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

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370 views

Anharmonic oscillator solution function

I am solving a CLASSICAL an-harmonic oscillator problem with Hamiltonian given by $H= (1/2)\dot{x}^2+(1/2)x^2-(1/2)x^4$ with all the constants (k's) and mass being taken as 1 (one). I find that $x= ...
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87 views

What is the difference between broken and unbroken sypersymmetry?

In a physics article I read recently, the author introduces the notion of supersymmetry by saying basically that the system described by the Hamiltonian $H$ is supersymmetric if $H$ can be decomposed ...
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174 views

What is the energy scale of a Hamiltonian?

On the second page of this paper a term 'fundamental energy scale' is used while talking about a Hamiltonian. The context is implementing Deutsch's algorithm using Adiabatic Quantum Computation. What ...
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39 views

Delivered/Reflected Power by Drive on a Hamiltonian System

Imagine a SHO with a drive F(t). (or in general a Hamiltonian system) What is the power delivered to the system and can we talk about the power reflected? is i am imagining say a MW oscillator ...
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117 views

Measurement in Quantum mechanics

I have got a quantum conservative system whose Hamiltonian is $H$. I consider an selfandjoint operator $O$ whose eigenvalues and eigenvectors are: $$O|\psi _{n}\rangle = \lambda _{n}|\psi ...
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59 views

Length of the orbit (semiclassical orbits)

The Gutzwiller trace is about $$ d(E)=d_{0} (E)+ \sum_{p.o}A_{p}Cos(S(E)\ell_{p}) $$ and $ \ell_{p} $ are the length of the orbit. However my question is, how can one derive the length of the orbit ...
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3answers
164 views

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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220 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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104 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
586 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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2answers
74 views

Energy conservation $\iff \frac{dE}{dt} = 0\ $?

If I'm asked to prove that a system is/ isn't conservative and compare it to whether or not the Hamiltonian is conserved, does that mean I need to compute the time derivative of energy $(T+U)$? Doing ...
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1answer
91 views

Commutator and Hamiltonian [closed]

Assume that $[\hat{A},\hat{H}]_-=0$ and $[\hat{B},\hat{H}]_-=0$ but we know that $[\hat{A},\hat{B}]_-\neq 0$. Then there exists degenerate stationary states of $H$. How to prove it?
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1answer
91 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
408 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
52 views

Commuting with time evolution operator implies commuting with Hamiltonian

Consider a quantum system (finite dimensional) has overall Hamiltonian: $H_t = H_0 + w(t)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of time. ...
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1answer
46 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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1answer
87 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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2answers
40 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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1answer
62 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
94 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
57 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
76 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
71 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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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
93 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
65 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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366 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
342 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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516 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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32 views

Total Hamiltonian of a system of two particles

In my QM class, we talked about the total Hamiltonian of a system of two spin $1/2$ particles: \begin{equation} H^T=H_1\otimes\mathbb{1}+\mathbb{1}\otimes H_2. \end{equation} It was suggested that ...
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36 views

Time evolution of interaction Hamiltonian in the Heisenberg picture

How does the interaction Hamiltonian of a (finite dim) quantum system with Hamiltonian: $H(t) = H_0 + w(t) H_I$ evolve in time in the Heisenberg picture. Is there anything special about the way ...
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50 views

Unitary evolution operator

Assume we have a system in a state $\psi$ that is a superposition of eigenvectors of some observable $A$. Now we make a measurement of the observable $A$; the state after the measurement $\phi$ is a ...
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51 views

Computing the probability density of wavefunctions

Suppose I am given a Hamiltonian operator $\hat{H}$ that satisfies the time-independent Schrödinger equation $$\hat{H} \psi = E\psi$$ I can compute energy eigenvalues and their associated ...
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21 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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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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50 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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39 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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56 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
66 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?: ...
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1answer
41 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
84 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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71 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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49 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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32 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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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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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
249 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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63 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
2k 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 ...