The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

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About the double orthogonality of the eigenfunctions of the Hamiltonian

Consider the usual Hamiltonian describing the motion of a particle, $$\hat H = \frac {\hat P^2}{2m} + V(r),$$ where for simplicity the problem can be considered in 1D on the semiaxis $[a, \infty)$ , ...
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189 views

Equation $H(q,p)=E$ is the equation of motion or energy-conservation law?

I do not completely understand, why do we consider Hamilton–Jacobi equation $H(q,p)=E$ as equation of motion, whereas it is looks like energy-conservation law?
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83 views

Is a gapless system always conducting and a gapped system insulating?

In an answer to this question, @user566 mentioned that there is a qualitative difference between gapped and gapless systems; that gapless systems are conducting and gapped system are insulating. Is ...
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3answers
552 views

Why can the Schroedinger equation be used with a time-dependent Hamiltonian?

I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems. However, one of the axioms in quantum mechanics postulates that ...
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1answer
89 views

Hamiltonian Operator for Harmonic Oscillator

I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both ...
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1answer
77 views

Adiabatic quantum Hamiltonian of variable dimension

Is adiabatic quantum Hamiltonian of variable dimension possible? This is very hypothetical and I am afraid may not have enough merit to belong to this forum. I would still like to elaborate. Here is ...
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2answers
390 views

Hamiltonian and non conservative force

I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$. I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
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1answer
45 views

Definition of a “variable” in a Hamiltonian? [closed]

What does this mean $c^+c$ when included in a Hamiltonian? I saw it in a Hamiltonian with Sigma Notation and I want to know what it means. Or is its definition specific to the given Hamiltonian?
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86 views

How much information does the Hamiltonian contain in quantum mechanics? [closed]

Given a Hamiltonian, let's say of a many-body system, through the Schrodinger equation,in principle we can find the eigenfunctions and their corresponding eigenvalues (spectrum). Now given an ...
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1answer
47 views

Eigenvalues of a nearest-neighbour tight-binding Hamiltonian in (Mahan, 2003)

In this paper by G. D. Mahan, he obtains the following electron Hamiltonian in a nearest-neighbour tight binding scheme: (page 2 of the paper, top of the right column) \begin{align} H_0 &= J_0 ...
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1answer
66 views

Commutation of Hamiltonian with momentum

In which case does the Hamiltonian $H$ commutes with the momentum $P$? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ...
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1answer
56 views

How can I write the anderson hamiltonian as a matrix? [closed]

How can I write this Hamiltonian: $$ H = \sum E_d \hat{n}_d + \sum_k \epsilon_k\hat{n}_k + \sum_k V_{kd} (\hat{a}^\dagger_k \hat{a}_d + \hat{a}^\dagger_d \hat{a}_k) $$ in matrix form using its ...
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1answer
71 views

Unitary transformation of the Hamiltonian with spin-orbital coupling

I am reading this Paper recently. The author says that: for this Hamiltonian: $$H(t) = \frac{p^2}{2m} + \frac{m\omega^2}{2}x^2 + \alpha p_x \sigma_y$$ If we make a unitary transformation ...
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1answer
63 views

The Hamiltonian and Energy

if anyone can give assistance on this question it is much appreciated! Suppose I have a Hamiltonian $$H=\frac{p^{2}}{2m}+V(r)+F(r,t) $$ where $$F(r,t) = Q(r)\Re{(e^{{iT(t)}})}:\quad Q(r), T(t),T'(t) ...
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2answers
67 views

Finding the spectrum of a curious hamiltonian

I wish to analyse the following hamiltonian, i.e. find its eigenvalues and eigenstates. $$H = \frac{1}{2}\epsilon(\sigma _z \otimes \mathbb{1} + 1\otimes \sigma _z) - \Delta (\sigma _x \otimes \sigma ...
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1answer
85 views

Vanishing diagonal matrix elements of pertubation

In time-dependent pertubation theory we can denote the Schrödinger equation by a set of two equations $$\dot{c_a} = -\frac{i}{\hbar}\Big[c_aH'_{aa}+c_bH'_{ab}e^{-i(E_b-E_a)t/\hbar}\Big] \\ \dot{c_b} ...
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1answer
61 views

Expectation value of Hamiltonian on number state [closed]

Hamiltonian is defined by $H_I = \hbar \omega (\hat{a}^+ \hat{a} + 1/2)$ What is the expectation value of the energy on the number state $$\vert \psi \rangle = \frac{1}{\sqrt{2}} ( \vert 1 \rangle ...
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1answer
334 views

Proof of the time-independent Schrödinger equation

I have a question regarding the proof of the time-independent Schrödinger equation. So if we have a time-Independent Hamiltonian, we can solve the Schrödinger equation by adopting separation of ...
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1answer
331 views

Hamiltonian operator apply to a wavefunction

When a Hamiltonian operator apply to a wavefunction, how could we write the hamiltonian as, $$H \psi = (E_n-\hbar \omega_0) \psi \ \ ? $$ Is this because $E_n= H+ \hbar \omega_0$? where ...
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2answers
502 views

Base states with hamiltonian matrix

It is better for you to have studied "Feynman lectures on Physics Vol.3", because I cannot distinguish whether the words or expressions are what Feynman uses only or not and in order to summarize my ...
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0answers
16 views

CSCO and Hamiltonian

Is there a relationship between the Hamiltonian and the number of observables needed in a CSCO to fully describe a quantum system? I am thinking that a new observable is needed each time there is a ...
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0answers
30 views

Why doesn't a quantum pairwise Hamiltonian couple states in which more than one interaction occurs?

This question is about the standard quantum mechanical pairwise interaction Hamiltonian. I'll phrase it in terms of an example using Rydberg atoms, but you could just as well imagine spins (for ...
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0answers
46 views

Does the additivity property of Integrals of motion and Lagrangians valid in all situations?

I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
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37 views

Hamiltonian for semiconductor

I was wondering which terms we need in a semiconductor Hamiltonian where no transition between the valence and conduction band occur? First we would have a term describing the energy of the full ...
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2answers
36 views

Allowed system energies from quantized Hamiltonian spectra

To find the allowed energies for a system, I can find the spectrum of the Hamiltonian $\hat{H}_{\psi}$ given a wavefunction $\psi$ representing the state of the system. 3 cases might happen: either ...
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0answers
50 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
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1answer
34 views

Trouble getting the matrix representation of a 4-state Hamiltonian

$\newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\ket}[1]{\left| #1 \right\rangle}$I have a simple 4-state Hamiltonian and am trying to find the matrix representation (in order to determine ...
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1answer
76 views

Where does this commutator relation come from?

What is the origin of this relation: $$ [H,a_n^\dagger] = \epsilon_n a_n^\dagger $$ for Hamiltonian $H$, creation operator $ a_n^\dagger $, and eigenvalue $ \epsilon_n $. The square brackets denote ...
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0answers
49 views

$N$ classical Harmonic Oscillators in microcanonical ensemble in three dimensions

Is my expression for the Hamiltonian for $N$ classical harmonic oscillators correct in three dimension as I am trying to solve it in microcanonical ensemble ...
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0answers
44 views

Symmetry Group of system to a given Hamiltonian

I want to determine the symmetry group of the following system: I consider a charged particle in a spherically symmetric potential $V$ and a homogeneous electric field of magnitude $E$ in ...
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0answers
29 views

Find port-Hamiltonian representation for DC motor connected to a load

We are modelling a DC motor connected to a rotating load via a rod that acts like a torsion spring, i.e., a rod that can have different angular velocities at its ends. Now the idea is to split the ...
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0answers
25 views

Dynamics of modifing a Hamiltonian

I am working on quantum mechanics right now, and I am wondering if there is a qualitative way to think about off diagonal term of a Hamiltonian matrix? I know that we can diagonalize a matrix, then ...
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0answers
14 views

Bounding the energy gap of a local spin Hamiltonian

What are some common mathematical techniques for bounding the gap between the ground state and first excited state of a local spin Hamiltonian? Does anyone have any good references for this?
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1answer
227 views

Two-state Hamiltonian matrix in basis

I have a homework problem as following: Write the two-state Hamiltonian matrix in a certain basis |1>, |2> in a general form as \begin{array}{ccc} H_{11} & H_{12} \\ H_{21} & H_{22} ...
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0answers
62 views

Finding interaction Hamiltonian for nano particles

I am studying Quantum electrodynamics (QED) and the core part of this theory is the light-matter interaction (quantized light and matter). I referred many papers and books (Molecular quantum ...
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1answer
1k views

Calculating the expectation value of a Hamiltonian

I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$ I want to know if I set this up properly. The ...
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0answers
142 views

Quantum Hamiltonian commuting with the Pauli-Runge vector

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
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0answers
77 views

antiferromagnetic spin wave

I have a hamiltonian that is derived from a spin wave energy dispersion calculation for a nearest neighbor interacting cubic antiferromagnet. After a Holstein-Primakoff transformation and making a ...
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1answer
287 views

Stationary states with a pair of hamiltonian equations

I read some derivation related with probability amplitudes and hamiltonian matrix in some book, and have a few questions. Here what the book says is. We want the general solution of the pair of ...
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1answer
118 views

Apply the Heisenberg Equation to the Hamiltonian [closed]

$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$ That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
-1
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2answers
127 views

Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]

I have a problem with the Hamiltonian, I don't think anything to solve it!! So could you give me some hints! Knowing that: $$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
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1answer
62 views

Two site Hamiltonian [closed]

So my question is I have this Hamiltonian: $$ H = \sum_i \epsilon_i \sigma^+_i\sigma^-_i + \sum_{i\neq j} V_{ij} \sigma^+_i \sigma^-_j, $$ and I want to write it out for two site. Is this correct?: ...