Hamiltonian Definition and 901 Threads

Hello! I read today, in the context of DM, about the Boltzmann equation: $$L[f]=C[f]$$ where ##L[f]## is the Liouville operator (basically ##\frac{df}{dt}##), with ##f(x,v,t)## being the phase-space distribution of the system and ##C[f]## being the collision operator. I am a bit confused about...

Homework Statement [/B] A 1D spin chain corresponds to the following figure: Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin...

Homework Statement Show that if the Lagrangian does not explicitly depend on time that the Hamiltonian is a constant of motion. Homework Equations see below The Attempt at a Solution method attached here: Apologies this is probably a bad question, but just on going from the line ##dH## to...

Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space. Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space. Could anyone please explain the difference between configuration space and phase space. Thank you in advance for...

A single particle Hamitonian ##H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}## can be expressed as: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}## or even: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}##...

Hi all, I've always regarded the coupling Hamiltonian for a bosonic cavity mode coupled to a two-level fermionic gain medium chromophore to be of the form, $$H_{coupling}=\hbar g(\sigma_{10}+\sigma_{01})(b+b^{\dagger})$$, where ##b## and ##b^{\dagger}## and annihilation and creation operators...

Homework Statement Hello Everyone I'm wondering, why in below product rule was not used for gradient of A where exponential is treated as constant for divergent of A and only for first term of equation we used the product rule? Homework Equations https://ibb.co/gHOauJ The Attempt at a Solution

In his article 'Quantum theory of radiation', Reviews of Modern Physics, Jan 1932, volume 4, Fermi gives the relativistic hamiltonian function ##W_a## for a point charge by equation (13), ## 0 = - \frac 1 {2m} \left( \left[ mc +\frac {W_a - ~eV} c \right ]^2 - \left[ p -\frac {eU} c...

Homework Statement A particle moves on the ##xy## plane having it's trajectory described by the Hamiltonian $$ H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t) $$ a) Find a complete integral for the Hamilton-Jacobi Equation b) Solve for ##x(t)## and ##y(t)## with...

Homework Statement Consider a point mass m attached to a string of slowly increasing length ##l(t)##. Them motion is confined to a plane. Find L and H. Is H conserved? Is H equal to the total energy? Is the total energy conserved? Assume ##|\dot{l}/l|<<\omega## Homework EquationsThe Attempt at...

Homework Statement This is derivation 2 from chapter 8 of Goldstein: It has been previously noted that the total time derivative of a function of ## q_i## and ## t ## can be added to the Lagrangian without changing the equations of motion. What does such an addition do to the canonical momenta...

Hello all, I am reading through the Jackson text as a hobby and have reached a question regarding the Hamiltonian transformation properties. I will paste the relevant section from the text below: I don't understand what he's getting at in the sentence I highlighted. To attempt to see what...

Homework Statement [/B] Parts (c) and (f) are the ones I'm having trouble with; Homework EquationsThe Attempt at a Solution [/B] For (c), I assume the problem is meant to involve using the result from part (b), which was H = g(J2 - L2 - S2)/2 . I was trying just to do it by first showing...

Hello Everyone This question is motivated by a small calculation I am doing on polarization of bodies in external electric field. What I wanted to do is this: 1) Mesh the region 2) Prescribe uniform (and non-changing) positive charge distribution 3) Prescribe (initially) uniform negative...

In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...

Homework Statement I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point. For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube. I'm then constructing the Hamiltonian...

Homework Statement Question attached here: I am just stuck on the first bit. I have done the second bit and that is fine. This is a quantum field theory course question but from what I can see this is a question solely based on QM knowledge, which I've probably forgot some of. Homework...

Hey everybody, I am trying to expand a system of seven qubits from one dimensional Hamiltonian to the two dimensional representation. I have the one dimensional representation and I don't know what to add to transform it from 1D to 2D representation. I would really appreciate your help and...

I am looking for an undergraduate textbook on Classical Mechanics that includes Hamiltonian and Lagrangian formulations. One reason for this is that I am interested in quantization and second quantization. It should include treatment of harmonics oscillators. Thanks!

Hamiltonian of XY model is defined by ##H=J\sum_i (\sigma_i^x \sigma_{i+1}^x+\sigma_i^y \sigma_{i+1}^y)## and because it is isotropic it is sometimes called XX model. If we do some unitary transformation, and get hamiltonian ##H=J\sum_i (\sigma_i^x \sigma_{i+1}^x-\sigma_i^y \sigma_{i+1}^y)##...

This is a basic question, so probably easy to answer. The following from Wikipedia seems pretty standard while describing the Schrödinger equation: "...and Ĥ is the Hamiltonian operator (which characterises the total energy of the system under consideration)." On the other hand, from page 100 of...

I am puzzled by the fact that a "single-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a multi-particle case (non-interacting particles) or that (only) a "two-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a...

Homework Statement I am given the Hamiltonian of the relativistic free particle. H(q,p)=sqrt(p^2c^2+m^2c^4) Assume c=1 1: Find Ham-1 and Ham-2 for m=0 2: Show L(q,q(dot))=-msqrt(1-(q(dot))^2/c^2) 3: Consider m=0, what does it mean? Homework Equations Ham-1: q(dot)=dH/dp Ham-2: p(dot)=-dH/dq...

Given a Hamiltonian in the position representation how do I represent it in operator form? for example I was asked to calculate the expectancy of the Darwin correction to the Hydrogen Hamiltonian given some eigenstate (I think it was |2,1> or something bu that doesn't matter right now), now I...

Homework Statement Matrices ##\alpha_k=\gamma^0 \gamma^k##, ##\beta=\gamma^0## and ##\alpha_5=\alpha_1\alpha_2\alpha_3 \beta##. If we know that for Dirac Hamiltonian H_D\psi(x)=E \psi(x) then show that \alpha_5 \psi(x)=-E \psi(x) Homework EquationsThe Attempt at a Solution I tried to...

Homework Statement The Hamiltonian of the positronium atom in the ##1S## state in a magnetic field ##B## along the ##z##-axis is to good approximation, $$H=AS_1\cdot S_2+\frac{eB}{mc}(S_{1z}-S_{2z}).$$ Using the coupled representation in which ##S^2=(S_1+S_2)^2##, and ##S_z=S_{1z}+S_{2z}## are...

The Hamiltonian of the Qi, Wu, Zhang model is given by(in momentum space): ## H(\vec{k})=(sink_x) \sigma_{x}+(sink_y) \sigma_{y}+(m+cosk_x+cosk_y)\sigma_{z} ## . What is the physical meaning of each component of this Hamiltonian? Note: for the real space Hamiltonian(where maybe the analysis of...

Homework Statement Question attached: Hi, To me this looks like a classical, continuous system, as a pose to a quantum, discrete system, so I am confused as to how to work the system in the grand canonical ensemble since , in my notes it has only been introduced as a quantum...

I wondered if anyone might know of any open access materials, possibly lecture notes, on the content of the following papers or books. P.A.M Dirac, 1950, Can. J. Math. 2,147 "Generalized Hamiltonian Dynamics" P.A.M Dirac, 1933, Proc. Camb. Phil. Soc., 29, 389 "Homogenous variables in classical...

What does it means for a physical theory to have hamiltonian, if it is formulated in lagrangian form? Why doesn't someone just apply the lagrangian transformation to the theory, and therefore its hamiltonian is automatically gotten?

Is the following logic correct?: If you have an hamiltonian, that has time has a variable explicitly, and you get the lagrangian,L, from it, and then you get an equivalent L', since L has the total time derivate of a function, both lagrangians will lead to the same equations euler-lagrange...

Dear physics forums, What is the physical interpretation of imposing the following constrain on a Hamiltonian: Tr(\hat H^2)=2\omega ^2 where \omega is a given constant. I am not very familiar with why is the trace of the hamiltonian there. Thanks in advance, Alex

Hello, The hydrogen atom Hamiltonian is $$H=\frac{p^2}{2m} -\frac{e^2}{r}\tag{1}$$ with e the elementary charge,m the mass of the electron,r the radius from the nucleus and p,the momentum. Apparently we can factorize H $$H=\gamma +\frac{1}{2m}\sum_{k=1}^{3}\left(\hat p_k+i\beta\frac{\hat...

Hello, I Have a non-Hermitian Hamiltonian, which is defined as an ill-condition numbered complex matrix, with non-orthogonal elements and linearily independent vectors spanning an open subspace. However, when accurate initial conditions are given to the ODE of the Hamiltoanian, it appears to...

Homework Statement Show that the mean value of a time-independent operator over an energy eigenstate is constant in time. Homework Equations Ehrenfest theorem The Attempt at a Solution I get most of it, I'm just wondering how to say/show that this operator will commute with the Hamiltonian...

Homework Statement I am given the Rashba Hamiltonian which describes a 2D electron gas interacting with a perpendicular electric field, of the form $$H = \frac{p^2}{2m^2} + \frac{\alpha}{\hbar}\left(p_x \sigma_y - p_y \sigma_x\right)$$ I am asked to find the energy eigenvalues and...

I have a question connected with the problem: https://www.physicsforums.com/threads/continuity-equation-in-an-electromagnetic-field.673312/ Why don’t we assume H=H*? Isn’t hamiltonian in magnetic field a self-adjoint operator? Why? Why do we use (+iħ∇-e/c A)2 instead of (-iħ∇-e/c A)2 two times?

Homework Statement Hi i got a problem in lc circuit, I need to find the hamiltonian to this circuit , I think that I did well but I am not sure, the problem and my attempt in the following file. Homework EquationsThe Attempt at a Solution

Hey all, I was reading Efficient Linear Optics Quantum Computation by Knill, Laflamme, and Milburn, when I came across their expression for the Hamiltonian for a phase shifter, given as ##\textbf{n}^{(\ell)} = \textbf{a}^{(\ell)\dagger} \textbf{a}^{(\ell)}##, where ##\ell## indicates the mode...

Homework Statement I'm having some issues understanding a number of concepts in this section here. I attached the corresponding figure at the end of the post for reference. Issue 1) 1st of all, I understand that a Hamiltonian can be written as such $$H = T_2 - T_0 + U$$ whereby ##T_2##...

Homework Statement Show that in the chiral (massless) limit, Gamma 5 commutes with the Dirac Hamiltonian in the presence of an electromagnetic field. Homework EquationsThe Attempt at a Solution My first question is whether my Dirac Hamiltonian looks correct, I constructed it by separating the...

I am in the process of writing a lecture out for my Graduate modeling class I teach. I normally don't write lectures out in LaTex or use PDF's because I write on the dry erase board, but if anyone is interested I wouldn't mind spending the time to type out some notes on the topic. The topic...

Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics. In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle? YES; I KNOW about Hamilton's Second...

Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...

1. The problem statement Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by H = \frac{p^2}{2m} + \frac{m \omega ^2 x^2}{2} Knowing that the ground state of the particle at a certain instant is described by the wave...

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian, see Peskin and Schroeder $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 \Big( a^{s\dagger}_\textbf{p}a^s_\textbf{p}...

The question is very general and could belong to another topic, but here it is. Suppose one wants to solve the set of differential equations $$ \frac{\partial x}{\partial t}=\frac{\partial H(x,p)}{\partial p},$$ $$\frac{\partial p}{\partial t}=-\frac{\partial H(x,p)}{\partial x},$$ with some...

The Dirac Hamiltonian is essentially ##H = m + \vec{p}##. I found a issue with this relation, because we know from relativity that ##E^2 = m^2 + p^2## and there seems to be no way of ##E = \pm \sqrt{m^2 + p^2} = m + p##. To get out of this issue, I tried the following. I considered ##E## as a...

The third law of quantum mechanics states that a system at absolute zero temperature has zero entropy. Entropy can be conceived as an expression of the number of possible microstates that can produce an identical macrostate. At zero entropy, there should be exactly *one* microstate configuration...

See attached photo please. So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.