The Electromagnetic Hamiltonian
Introduction
We know that the Hamiltonian is very important in non-relativistic quantum mechanics. However, we sometimes forget that the Hamiltonian is also extremely useful in the realm of classical mechanics. For example, the classical equations for the motion of a particle in an electromagnetic field can be derived using the Hamiltonian formalism.
Derivation
Consider a classical Hamiltonian for a single particle:
\[\begin{equation} \mathcal{H} = \frac{\vec{p}^2}{2m} + U(\vec{x},t) \end{equation}\]The first term represents the total kinetic energy of the particle, and the second term represents the total potential energy of the particle. We want to update this Hamiltonian to include the effects of an external electromagnetic field. Although we do not know the exact form of the Hamiltonian just yet, we do know a few properties that must be satisfied:
- The Hamiltonian should be reasonably simple since Maxwell’s equations are all linear.
- Electromagnetic fields can be expressed in terms of a scalar and vector potentials (\(\phi\) and \(\vec{A}\)). Thus, our Hamiltonian should have additional \(\phi\) and \(\vec{A}\) terms.
- Electromagnetic fields have momentum and energy. Thus, we have to alter both \(\vec{p}\) and \(U\) in the expression above.
Let’s try something simple; we will insert the scalar and vector potential into the expression by making the following changes:
\[\begin{equation} \vec{p} \rightarrow \vec{p} - e\vec{A} \end{equation}\] \[\begin{equation} U = e\phi \end{equation}\]In the first expression, we subtract the momentum of the electromagnetic field from the momentum of the particle to get the total momentum of the system. In the second expression, we state that the total potential energy of the system is equal to the charge on the particle times the scalar potential. In both expressions, the charge on the particle is \(e\). Note that this is a general expression - the variable \(e\) is not necessarily the negative charge on an electron. Then, the Hamiltonian becomes:
\[\begin{equation} \mathcal{H} = \frac{\left ( \vec{p} - e\vec{A} \right )^2}{2m} + e\phi \end{equation}\]How do we know that this Hamiltonian is correct? Well, if we analyze it using Hamilton’s equations, we had better recover the classical equations of motion for a charged particle in an electromagnetic field.
Let’s start with the time derivative of canonical position. In this case, the canonical position \(q_i\) is equal to the coordinate position \(x_i\). Taking the derivative with respect to momentum yields:
\[\begin{align} \dot{q_i} &= \frac{d\mathcal{H}}{dp_i} \\[0.1cm] &= \frac{d}{dp_i} \left [ \frac{\left ( p_i - eA_i \right )^2}{2m} + e\phi \right ] \\[0.1cm] &= \frac{2 \left ( p_i - eA_i \right ) }{2m} \\[0.1cm] &= \frac{1}{m} \left ( p_i - eA_i \right ) \end{align}\]Multiplying each side by the mass of the particle and solving for the particle momentum gives:
\[\begin{align} m \dot{q_i} &= p_i - eA_i \\[0.1cm] m \dot{x_i} &= p_i - eA_i \\[0.1cm] p_i &= m\dot{x}_i + eA_i \end{align}\]Now we can compute the time derivative of the momentum of the particle.
\[\begin{align} \dot{p}_i &= -\frac{d\mathcal{H}}{dq_i} \\[0.1cm] &= -\frac{d\mathcal{H}}{dx_i} \\[0.1cm] &= -\frac{d}{dx_i} \left [ \frac{\left ( p_i - eA_i \right )^2}{2m} + e\phi \right ] \\[0.1cm] &= - \frac{2 \left ( p_i - eA_i \right ) \frac{d}{dx_i} \left [ p_i - eA_i \right ] }{2m} - e\frac{d\phi}{dx_i} \\[0.1cm] &= - \frac{1}{m} \left ( p_i - eA_i \right ) \left ( \frac{dp_i}{dx_i} - e\frac{dA_i}{dx_i} \right ) - e\frac{d\phi}{dx_i} \end{align}\]In our system, we will assume that the momentum \(\vec{p}\) does not directly depend on the position (translational invariance). Thus, the derivative of momentum with respect to position is zero.
\[\begin{align} \dot{p}_i &= - \frac{1}{m} \left ( p_i - eA_i \right ) \left ( - e \frac{dA_i}{dx_i} \right ) - e\frac{d\phi}{dx_i} \\[0.1cm] &= \frac{e}{m} \left ( p_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e\frac{d\phi}{dx_i} \end{align}\]We can now substitute \(p_i = m\dot{x}_i + eA_i\) into this expression. First, we substitute it on the left side of the equation:
\[\begin{align} \frac{d}{dt} \left [ m\dot{x}_i + eA_i \right ] &= \frac{e}{m} \left ( p_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e\frac{d\phi}{dx_i} \\[0.1cm] m\ddot{x}_i + e \frac{dA_i}{dt} &= \frac{e}{m} \left ( p_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e\frac{d\phi}{dx_i} \\[0.1cm] m\ddot{x}_i &= \frac{e}{m} \left ( p_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e \frac{dA_i}{dt} - e\frac{d\phi}{dx_i} \\[0.1cm] F_i &= \frac{e}{m} \left ( p_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e \frac{dA_i}{dt} - e\frac{d\phi}{dx_i} \\[0.1cm] \end{align}\]In the last line, we replace \(m\ddot{x}_i\) with the force \(F_i\). Now we can replace \(p_i\) on the right side of the equation.
\[\begin{align} F_i &= \frac{e}{m} \left ( m\dot{x}_i + eA_i - eA_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e \frac{dA_i}{dt} - e\frac{d\phi}{dx_i} \\[0.1cm] &= \frac{e}{m} \left ( m\dot{x}_i \right ) \left ( \frac{dA_i}{dx_i} \right ) - e \frac{dA_i}{dt} - e\frac{d\phi}{dx_i} \\[0.1cm] &= e v_i \left ( \frac{dA_i}{dx_i} \right ) - e \frac{dA_i}{dt} - e\frac{d\phi}{dx_i} \\[0.1cm] \end{align}\]Remember, the derivative of the vector potential with respect to time is a gauge covariant derivative:
\[\begin{align} F_i &= e v_i \left ( \frac{dA_i}{dx_i} \right ) - e \left [ \frac{\partial \vec{A}}{\partial t} + e \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i - e\frac{d\phi}{dx_i} \\[0.1cm] &= e v_i \frac{dA_i}{dx_i} - e \frac{\partial A_i}{\partial t} - \left [ e \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i - e\frac{d\phi}{dx_i} \\[0.1cm] &= e \frac{d(v_i A_i)}{dx_i} - e \frac{\partial A_i}{\partial t} - \left [ e \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i - e\frac{d\phi}{dx_i} \\[0.1cm] &= e \frac{d \left ( \vec{v} \cdot \vec{A} \right )_i }{dx_i} - \left [ e \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i - e \frac{\partial A_i}{\partial t} - e\frac{d\phi}{dx_i} \\[0.1cm] &= e \left [ \vec{\nabla} \left ( \vec{v} \cdot \vec{A} \right ) - \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i + e \left [ - \frac{\partial \vec{A}}{\partial t} - \vec{\nabla} \phi \right ]_i \\[0.1cm] \end{align}\]We recognize the second term in the brackets on the right hand side of the equation as the electric field:
\[\begin{equation} F_i = e \left [ \vec{\nabla} \left ( \vec{v} \cdot \vec{A} \right ) - \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \right ]_i + e E_i \end{equation}\]And we also notice that the other term in brackets can be simplified by using a property of the triple cross product and a property of the magnetic field:
\[\begin{equation} \vec{v} \times \left ( \vec{\nabla} \times \vec{A} \right ) = \vec{\nabla} \left ( \vec{v} \cdot \vec{A} \right ) - \left ( \vec{v} \cdot \vec{\nabla} \right ) \vec{A} \end{equation}\] \[\begin{equation} \vec{B} = \vec{\nabla} \times \vec{A} \end{equation}\]Plugging this in yields:
\[\begin{align} F_i &= e \left [ \vec{v} \times \left ( \vec{\nabla} \times \vec{A} \right ) \right ]_i + eE_i \\[0.1cm] &= e \left [ \vec{v} \times \vec{B} \right ]_i + eE_i \\[0.1cm] \end{align}\]Writing the final vector expression yields:
\[\begin{equation} \vec{F} = e \left [ \vec{E} + \vec{v} \times \vec{B} \right ] \end{equation}\]As we see here, we have derived the Lorentz force using Hamilton’s equations of motion.


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{% include head.html %} {% include nav.html %}About Jonti Talukdar
Jonti is a fun loving and passionate engineer who has rich experience working in a wide variety of multi-disciplinary projects relevant to- Digital Design, Hardware Verification, Deep Learning, VLSI Testing and Circuit Design.
Currently in his second year as a Master's candidate in the field of Computer Engineering at Duke University (Durham, NC), he also has experience working at semiconductor giants like Intel Corporation and NXP Semiconductors.
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Pushout: This term is often used instead of “dropout,” since it can be argued that students do not drop out of school unprompted, but as a consequence of policies and actions that make it difficult or impossible for them to remain in school.
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Arguably the simplest and cleanest way to draw out a coffee’s best qualities, the pour-over method is elegant without being prohibitively difficult. For those accustomed to coffee from a drip machine, this method will produce something similar but noticeably more delicate and complex.
Step #1 Boil Water
The first thing we want to do is fill up our kettle and heat up the water. We want the water to be anywhere from 195℉ - 205℉
Step #2 Fold Filter
Once our water is heated we will want to fold our coffee filter.Fold a Melitta #4 filter down the scalloped edge.Fold the edegs opposite of each other. Insert it into the Bee House Brewer and place it on top of your mug or pitcher.
Step #3 Pre Heat filter
Once our dripper is prepped we want rinse the filter with our hot water to remove any paper filter taste that we might get from the filter.
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Step #6 Continue Pour
Starting in the middle of the grounds and spiraling outwards continue to pour your water untnil you reached your recommened amount of water


Whats the place you folks call home
BRISBANE! Hang loose!
The big apple.
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070146 KU Methodenkurs (2018S)—TNT: Tools & Techniques for Digital Humanities
- Instructor: Dr. Maxim Romanov<EMAIL_ADDRESS>- Office hours: We 2–3pm (or by appointment)
- Office: Hauptgebäude, Room O2.121 (2.Stock, Stiege 9)
Course
- u:find Link: https://ufind.univie.ac.at/en/course.html?lv=070146&semester=2019S
- Meeting time: Tu 11:30-13:00
- Meeting place: Hörsaal 29 Hauptgebäude, 1.Stock, Stiege 7
Aims, contents and method of the course
The class will introduce the students to a variety of software tools and methods used in the Digital Humanities, primarily using the Python programming language. No prior programming experience is expected.
Course Evaluation
Course evaluation will be a combination of in-class participation (30%), weekly homework assignments (50%), and the final project (20%).
Class Participation
Each class session will consist in large part of practical hands-on exercises led by the instructor. BRING YOUR LAPTOP! We will accommodate whatever operating system you use, but it should be a laptop rather than a tablet. Don’t forget that asking for help counts as participation!
Homework
Just as in research and real life, collaboration is a very good way to learn and is therefore encouraged. If you need help with any assignment, you are welcome to ask a fellow student. If you do work together on homework assignments, then when you submit it please include a brief note (just a sentence or two) to indicate who did what.
NB: On submitting homework, see below.
Final Project
The final project is your website built in Jekyll and hosted on Github. Your website must have blogposts about all homework assignments. You are most welcome to work on this in groups, but every one will have to submit their own website.
Study materials:
- https://www.codecademy.com
- Please, register at this website. You will be assigned to complete free versions of several courses on Codecademy:
- https://programminghistorian.org/lessons/
- “Programming Historian” offers a number of tutorials for aspiring digital humanists. These will be assigned to you as reference materials. You also are encouraged to explore those tutorials that are not included into the course.
- https://www.youtube.com/playlist?list=PL6kqrM2i6BPIpEF5yHPNkYhjHm-FYWh17
- Paul Vierthaler’s “Hacking the Humanities Tutorials” (Python)
Software, Tools, & Technologies:
The following is the list of software, applications and packages that we will be using in the course. Make sure to have them installed by the class when we are supposed to use them.
- Zotero, https://www.zotero.org/
- MS Word or Apache OpenOffice (you most likely already have one of these)
- [Mac] Terminal / [Windows] Powershell (both are already on your machines)
- Python https://www.python.org/
- R https://www.r-project.org/ and RStudio https://www.rstudio.com/
- git and https://github.com/, version control system
- Jekyll (https://jekyllrb.com/), a simple, blog-aware, static site generator
- pandoc (https://pandoc.org/), markdown, bibTex (bibliographical format for LaTeX)
- QGIS, a Free and Open Source Geographic Information System (https://qgis.org/en/site/)
- Regular expressions, EditPad Pro/Sublime Text
- Wget (https://www.gnu.org/software/wget/), a free software package for retrieving files
- [TEI] XML, csv/tsv, json, yml, etc.
- Gephi (https://gephi.org/)
Submitting Homework:
- Homework assignments are to be submitted by the beginning of the next class;
- For the first few classes you must email them to the instructor (as attachments)
- Later, you will be publishing your homework assignments on your websites.
- For Codecademy.com assignments:
- take screenshots with your latest progress and email the image to the instructor.
- [Mac] Taking a screenshot: https://support.apple.com/en-us/HT201361
- [Win] Taking a screenshot: https://www.howtogeek.com/226280/how-to-take-screenshots-in-windows-10/
- Later, you will be posting these progress screenshots on your websites.
- take screenshots with your latest progress and email the image to the instructor.
- DH is a collaborative field, so you are most welcome to work on your homework assignments in groups, however:
- You must still submit it. That is, if a groups of three works on one assignment, there must be three separate submissions: either emailed from each member’s email; or, later, published on each member’s website.
- Codecademy.com assignments must be completed individually.
Schedule
Tu 05.03 - [ #01 ]
Citation Management and Academic Writing ITu 19.03 - [ #02 ]
“Off with the Interface!” Getting to know the command lineTu 26.03 - [ #03 ]
Version Control and CollaborationTu 02.04 - [ #04 ]
Citation Management and Academic Writing IITu 09.04 - [ #05 ]
Georeferencing with QGISTu 30.04 - [ #06 ]
Regular expressionsTu 07.05 - [ #07 ]
WebscrapingTu 14.05 - [ #08 ]
Text Markup [TEI XML], and how to remove it…Tu 21.05 - [ #09 ]
Structuring dataTu 28.05 - [ #10 ]
Text to Map (1/2)Tu 04.06 - [ #11 ]
Text to Map (2/2)Tu 18.06 - [ #12 ]
Topic modeling with RTu 25.06 - [ #13 ]
Social Network Analysis with Gephi
Please, see Lessons in the menue at the top of the page for detailed information on each lesson.


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