus relation $$ \int_{-\infty}^{\infty} \frac{dk}{k-k_0} =0. $$ It means that the curve \( 1/(k-k_0) \) has equal and opposite areas on both sides of the singular point \( k_0 \). If we break the integral into one over positive \( k \) and one over negative \( k \), a change of variable \( k\rightarrow -k \) allows us to rewrite the last equation as $$ \int_{0}^{\infty} \frac{dk}{k^2-k_0^2} =0. $$
We can then express a principal values integral as $$ \begin{equation} {\cal P}\int_{0}^{\infty} \frac{f(k)dk}{k^2-k_0^2} = \int_{0}^{\infty} \frac{(f(k)-f(k_0))dk}{k^2-k_0^2}, \tag{18} \end{equation} $$ where the right-hand side is no longer singular at \( k=k_0 \), it is proportional to the derivative \( df/dk \), and can be evaluated numerically as any other integral.
We can then use this trick to obtain $$ \begin{equation} R(k,k') = V(k,k') +\frac{2}{\pi} \int_0^{\infty}dq \frac{q^2V(k,q)R(q,k')-k_0^2V(k,k_0)R(k_0,k') } {(k_0^2-q^2)/m}. \tag{19} \end{equation} $$ This is the equation to solve numerically in order to calculate the phase shifts. We are interested in obtaining \( R(k_0,k_0) \).
How do we proceed?
Using the mesh points \( k_j \) and the weights \( \omega_j \), we reach $$ R(k,k') = V(k,k') +\frac{2}{\pi} \sum_{j=1}^N\frac{\omega_jk_j^2V(k,k_j)R(k_j,k')} {(k_0^2-k_j^2)/m} -\frac{2}{\pi}k_0^2V(k,k_0)R(k_0,k') \sum_{n=1}^N\frac{\omega_n} {(k_0^2-k_n^2)/m}. $$
This equation contains now the unknowns \( R(k_i,k_j) \) (with dimension \( N\times N \)) and \( R(k_0,k_0) \).
We can turn it into an equation with dimension \( (N+1)\times (N+1) \) with a mesh which contains the original mesh points \( k_j \) for \( j=1,N \) and the point which corresponds to the energy \( k_0 \). Consider the latter as the 'observable' point. The mesh points become then \( k_j \) for \( j=1,n \) and \( k_{N+1}=k_0 \).
With these new mesh points we define the matrix $$ \begin{equation} A_{i,j}=\delta_{i,j}-V(k_i,k_j)u_j, \tag{20} \end{equation} $$ where \( \delta \) is the Kronecker \( \delta \) and $$ u_j=\frac{2}{\pi}\frac{\omega_jk_j^2}{(k_0^2-k_j^2)/m}\hspace{1cm} j=1,N $$ and $$ u_{N+1}=-\frac{2}{\pi}\sum_{j=1}^N\frac{k_0^2\omega_j}{(k_0^2-k_j^2)/m}. $$
The first task is then to set up the matrix \( A \) for a given \( k_0 \). This is an \( (N+1)\times (N+1) \) matrix. It can be convenient to have an outer loop which runs over the chosen observable values for the energy \( k_0^2/m \). {\em Note that all mesh points \( k_j \) for \( j=1,N \) must be different from \( k_0 \). Note also that \( V(k_i,k_j) \) is an \( (N+1)\times (N+1) \) matrix}.
With the matrix \( A \) we can rewrite the problem as a matrix problem of dimension \( (N+1)\times (N+1) \). All matrices \( R \), \( A \) and \( V \) have this dimension and we get $$ A_{i,l}R_{l,j}=V_{i,j}, $$ or just $$ AR=V. $$
Since you already have defined \( A \) and \( V \) (these are stored as \( (N+1)\times (N+1) \) matrices) The final equation involves only the unknown \( R \). We obtain it by matrix inversion, i.e., $$ \begin{equation} R=A^{-1}V. \tag{21} \end{equation} $$ Thus, to obtain \( R \), you will need to set up the matrices \( A \) and \( V \) and invert the matrix \( A \). With the inverse \( A^{-1} \), perform a matrix multiplication with \( V \) results in \( R \).
With \( R \) you can then evaluate the phase shifts by noting that $$ R(k_{N+1},k_{N+1})=R(k_0,k_0)=-\frac{tan\delta}{mk_0}, $$ where \( \delta \) are the phase shifts.
For elastic scattering, the scattering potential can only change the outgoing spherical wave function up to a phase. In the asymptotic limit, far away from the scattering potential, we get for the spherical bessel function $$ j_l(kr) \xrightarrow[]{ r \gg 1} \frac{\sin(kr -l\pi/2)}{kr} = \frac{1}{2ik}\left( \frac{e^{i(kr-l\pi/2)}}{r} - \frac{e^{-i(kr-l\pi/2)}}{r}\right) $$ The outgoing wave will change by a phase shift \( \delta_l \), from which we can define the S-matrix \( S_l(k) = e^{2i\delta_l(k)} \). Thus, we have $$ \frac{e^{i(kr-l\pi/2)}}{r} \xrightarrow[]{\mathrm{phase change}} \frac{S_l(k)e^{i(kr-l\pi/2)}}{r} $$
The solution to the Schrodinger equation for a spherically symmetric potential, will have the form $$ \psi_k(r) = e^{ikr} + f(\theta)\frac{e^{ikr}}{r} $$ where \( f(\theta) \) is the scattering amplitude, and related to the differential cross section as $$ \frac{d\sigma}{d\Omega} = |f(\theta)|^2 $$ Using the expansion of a plane wave in spherical waves, we can relate the scattering amplitude \( f(\theta) \) with the partial wave phase shifts \( \delta_l \) by identifying the outgoing wave $$ \psi_k(r) = e^{ikr} + \left[\frac{1}{2ik}\sum_l i^l (2l+1) (S_l(k)-1)P_l(\cos(\theta))e^{-il\pi/2}\right] \frac{e^{ikr}}{r} $$ which can be simplified further by cancelling \( i^l \) with \( e^{-il\pi/2} \)
We have $$ \psi_k(r) = e^{ikr} + f(\theta) \frac{e^{ikr}}{r} $$ with $$ f(\theta) = \sum_l (2l+1)f_l(\theta) P_l(\cos(\theta)) $$ where the partial wave scattering amplitude is given by $$ f_l(\theta) = \frac{1}{k}\frac{(S_l(k)-1)}{2i} = \frac{1}{k}\sin\delta_l(k) e^{i\delta_l(k)} $$ With Eulers formula for the cotangent, this can also be written as $$ f_l(\theta) = \frac{1}{k}\frac{1}{\cot \delta_l(k) - i}. $$
The integrated cross section is given by \[ \sigma = 2\pi \int_0^{\pi} |f(\theta)|^2 \sin \theta d\theta \] \[ =2\pi \sum_l |\frac{(2l+1)}{k} \sin(\delta_l)|^2 \int_0^{\pi} (P_l(\cos(\theta)))^2 \sin(\theta) d\theta\] \[ = \frac{4\pi}{k^2} \sum_l (2l+1) \sin^2\delta_l(k) = 4\pi \sum_l (2l+1)|f_l(\theta)|^2, \] where the orthogonality of the Legendre polynomials was used to evaluate the last integral \[ \int_0^{\pi} P_l(\cos \theta)^2 \sin \theta d\theta = \frac{2}{2l+1}. \] Thus, the total cross section is the sum of the partial-wave cross sections. Note that the differential cross section contains cross-terms from different partial waves. The integral over the full sphere enables the use of the legendre orthogonality, and this kills the cross-terms.
At low energy, \( k \rightarrow 0 \), S-waves are most important. In this region we can define the scattering length \( a \) and the effective range \( r \). The $S-$wave scattering amplitude is given by \[ f_l(\theta) = \frac{1}{k}\frac{1}{\cot \delta_l(k) - i}. \] Taking the limit \( k \rightarrow 0 \), gives us the expansion \[ k \cot \delta_0 = -\frac{1}{a} + \frac{1}{2}r_0 k^2 + \ldots \] Thus the low energy cross section is given by \[ \sigma = 4\pi a^2. \] If the system contains a bound state, the scattering length will become positive (neutron-proton in \( ^3S_1 \)). For the \( ^1S_0 \) wave, the scattering length is negative and large. This indicates that the wave function of the system is at the verge of turning over to get a node, but cannot create a bound state in this wave.
It is important to realize that the phase shifts themselves are not observables. The measurable scattering quantity is the cross section, or the differential cross section. The partial wave phase shifts can be thought of as a parameterization of the (experimental) cross sections. The phase shifts provide insights into the physics of partial wave projected nuclear interactions, and are thus important quantities to know.
The nucleon-nucleon differential cross section have been measured at almost all energies up to the pion production threshold (290 MeV in the Lab frame), and this experimental data base is what provides us with the constraints on our nuclear interaction models. In order to pin down the unknown coupling constants of the theory, a statistical optimization with respect to cross sections need to be carried out. This is how we constrain the nucleon-nucleon interaction in practice!
The \( pp \)-data is more accurate than the \( np \)-data, and for \( nn \) there is no data. The quality of a potential is gauged by the $\chi^2$/datum with respect to the scattering data base
| \( T_{\mathrm{lab}} \) bin (MeV) | N3LO$^1$ | NNLO$^2$ | NLO$^2$ | AV18$^3$ |
|---|---|---|---|---|
| 0-100 | 1.05 | 1.7 | 4.5 | 0.95 |
| 100-190 | 1.08 | 22 | 100 | 1.10 |
| 190-290 | 1.15 | 47 | 180 | 1.11 |
| \( \mathbf{0-290} \) | \( \mathbf{1.10} \) | \( \mathbf{20} \) | \( \mathbf{86} \) | \( \mathbf{1.04} \) |
a) List all allowed according to the Pauli principle partial waves with isospin \( T \), their projection \( T_z \), spin \( S \), orbital angular momentum \( l \) and total spin \( J \) for \( J\le 3 \). Use the standard spectroscopic notation \( ^{2S+1}L_J \) to label different partial waves. A proton-proton state has \( T_Z=-1 \), a proton-neutron state has \( T_z=0 \) and a neutron-neutron state has \( T_z=1 \).
a) Find the closed form expression for the spin-orbit force. Show that the spin-orbit force {\bf LS} gives a zero contribution for \( S \)-waves (orbital angular momentum \( l=0 \)). What is the value of the spin-orbit force for spin-singlet states (\( S=0 \))?
b) Find thereafter the expectation value of \( \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 \), where \( \mathbf{\sigma}_i \) are so-called Pauli matrices.
c) Add thereafter isospin and find the expectation value of \( \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\mathbf{\tau}_1\cdot\mathbf{\tau}_2 \), where \( \mathbf{\tau}_i \) are also so-called Pauli matrices. List all the cases with \( S=0,1 \) and \( T=0,1 \).
A simple parametrization of the nucleon-nucleon force is given by what is called the \( V_8 \) potential model, where we have kept eight different operators. These operators contain a central force, a spin-orbit force, a spin-spin force and a tensor force. Several features of the nuclei can be explained in terms of these four components. Without the Pauli matrices for isospin the final form of such an interaction model results in the following form: $$ V(\mathbf{r})= \left\{ C_c + C_\mathbf{\sigma} \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 + C_T \left( 1 + {3\over m_\alpha r} + {3\over\left(m_\alpha r\right)^2}\right) S_{12} (\hat r)\right. $$ $$ \left. + C_{SL} \left( {1\over m_\alpha r} + {1\over \left( m_\alpha r\right)^2} \right) \mathbf{L}\cdot \mathbf{S} \right\} \frac{e^{-m_\alpha r}}{m_\alpha r} $$ where \( m_{\alpha} \) is the mass of the relevant meson and \( S_{12} \) is the familiar tensor term. The various coefficients \( C_i \) are normally fitted so that the potential reproduces experimental scattering cross sections. By adding terms which include the isospin Pauli matrices results in an interaction model with eight operators.
The expectaction value of the tensor operator is non-zero only for \( S=1 \). We will show this in a forthcoming lecture, after that we have derived the Wigner-Eckart theorem. Here it suffices to know that the expectaction value of the tensor force for different partial values is (with \( l \) the orbital angular momentum and \( {\cal J} \) the total angular momentum in the relative and center-of-mass frame of motion) $$ \langle l {\cal J}S=1| S_{12} | l' {\cal J}S=1\rangle = -\frac{2{\cal J}({\cal J}+2)}{2{\cal J}+1} \hspace{0.5cm} l= {\cal J}+1 \hspace{0.1cm}\mathrm{and} \hspace{0.1cm} l'={\cal J}+1, $$ $$ \langle l {\cal J}S=1| S_{12} | l' {\cal J}S=1\rangle = \frac{6\sqrt{{\cal J}({\cal J}+1)}}{2{\cal J}+1} \hspace{0.5cm} l= {\cal J}+1 \hspace{0.1cm}\mathrm{and} \hspace{0.1cm} l'={\cal J}-1, $$ $$ \langle l {\cal J}S=1| S_{12} | l' {\cal J}S=1\rangle = \frac{6\sqrt{{\cal J}({\cal J}+1)}}{2{\cal J}+1} \hspace{0.5cm} l= {\cal J}-1 \hspace{0.1cm}\mathrm{and} \hspace{0.1cm} l'={\cal J}+1, $$ $$ \langle l {\cal J}S=1| S_{12} | l' {\cal J}S=1\rangle = -\frac{2({\cal J}-1)}{2{\cal J}+1} \hspace{0.5cm} l= {\cal J}-1 \hspace{0.1cm}\mathrm{and} \hspace{0.1cm} l'={\cal J}-1, $$ $$ \langle l {\cal J}S=1| S_{12} | l' {\cal J}S=1\rangle = 2 \hspace{0.5cm} l= {\cal J} \hspace{0.1cm}\mathrm{and} \hspace{0.1cm} l'={\cal J}, $$ and zero else.
In this exercise we will focus only on the one-pion exchange term of the nuclear force, namely $$ V_{\pi}(\mathbf{r})= -\frac{f_{\pi}^{2}}{4\pi m_{\pi}^{2}}\mathbf{ \tau}_1\cdot\mathbf{\tau}_2 \frac{1}{3}\left\{\mathbf{ \sigma}_1\cdot\mathbf{ \sigma}_2+\left( 1 + {3\over m_\pi r} + {3\over\left(m_\pi r\right)^2}\right) S_{12} (\hat r)\right\} \frac{e^{-m_\pi r}}{m_\pi r}. $$ Here the constant \( f_{\pi}^{2}/4\pi\approx 0.08 \) and the mass of the pion is \( m_\pi\approx 140 \) MeV/c${}^{2}$.
a) Compute the expectation value of the tensor force and the spin-spin and isospin operators for the one-pion exchange potential for all partial waves you found previously. Comment your results. How does the one-pion exchange part behave as function of different \( l \), \( {\cal J} \) and \( S \) values? Do you see some patterns?
b) For the binding energy of the deuteron, with the ground state defined by the quantum numbers \( l=0 \), \( S=1 \) and \( {\cal J}=1 \), the tensor force plays an important role due to the admixture from the \( l=2 \) state. Use the expectation values of the different operators of the one-pion exchange potential and plot the ratio of the tensor force component over the spin-spin component of the one-pion exchange part as function of \( x=m_\pi r \) for the \( l=2 \) state (that is the case \( l,l'={\cal J}+1 \)). Comment your results.
The aim here is to develop a program which solves the Lippman-Schwinger equation for a simple parametrization for the \( ^1S_0 \) partial wave. This partial wave is given by a central force only and is parametrized in coordinate space as $$ V(r)=V_a \frac{e^{-ax}}{x}+V_b \frac{e^{-bx}}{x}+V_c \frac{e^{-cx}}{x} $$ with \( x=\mu r \), \( \mu=0.7 \) fm (the inverse of the pion mass), \( V_a=-10.463 \) MeV and \( a=1 \), \( V_b=-1650.6 \) MeV and \( b=4 \) and \( V_c=6484.3 \) MeV and \( c=7 \).
a) Find an analytical expression for the Fourier-Bessel transform (Hankel transform) to momentum space for \( l=0 \) using $$ \left\langle k \right | V_{l} \left | k' \right\rangle = \int j_l(kr)V(r)j_l(k'r)r^2dr. $$
b) Write a small program which calculates the latter expression and use this potential to compute the \( T \)-matrix at positive energies for \( l=0 \). Compare your results to those obtained with a box potential given by $$ V(r)=\left\{ \begin{array}{cc} -V_0& r < R_0 \\ 0 & r > R_0 \end{array} \right. $$ Make a plot of the two \( T \)-matrices for energies up to 300 MeV in the lab frame and comment your results.
Finally, a warning, the above central potential is fitted to data from approximately 20 MeV to some 300 MeV. This means that results outside the data set should be taken seriously.
Solution. The following Fortran 95 program solves the above Lippmann-Schwinger equation. Python and C++ codes will be added later. Discussions of the results will also be added.
C *******************************************************
C Example program used to evaluate the
C T-matrix following Kowalski's method (eqs V88 & V89
C in Brown and Jackson)
C for positive energies only
C The program is set up for S-waves only
C Coded by : Morten Hjorth-Jensen
C Language : FORTRAN 90
C *******************************************************
C ******************************
C Def of global variables
C ******************************
MODULE constants
DOUBLE PRECISION , PUBLIC :: p_mass, hbarc
PARAMETER (p_mass =938.926D0, hbarc = 197.327D0)
END MODULE constants
MODULE mesh_variables
INTEGER, PUBLIC :: n_rel
PARAMETER(n_rel=48)
DOUBLE PRECISION, ALLOCATABLE, PUBLIC :: ra(:), wra(:)
END MODULE mesh_variables
C ******************************
C Begin of main program
C ******************************
PROGRAM t_matrix
USE mesh_variables
IMPLICIT NONE
INTEGER istat
ALLOCATE( ra (n_rel), wra (n_rel), STAT=istat )
CALL rel_mesh ! rel mesh & weights
CALL t_channel ! calculate the T-matrix
DEALLOCATE( ra,wra, STAT=istat )
END PROGRAM t_matrix
C *********************************************************
C obtain the t-mtx
C vkk is the box potential
C f_mtx is equation V88 og Brown & Jackson
C *********************************************************
SUBROUTINE t_channel
USE mesh_variables
IMPLICIT NONE
INTEGER istat, i,j
DIMENSION vkk(:,:),f_mtx(:),t_mtx(:)
DOUBLE PRECISION, ALLOCATABLE :: vkk,t_mtx,f_mtx
DOUBLE PRECISION t_shell
ALLOCATE(vkk (n_rel,n_rel), STAT=istat)
CALL v_pot_yukawa(vkk) ! set up the box potential in routine vpot
ALLOCATE(t_mtx (n_rel), STAT=istat) ! allocate space in heap for T
ALLOCATE(f_mtx (n_rel), STAT=istat) ! allocate space for f
DO i=1,n_rel ! loop over positive energies e=k^2
CALL f_mtx_eq(f_mtx,vkk,i) ! solve eq. V88
CALL principal_value(vkk,f_mtx,i,t_shell) ! solve Eq. V89
DO j=1,n_rel ! the t-matrix
t_mtx(j)=f_mtx(j)*t_shell
IF(j == i) WRITE(6,*) ra(i) ,t_mtx(i)
c & DATAN(-ra(i)*t_mtx(i))
ENDDO
ENDDO
DEALLOCATE(vkk , STAT=istat)
DEALLOCATE(t_mtx, f_mtx, STAT=istat)
1000 FORMAT( I3, 2F12.6)
END SUBROUTINE t_channel
C ***********************************************************
C The analytical expression for the box potential
C of exercise 1 and 12
C vkk is in units of fm^-2 (14 MeV/41.47Mevfm^2, where
C 41.47= \hbarc^2/mass_nucleon),
C ra are mesh points in rel coordinates, units of fm^-1
C ***********************************************************
SUBROUTINE v_pot_box(vkk)
USE mesh_variables
USE constants
IMPLICIT NONE
INTEGER i,j
DOUBLE PRECISION vkk, r_0, v_0, a, b, fac
PARAMETER(r_0=2.7d0,v_0=0.33759d0) !r_0 in fm, v_0 in fm^-2
DIMENSION vkk(n_rel,n_rel)
DO i=1,n_rel ! set up the free potential
DO j=1,i-1
a=ra(i)+ra(j)
b=ra(i)-ra(j)
fac=v_0/(2.d0*ra(i)*ra(j))
vkk(j,i)=fac*(DSIN(a*r_0)/a-DSIN(b*r_0)/b)
vkk(i,j)=vkk(j,i)
ENDDO
fac=v_0/(2.d0*(ra(i)**2))
vkk(i,i)=fac*(DSIN(2.d0*ra(i)*r_0)/(2.d0*ra(i))-r_0)
ENDDO
END SUBROUTINE v_pot_box
C ***********************************************************
C The analytical expression for a Yukawa potential
C in the l=0 channel
C vkk is in units of fm^-2,
C ra are mesh points in rel coordinates, units of fm^-1
C The parameters here are those of the Reid-Soft core
C potential, see Brown and Jackson eq. A(4)
C ***********************************************************
SUBROUTINE v_pot_yukawa(vkk)
USE mesh_variables
USE constants
IMPLICIT NONE
INTEGER i,j
DOUBLE PRECISION vkk, mu1, mu2, mu3, v_1, v_2, v_3, a, b, fac
PARAMETER(mu1=0.49d0,v_1=-0.252d0)
PARAMETER(mu2=7.84d0,v_2=-39.802d0)
PARAMETER(mu3=24.01d0,v_3=156.359d0)
DIMENSION vkk(n_rel,n_rel)
DO i=1,n_rel ! set up the free potential
DO j=1,i
a=(ra(j)+ra(i))**2
b=(ra(j)-ra(i))**2
fac=1./(4.d0*ra(i)*ra(j))
vkk(j,i)=v_1*fac*DLOG((a+mu1)/(b+mu1))+
& v_2*fac*DLOG((a+mu2)/(b+mu2))+
& v_3*fac*DLOG((a+mu3)/(b+mu3))
vkk(i,j)=vkk(j,i)
ENDDO
ENDDO
END SUBROUTINE v_pot_yukawa
C **************************************************
C Solves eq. V88
C and returns < p | f_mtx | n_pole =k>
C **************************************************
SUBROUTINE f_mtx_eq(f_mtx,vkk,n_pole)
USE mesh_variables
USE constants
IMPLICIT NONE
INTEGER i, j, int, istat, n_pole
DOUBLE PRECISION f_mtx,vkk,dp,deriv,pih,xsum
DIMENSION dp(1),deriv(1)
DIMENSION f_mtx(n_rel),vkk(n_rel,n_rel),a(:,:),fu(:),q(:),au(:)
DOUBLE PRECISION, ALLOCATABLE :: fu, q, au, a
pih=2.D0/ACOS(-1.D0)
ALLOCATE( a (n_rel,n_rel), STAT=istat)
DO i=1,n_rel
ALLOCATE(fu(n_rel), q(n_rel), au(n_rel), STAT=istat)
DO j=1,n_rel
fu(j)=vkk(i,j)-vkk(i,n_pole)*vkk(n_pole,j)/
& vkk(n_pole,n_pole)
ENDDO
DO j=1,n_rel
IF(j /= n_pole ) THEN ! regular part
a(j,i)=pih*fu(j)*wra(j)*(ra(j)**2)/
& (ra(j)**2-ra(n_pole)**2)
ELSEIF(j == n_pole) THEN ! use l'Hopitals rule to get pole term
dp(1)=ra(j)
CALL spls3(ra,fu,n_rel,dp,deriv(1),1,q,au,2,0)
a(j,i)=pih*wra(j)*ra(j)/2.d0*deriv(1)
ENDIF
ENDDO
DEALLOCATE(fu, q, au, STAT=istat) ! free space in heap
a(i,i)=a(i,i)+1.D0
ENDDO
CALL matinv(a, n_rel) ! Invert the matrix a
DO j=1,n_rel ! multiply inverted matrix a with dim less pot
xsum=0.D0
DO i=1,n_rel
xsum=xsum+a(i,j)*vkk(i,n_pole)/vkk(n_pole,n_pole) ! gives f-matrix in V88
ENDDO
f_mtx(j)=xsum
ENDDO
DEALLOCATE (a, STAT=istat)
END SUBROUTINE f_mtx_eq
C **************************************************
C Solves the principal value integral of V89
C returns the t-matrix for k=k, t_shell
C **************************************************
SUBROUTINE principal_value(vkk,f_mtx,n_pole,t_shell)
USE mesh_variables
IMPLICIT NONE
DOUBLE PRECISION vkk, f_mtx, t_shell, sum, pih, deriv, term
DIMENSION deriv(1)
DIMENSION vkk(n_rel, n_rel), f_mtx(n_rel),fu(:), q(:), au(:)
DOUBLE PRECISION, ALLOCATABLE :: fu, q, au
INTEGER n_pole, i, istat
ALLOCATE(fu(n_rel), q(n_rel), au(n_rel), STAT=istat)
sum=0.D0
pih=2.D0/ACOS(-1.D0)
DO i=1,n_rel
fu(i)=vkk(n_pole,i)*f_mtx(i)
ENDDO
DO i=1,n_pole-1 ! integrate up to the pole - 1 mesh
term=fu(i)*(ra(i)**2)-fu(n_pole)*(ra(n_pole)**2)
sum=sum+pih*wra(i)*term/(ra(i)**2-ra(n_pole)**2)
ENDDO ! here comes the pole part
CALL spls3(ra,fu,n_rel,ra(n_pole),deriv,1,au,q,2,0)
sum=sum+pih*wra(n_pole)*(fu(n_pole)+ra(n_pole)*deriv(1)/2.d0)
DO i=n_pole+1,n_rel ! integrate from pole + 1mesh pt to infty
term=fu(i)*(ra(i)**2)-fu(n_pole)*(ra(n_pole)**2)
sum=sum+pih*wra(i)*term/(ra(i)**2-ra(n_pole)**2)
ENDDO
t_shell=vkk(n_pole,n_pole)/(1.d0+sum)
DEALLOCATE (fu, q, au, STAT=istat)
END SUBROUTINE principal_value
C ***********************************************
C Set up of relative mesh and weights
C ***********************************************
SUBROUTINE rel_mesh
USE mesh_variables
IMPLICIT NONE
INTEGER i
DOUBLE PRECISION pih,u,s,xx,c,h_max
PARAMETER (c=0.75)
DIMENSION u(n_rel), s(n_rel)
pih=ACOS(-1.D0)/2.D0
CALL gausslegendret (0.D0,1.d0,n_rel,u,s)
DO i=1,n_rel
xx=pih*u(i)
ra(i)=DTAN(xx)*c
wra(i)=pih*c/DCOS(xx)**2*s(i)
ENDDO
END SUBROUTINE rel_mesh
C *********************************************************
C Routines to do mtx inversion, from Numerical
C Recepies, Teukolsky et al. Routines included
C below are MATINV, LUDCMP and LUBKSB. See chap 2
C of Numerical Recepies for further details
C Recoded in FORTRAN 90 by M. Hjorth-Jensen
C *********************************************************
SUBROUTINE matinv(a,n)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION a(n,n)
INTEGER istat
DOUBLE PRECISION, ALLOCATABLE :: y(:,:)
INTEGER, ALLOCATABLE :: indx(:)
ALLOCATE (y( n, n), STAT =istat)
ALLOCATE ( indx (n), STAT =istat)
DO i=1,n
DO j=1,n
y(i,j)=0.
ENDDO
y(i,i)=1.
ENDDO
CALL ludcmp(a,n,indx,d)
DO j=1,n
call lubksb(a,n,indx,y(1,j))
ENDDO
DO i=1,n
DO j=1,n
a(i,j)=y(i,j)
ENDDO
ENDDO
DEALLOCATE ( y, STAT=istat)
DEALLOCATE ( indx, STAT=istat)
END SUBROUTINE matinv
SUBROUTINE LUDCMP(A,N,INDX,D)
IMPLICIT REAL*8(A-H,O-Z)
PARAMETER (TINY=1.0E-20)
DIMENSION A(N,N),INDX(N)
INTEGER istat
DOUBLE PRECISION, ALLOCATABLE :: vv(:)
ALLOCATE ( vv(n), STAT = istat)
D=1.
DO I=1,N
AAMAX=0.
DO J=1,N
IF (ABS(A(I,J)) > AAMAX) AAMAX=ABS(A(I,J))
ENDDO
IF (AAMAX == 0.) PAUSE 'Singular matrix.'
VV(I)=1./AAMAX
ENDDO
DO J=1,N
IF (J > 1) THEN
DO I=1,J-1
SUM=A(I,J)
IF (I > 1)THEN
DO K=1,I-1
SUM=SUM-A(I,K)*A(K,J)
ENDDO
A(I,J)=SUM
ENDIF
ENDDO
ENDIF
AAMAX=0.
DO I=J,N
SUM=A(I,J)
IF (J > 1)THEN
DO K=1,J-1
SUM=SUM-A(I,K)*A(K,J)
ENDDO
A(I,J)=SUM
ENDIF
DUM=VV(I)*ABS(SUM)
IF (DUM >= AAMAX) THEN
IMAX=I
AAMAX=DUM
ENDIF
ENDDO
IF (J /= IMAX)THEN
DO K=1,N
DUM=A(IMAX,K)
A(IMAX,K)=A(J,K)
A(J,K)=DUM
ENDDO
D=-D
VV(IMAX)=VV(J)
ENDIF
INDX(J)=IMAX
IF(J /= N)THEN
IF(A(J,J) == 0.) A(J,J)=TINY
DUM=1./A(J,J)
DO I=J+1,N
A(I,J)=A(I,J)*DUM
ENDDO
ENDIF
ENDDO
IF(A(N,N) == 0.) A(N,N)=TINY
DEALLOCATE ( vv, STAT = istat)
END SUBROUTINE LUDCMP
SUBROUTINE LUBKSB(A,N,INDX,B)
implicit real*8(a-h,o-z)
DIMENSION A(N,N),INDX(N),B(N)
II=0
DO I=1,N
LL=INDX(I)
SUM=B(LL)
B(LL)=B(I)
IF (II /= 0)THEN
DO J=II,I-1
SUM=SUM-A(I,J)*B(J)
ENDDO
ELSE IF (SUM /= 0.) THEN
II=I
ENDIF
B(I)=SUM
ENDDO
DO I=N,1,-1
SUM=B(I)
IF (I < N)THEN
DO J=I+1,N
SUM=SUM-A(I,J)*B(J)
ENDDO
ENDIF
B(I)=SUM/A(I,I)
ENDDO
END SUBROUTINE lubksb
c) The parameters of the box potential are chosen to fit a potential with a bound state at zero energy. What does this mean for your \( T \)-matrix with this potential when \( k\rightarrow 0 \)?


Who We Are?
Here at Red Cards, we are a unique community of artists, writers, musicians, marketing and technology professionals, dedicated to providing you with fun and imaginative card designs and an engaging site experience.
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We are passionate about what we do. We work hard every day to make the world a more thoughtful and caring place. Together with our members, we share the conviction that nurturing relationships and making a difference in people’s lives, is the most important way we can spend our time.
What We Believe?
We believe that wishes are granted, dreams come true and anything is possible. We know that not every day is a holiday, but every day is a good day and so we celebrate, comfort and commemorate.
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{% extends "base.html" %} {% load static %} {% block title %}
Low cost stations in the Mara River Basin collect real time data. Here the open access data can be viewed. Click one of the channels to view data of your interest.
Select Display Mode: Weather Station River Depth Rainfall and Temperature


{{/* Use this method to have Parcel process CSS */}}
{{ $css := resources.Get "output/index.css" | minify }}
{{ with $css }}
{{ $securecss := . | resources.Fingerprint }}
{{ end }}
{{/*
Use this method to have Hugo process PostCSS
{{ $css := resources.Get "css/styles.css" | postCSS (dict "config" "./assets/css/postcss.config.js") | minify }}
{{ with $css }}
{{ $securecss := . | resources.Fingerprint }}
{{ end }}
*/}}
{{ if (fileExists "./assets/output/index.css") }}
{{/* If we have an index.css file, it's for fonts. So process it with Pipes */}}
{{ $fonts_css := resources.Get "output/index.css" }}
{{ with $fonts_css }}
{{ $fonts_securecss := . | resources.Fingerprint }}
{{ end }}
{{/* And, Parcel generates the files to the output directory. Grab those files to create Prefetch links, and using Pipes, also puts them in our public folder */}}
{{ $path := "./assets/output" }}
{{ $files := readDir $path }}
{{ range $files }}
{{/* You may want to narrow the range to something more specific */}}
{{ if in .Name "woff" }}
{{ $file := resources.Get (printf "%s%s" "output/" .Name) }}
{{ end }}
{{ end }}
{{ end }}


GOVERNMENT OF THE DISTRICT OF COLUMBIA Serve DC – The Mayor’s Office on Volunteerism Open Government Report 2014 To institutionalize a culture of transparent and open government, accountability, and to expand opportunities for resident participation and collaboration, this Report describes how the Serve DC – The Mayor’s Office on Volunteerism has and will continue to develop and enhance transparency, public participation, and collaboration. In accordance with Mayor’s Order 2014-170, this report addresses the following topics: 1) Transparency Serve DC makes information regarding all funding opportunities available via the agency website, serve.dc.gov, posts in the DC registry, and the Office of Partnerships & Grant Services funding alert. Serve DC posts all scheduled meetings and the subsequent meeting minutes of the DC Commission for National & Community Service within the Board of Ethics & Government Accountability’s website. As an office within the Executive Office of the Mayor, Serve DC – The Mayor’s Office on Volunteerism is working closely with the Office of the City Administrator (OCA) and the Office of the Chief Technology Officer (OCTO) to identify a significant number of datasets that OCA and OCTO expect to publish online later this year. If no dataset from Serve DC – The Mayor’s Office on Volunteerism is selected to be included in this 2014 release, we will continue to work with OCA and OCTO to identify datasets appropriate to publish in 2015. In the future, requirements for open data reporting will be provided by the Chief Data Officer. 2) Public Engagement and Participation Serve DC archives all resources and information about agency programs and initiatives on the agency website. Serve DC has taken major steps in 2014 to increase access to information about government volunteer opportunities via the creation of NeighborGood, the Districts volunteer opportunity search engine. Serve DC has made information available on all AmeriCorps grant funded organizations through serve.dc.gov and dcamericorps.org which is linked to the agencies main website serve.dc.gov. Serve DC actively utilizes social media via Twitter and Facebook to engage the public in agency programs and initiatives. As the District Government Agency dedicated to promoting volunteerism as a sustainable solution to meeting community needs, Serve DC is consistently 1 3) engaged in efforts to involve the community in all agency programs and initiatives through outreach activities and events. Serve DC utilizes the DC Department of Health’s DC Responds communication system to notify volunteers, partners, and programs of important announcements and emergencies. Collaboration In 2013 Serve DC Launched the District of Columbia Cities of Service Task Force bringing together volunteer managers and program staff from across District Government agencies to facilitate collaboration, communication, and coordination around how District Government is leveraging volunteer and pro bono services. These efforts continued throughout 2014. In partnership with the DC Department of Health – Health Emergency Preparedness Administration, Serve DC has launched the DC Vulnerable Populations Community & Health Care Coalition, bringing together government agencies, for profit, and not profit service providers to develop plans and strategies for ensuring the safety and resilience of seniors, residents with access and functional needs, and other vulnerable populations before, during, and after a disaster or emergency. In 2014 Serve DC launched partnerships with the DC Department of Employment Services and the Department of Youth Rehabilitation Services to promote volunteerism as a pathway to employment exposing youth and other residents to careers within the emergency management and public safety sector. Serve DC convenes a monthly Service members, Veterans and Military Families brown bag bringing together government agencies and non-profits providing services to service members, veterans and military families in the District of Columbia. 2


| Trees | Indices | Help |
|
|---|
|
|
A distributed process implementation of the papyros API.
Both the TaskDispatcher and each Worker is a separate process, running on one or more interconnected machines.Requires: Pyro 3.7.
|
|||
|
NonRemoteMethodError Tried to call remotely a method allowed to be called locally only. |
|||
|
PyroTaskDispatcher A TaskDispatcher run as a (potentially remote) process. |
|||
|
PyroTaskDispatcherProxy A proxy class of PyroTaskDispatcher. |
|||
|
PyroTaskQueue A TaskQueue used for distributed execution. |
|||
|
PyroTaskQueueProxy A proxy class of PyroTaskQueue. |
|||
|
PyroWorker A Worker run as a (potentially remote) process. |
|
|||
|
|||
|
|||
|
|
|||
PYRO_DAEMON = None
|
|||
ROOT_GROUP =
|
|||
NAME_SERVER_LOCATOR = None
|
|
Specify the Pyro NameServer address. If this function is not called, the NameServer is attempted to be discovered automatically by a broadcast lookup.
|
obj to the PYRO_DAEMON with the
given name .
|
|
| Trees | Indices | Help |
|
|---|
| Generated by Epydoc 3.0beta1 on Tue Aug 12 05:39:44 2008 | http://epydoc.sourceforge.net |


First things first, as you will hear all the time when learning Ruby, everything is an object. Classes obey this golden rule as well. A class can be designed to initialize new objects that will all share similar traits. These traits can be attributes (instance variables) or functions (instance methods). Let's take a real-world object and translate it into its own class. For this example, we will be creating a class for Cheese! Cheese has several different attributes (color, consistency, stink, origin, name) but we are going to focus on the last two.
class Cheese attr_accessor :name, :origin def initialize(name,origin) @name = name @origin = origin end def display puts "I am #{name} cheese from #{@origin}" end end
This very simple class can now be used to generate an object that will have the built in attributes name and origin. When we make a call to create a new Cheese object, we simply pass in the name and origin and that information will be saved with the object. These attributes are known as instance variables and will be specific to each instance of the class that has been created. The instance variable is denoted by an @ sign to begin the name. Classes also have instance methods which are available only to objects of that class. In our example, Cheese objects will have the instance method display which will simply read out the name and origin of that instance. An idea for expanding this class could be to include a hash that consists of wines paired with the cheeses that go well with them and we could define a new method to return what wine pairs well with our Cheese object.


BTS Site Manager
You are about to access a private system. This system is for the use of authorized users only. All connections are logged. Any unauthorized access or attempts may be punished to the fullest extent possible under the applicable local legislation.
© 2016 Nokia. All rights reserved.


{{companyName}}
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Waiting an answer
-
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SRC
prepare
| Mirror | Hosting | Type | Main src_prepare overlay at |
|---|---|---|---|
| Gentoo | Github | project | https://github.com/gentoo-mirror/src_prepare-overlay |
| xgqt | Github | personal | https://github.com/xgqt/src_prepare-overlay |
| src_prepare | Codeberg | project | https://codeberg.org/src_prepare/src_prepare-overlay |
| src_prepare | Yorune | project | https://git.yorune.pl/src_prepare/src_prepare-overlay |


KEYBOARD_1: Widget elements must have keyboard event handlers
KEYBOARD_2: All functionality operable through the keyboard alone
KEYBOARD_3: Keyboard Trap


We do it all on Discord: PSR CRYPTO Discord.
Please remember that although we receive a small fee, we are volunteers with jobs and daytime responsibilities that we prioritize. We'll endeavor to respond to any support tickets within 24 hours. We ask for your short-term patience in exchange for a long-term commitment.
Our plan is to run our pool for one year, through June 30, 2019. The Internet is full of aspirations like that so take this one for what it's worth, it's not a guarantee but it's our plan of record. If you're like many miners in the decentralized world and are looking to "set & forget" your rigs, then welcome home, that's exactly why we built the pool for our use.
Although there can be software hiccups, the infrastructure is pretty solid, and we're banking on little support needed in this case. We run our pool on Google Cloud Platform and it's products. Our Turtle Pool is build on Google Compute Engine, which offers a Service Level Objective of 99.99%. Additionally, the DDoS protection of GCP means our pool has the same thick shell for DDoS that Google, itself, uses.
Although the gateways to GCP infrastructure are global, we chos to run our Pool on GCP infrastructure in Oregon. We chose this location because it had the lowest latency for users in Silicon Valley.


---
layout: altpage
title: "Teaching"
description: "Material for my courses"
header-img: "img/home-bg.jpg"
---
Course Listing
Below are courses I have been responsible for teaching. They include courses at both a
2 year and 4 year University Systems. I am also in the progress of creating my own content
around technologies I am passionate about.
Bakersfield College
Courses taught at Bakersfield College
- COMP B2 : Intro. to Computer Information Systems | Fall 2015
- COMP B3 : Computer Survival Skills | Spring 2015
- COMP B2 : Intro. to Computer Information Systems | Fall 2014
CSU, Bakersfield
Courses taught at CSU, Bakersfield
- CMPS 321 : Computer Architecture | Winter 2016
- CMPS 450 : Compilers | Spring 2016
Hacking .Net
These courses are being taught via Hacking .Net that I have developed, or are being developed. The purpose of these
courses is to bring concepts normally taught in a Computer Science curriculumn but using the .Net Core stack.
They're being hosted on Github.


|
Assimp
5.0.1, master @ cd42b955
|
| .ASS File formats | |
| Installation | |
| Usage | |
| Data Structures | The assimp library returns the imported data in a collection of structures |
| Material System | |
| Performance | |
| Threading | |
| Resources | This page lists some useful resources for assimp |
| Importer Notes | |
| Extending the Library | |
| Animation Overview | |
| 'version'-Command | Display the version/revision of Assimp used |
| 'help'-Command | Display a really helpful text |
| 'info'-Command | Load a model file and print basic statistics |
| 'cmpdump'-Command | Used for Assimp's internal regression testing |
| 'dump'-Command | Generate a text or binary dump of a model |
| 'extract'-Command | Extracts one or more embedded texture images from models |
| Common parameters | The parameters described on this page are commonly used by almost every assimp command |
| Todo List | |
| Deprecated List |


I write code, think about data, and create digital experiences. Currently an Engineering Undergrad doing B. Tech in Computer Science at Institute of Technical Education and Research.
Download CVIntroduction
Hey there! Let me introduce myself again, my name is Saheb Giri , and I am a Full-Stack JavaScript developer. I have over 3 years of experience developing websites and web applications. I enjoy working on both Front-end and Back-end, but I work mostly on Back-end part of apps. My go to library on Front-end is React, and NodeJS(Adonis) on Back-end.
I also build native android apps as well as native desktop apps using Qt, a C++ framework.
If you are looking to hire someone with my skills then you can drop me a mail at<EMAIL_ADDRESS>If you'd like, you can read more about me here, or scroll to see my development backpack (in other words the technologies I use daily).
Tech stacks
My go to programming language pretty much for everything. Be it the front-end, backend, or a CLI tool.
I use TypeScript whenever I can, it has a great complier and type safety is the way to go!
I also like playing with Python for side projects and keen to use this knowledge to learn ML and AI.
I use it mostly in competitive programming, building desktop application in Qt and anywhere performance is utmost important.
I use Java for building native android application.
When it comes to build any backend project I use Adonis JS, a Node js backend framework focused on developer experience.
My favorite library for building single page application or anything frontend related stuffs.
I use it when I want features like SSR, automatic code splitting, api routes with serverless framework and much more.
I don't actively use it but had few projects in vue, Nuxt JS to be specific.
For building eye candy user interfaces and experience. It is like CSS on steroids.
Everything I make on the internet lives on Vercel. Vercel is the gold standard of developer tools.
All of my Open Source projects dumped at a single platform.
Avoid environment setup while deployment, just pull the container and it just works.
Instant realtime GraphQL APIs for rapidly building apps without having to build, operate & scale a GraphQL server.
Serves all my relation databases need. It works like charm once set up.
Projects
Exploring better ways to identify golden ratio in everyday objects using 2D canvas.
Learn more →Meetwits is the most authoritative source for movie, TV, and celebrity information.
Learn more →Soundsky is a digital music service that gives you access to millions of 30s songs.
Learn more →Testimonial
We had an incredible experience working with Saheb and were impressed he made such a big difference in only three weeks. Our team is so grateful for the wonderful improvements he made and his ability to get familiar with the product concept so quickly. It acted as a catalyst to take our design to the next level and get more eyes on our product.


Article One
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There are quite a few things wrong here, can you find and fix them?
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Some sample text. Read this article


GSoC 2018 - Week 4 - Continuing with transolve Part-II
09 Jun 2018By the end of the previous week I had completed majority of the TODO’s of the PR #14736. This week I started on minor improvements suggested by Amit. This included:
-
Minor Documentation changes
Included proof of correctness, typo fixes, rephrasing lines etc. -
Removing
flag
variable from transolve.
flag
variable was passed as a function parameter intransolve
and was used to check on the recursive call. The thought was that there might be a case when equation could not be solved by any of the helpers but during the process the equation might get modified, so we wanted to double check whether the modified equation could be solved again, thats why we usedflag
but Amit suggested on removing it as there wasn’t a case as of now and it will only make things unnecessarily complicated. -
Also worked on XFAILS, mainly on solving equations containing only symbols. There was an exception raised in
_invert_real
, because of a relational comparison made with0
which caused the faiure of the tests. It has been fixed by handling the exception. -
Imported tests of log and lambert.
-
Another thing we discussed was about log solver. The helpers for solving logarithmic equations will be implemented in a seperated PR, to make the review and implementing task easy and independent.
After this I started a working on solving exponential equations in complex domain, though it turns out that we will be focussing on only the real domain as of now. But I have added tests for it as XFAIL.
To make exponents work in complex domain, we require the complex logarithm.
If we have equation as
z = e**w,
where w is complex number and z is non-zero complex number,
then
Ln(z) = ln|z| + I*(Arg(z) + 2*n*pi)
where Ln(z) is the complex logarithm and ln|z| is the
logarithm evaluated with real number argument.
Apart from this we also had a meeting yesterday in which we discussed about how we will be wrapping up things before the first evaluation.
I will be completing few leftovers of the current PR to make it merge and start working on implementing log solver (making a commit probably by tomorrow).
Amit also suggested on creating a coverage report so as to get an idea about the amount of tests covered by the code. The current coverage report can be found here. Other than this he also advised on reading Clean Code by Robert C. Martin
which will help me improve in writing clean and efficient code.
The main target for the coming week will be merging PR #14736
, implementing log solver
(will try to merge this as well), starting with implementing lambert solver
(try to add atleast a WIP PR) and reading the book.
Hope things go as planned!!
Follow @Yathartha22


Audience3
Parameters
Random seed
Rated movies
Unrated movies
Users
Suggestion count
User ratings (1~5)
Movie similarity (0 to 1) - Warning! Those values are not keeping when rebuild matrix.
Do suggestion >>
Movie suggest
User
Suggest(score)


Track your Ethereum Address. You will be notified of any transaction that has a direct impact on your ethereum address!


Biography
Born in the beautiful state of Montana, I made my way from Missoula to Kennewick, Washington when my father took a job at PNNL. There I grew up a happy child, excited to explore the world around me. I loved playing sports and working in teams, and had an analytical knack that put in place as the captain of serval of my teams. When I graduated highschool, always wanting to head into a big city, I moved to the sunny state of Arizona and attended ASU where I gradutated with a Bachelor's.
Education
- Arizona State University (Bachelors)
- Uniform Investment Law Examination (Series 65)
- C.F.A. Level 1 Examination (First Try Pass)
- Real Estate License Examination
Job Experience
- Realty One Group Preview
- Atomic Profits: Social Media Marketing Agency
Top 10 Books
- Harry Potter and the Order of the Phoenix by: J.K. Rowling
- How to Win Friends and Influence People by: Dale Carnegie
- Rich Dad Poor Dad by: Robert T. Kiyosaki
- Managing Oneself by: Peter Drucker
- Harry Potter and the Prisoner of Azkaban by: J.K. Rowling
- The Phantom Tollbooth by: Norton Juster
- Reminiscences of a Stock Operator. Reminiscences of a Stock Operator by: Edwin Lefèvre
- Harry Potter and the Goblet of Fire by: J.K. Rowling
- Harry Potter and the Halfbood Prince by: J.K. Rowling
- Harry Potter and the Sorcerer's Stone by: J.K. Rowling


Adjustable parameters¶
General¶
plankton_mask
| function: | spectra_mole.broaden_spectrum() |
|---|---|
| purpose: | ignore all bins with an ldr greater (optinally add a maximum range for this threshold) |
| standard: | -13 |
Q value
| function: | several |
|---|---|
| purpose: | factor between noise_level and noise_threshold |
| standard: | 3 |
plausible moments
| function: | spectra_mole.filter_moments() |
|---|---|
| purpose: | filter the list of rwp moments for physically unplausible values. Bragg peak has to be right of cr peak and has a certain width |
| standard: | (v_moment > v_cr + sigma_cr & width_moment > 0.08 and right_bound_moment > right_bound_cr) |
rwp calibration fine
| function: | |
|---|---|
| purpose: | if there is a peak in the cloud radar, check if the rwp particle peak is close to it |
| standard: |
|
compare given noise level with Hildebrand and Sehkon
| function: | spectra_mole.check_noise_level() |
|---|---|
| purpose: | ind elevated noise level (e.g. clutter, melting layer) by comparison with the noise level from the standard signal processing algorithm |
| standard: | noise level estimated with HS more than 4 dB larger than standard signal processing noise level |
velocity range for calibration
| function: | spectra_mole.estimate_calibration() |
|---|---|
| purpose: | use only power within a small spectral band for calibration to exclude Bragg and Mie contribution |
| standard: | bin 91 to 118 (vel -3.0 to -0.7) |
weighting function method¶
weighting function smoothing
| function: | spectra_mole.correct_with_weighting() |
|---|---|
| purpose: | supress noise in the weighting function |
| standard: | convol_kernel = [0.3, 0.5, 1., 0.5, 0.3]/2.6 |
weighting function threshold
| function: | spectra_mole.correct_with_weighting() |
|---|---|
| purpose: | weights below this threshold are supressed by the snr^-1 |
| standard: | 0.6 |
peak fitting method¶
fuzzy membership minimum peak width
| function: | spectra_mole.correct_with_fuzzy() |
|---|---|
| standard: | 3 bins |
peak right of cloud radar
| function: | spectra_mole.correct_with_fuzzy() |
|---|---|
| standard: | ight > cr_edge + 2 & left > cr_edge -1 & within the raw RWP peak |
boundaries in the fitting algorithm
| function: | |
|---|---|
| purpose: | constrain the fitting algorithm to the properties of clear air bragg peaks |
| standard: |
|


Test files without 2 passes: 4; Subtests without 2 passes: 12; Failure level: 12/162 (7.41%)
| Test | Ch45 | FF38 | IE11 |
|---|---|---|---|
| /webstorage/idlharness.html (8/33, 24.24%, 4.94% of total) | OK | OK | OK |
| Window interface: attribute sessionStorage | FAIL | PASS | FAIL |
| Window interface: attribute localStorage | FAIL | PASS | FAIL |
| Storage interface: attribute length | FAIL | PASS | FAIL |
| Storage interface: operation key(unsigned long) | FAIL | PASS | FAIL |
| Storage interface: operation getItem(DOMString) | FAIL | PASS | FAIL |
| Storage interface: operation setItem(DOMString,DOMString) | FAIL | PASS | FAIL |
| Storage interface: operation removeItem(DOMString) | FAIL | PASS | FAIL |
| Storage interface: operation clear() | FAIL | PASS | FAIL |
| /webstorage/storage_builtins.html (2/2, 100.00%, 1.23% of total) | OK | OK | OK |
| Builtins in localStorage | FAIL | PASS | FAIL |
| Builtins in sessionStorage | FAIL | PASS | FAIL |
| /webstorage/storage_local_setitem_quotaexceedederr.html (1/1, 100.00%, 0.62% of total) | OK | OK | OK |
| Throws QuotaExceededError when the quota of 800 has been exceeded | PASS | - | FAIL |
| /webstorage/storage_session_setitem_quotaexceedederr.html (1/1, 100.00%, 0.62% of total) | OK | TIMEOUT | OK |
| Throws QuotaExceededError when the quota of 800 has been exceeded | PASS | - | FAIL |