Questions tagged [hamiltonian]

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

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Convert a Lindbladian time evolution operator to the Kraus operator sum representation

I try to understand how I can convert a Lindbladian time evolution operator to the corresponding Kraus operator sum. Let's assume we have a time independent Hamiltonian $H$ and a set of time ...
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Significance of sign of perturbation matrix

I understand that the off-diagonal elements of a Hamiltonian denote the interaction between different states. The magnitude of off-diagonal elements therefore tells us how strong the interaction is. ...
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In QM perturbation theory, is the system generally in an eigenstate of the perturbing Hamiltonian, $\hat H_1$?

In my notes the derivation of the second order energy correction we don't do the following: $$\sum_k a_{nk}\langle\phi_n|\hat H_1|\phi_k\rangle=\sum_ka_{nk}E_k^{(1)}\langle\phi_n|\phi_k\rangle$$ ...
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Why can a partial derivative be added to a hamiltonian in canonical transformations?

In canonical transformations, how come we allow hamiltonian to change by a partial derivative of time? $$H'(P, Q, t) = H(p, q, t) + \frac{\partial F}{\partial t}.$$ Here $F$ is the generating function....
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Evolution of quantum state that cannot exist with accompanying Hamiltonian

So I am studying a certain Hamiltonian that has projection operators in its definition. To keep it simple, suppose our Hilbert space is a one particle system that can be spin up/spin down (excited, ...
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81 views

Eigenfrequencies of an Hamiltonian dynamical system in different bases

Consider the Hamiltonian $$ H=H(x,y,p_x,p_y) $$ which generates the dynamical system $$ \dot{x}=+\frac{\partial H}{\partial p_x} $$ $$ \dot{y}=+\frac{\partial H}{\partial p_y} $$ $$ \dot{...
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How come the eigenvalues of the Hamiltonian represent energy levels when the Hamilton doesn't represent the energy of the system?

Like in the Hamiltonian for a particle in an electromagnetic field. This is not a conservative field so the Hamiltonian doesn't represent the energy of the system. And yet the time independent ...
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Effective hamiltonian method for the quantum Ising model

I am reading Subir Sachdevs book on quantum phase transitions (second edition). In chapter 5 (page 58) he defines a hamiltonian $H=H_0+H_1$ where the eigenstates of $H_0$ are known and the influence ...
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How to argue on physical grounds that a function is the ground state of a Hamiltonian?

$u_l(r) = Ar^{l+1}e^{-\kappa r}$ is provided as a solution to the radial wave equation for the Coulomb potential $$-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}u_l(r)+\Bigl[\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2} -...
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Defining the number operator for a general Hamiltonian

I have a Hamiltonian of the following general form, $$H=\int \frac{d^{3} k}{(2 \pi)^{3}}\left[A_{\vec{k}}(t) a_{\vec{k}}^{\dagger} a_{\vec{k}}+B_{\vec{k}}(t) b_{\vec{k}}^{\dagger} b_{\vec{k}}+C_{\vec{...
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Derive hamiltonian from equations of motion

Is there a method for deriving the hamiltonian given that you know the equations of motion? For example given the equation (equation 5 in paper linked) they simply the derive the Hamiltonian in ...
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56 views

Spectrum of Dirac Hamiltonian

The Dirac Hamiltonian is given by, \begin{aligned} H &=\sum_s\int \frac{d^{3} p}{(2 \pi)^{3}} E_{\vec{p}}\left[b_{\vec{p}}^{s \dagger} b_{\vec{p}}^{s}+c_{\vec{p}}^{s \dagger} c_{\vec{p}}^{s}\right]...
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Electron-Phonon Hamiltonian and One-particle Approximation

Consider a hamiltonian with electron-phonon coupling, for instance, a very simple version of Holstein hamiltonian: $$t\sum_k \hat{c}^\dagger_k \hat{c}_{k+1}+\text{h.c.}+\hat{b}^\dagger \hat{b}+\alpha ...
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Hamiltonian of non-regular Lagrangian is well-defined on phase space

In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian $$H=\dot{q}^np_n-L,\tag{1.8}$$ although trivially a function of $q$ and $\dot{q}$, can ...
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Inversion symmetry in Bloch bands

In Bloch bands, when we consider the energy degeneracy caused by translation symmetry and spatial inversion symmetry, we use the standard procedure like in Bloch waves under space inversion (parity) ...
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Hamiltonian in explicit Spin z basis

For a time independent Hamiltonian $H = \frac{\mu}{\hbar}(\vec{S_1} + \vec{S_2})*\vec{B}$ and $B= (0,0,B_0)^T $, $\vec{S}= \frac{\hbar}{2}\vec{\sigma}$ I want to find the explicit Hamiltonoperator ...
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Exchange Momentum With Position Preserves Hamiltonian

I read a paper a while back that showed you can exchange the roles of p and q in a Hamiltonian and still retain its Hamiltonian nature. In some cases, this can simplify the problem. For the ...
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Minimal coupling to electric dipole form - II

In addition to link, may I ask you for details of the Hamiltonian transformation. Knowing that: $[\textbf{x},\textbf{p}]=i\hbar$, $[\textbf{x},\textbf{p}^2]=2i\hbar\textbf{p}$ and $e^{-i\alpha A}Be^{...
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This step in Griffiths' Introduction to Quantum Mechanics book

I am working my way through time-dependent perturbation theory at the moment. There is a derivation of the formula for determining the time-dependent coefficients, $c_a(t)$, $c_b(t)$, which I am stuck ...
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Extension of classical Liouville operator

Let us consider a classical Hamiltonian system described by the Hamiltonian \begin{equation} H(q,p) =\frac{p^2}{2m}+V(q) \end{equation} where we stick to the case of single particle for simplicity. I ...
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Scalar hamiltonian and electromagnetic transitions

How can a Hamiltonian be a scalar and allow transitions between states with different angular momentum at the same time? Electromagnetic induced transitions are usually represented as a perturbation ...
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The Hamiltonian and differentials

From Lifshitz and Landau Vol.$1$: From the equation in differentials $$ \mathrm{d} H=-\sum \dot{p}_{i} \mathrm{d} q_{i}+\sum \dot{q}_{i} \mathrm{d} p_{i} $$ in which the independent variables ...
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Acting state on pauli matrix and problem with dimension

I define local hamiltonian as: $ H_{\Omega_{k}}=\frac{\mathbb{1}- \sigma^{z}}{2} \frac{\mathbb{1}- \sigma^{z}}{2} \in M_{2{\times}2} $ where $\sigma^{z}$ is pauli operator. And prepare the state: ...
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Chaotic Hamiltonian system poincare surfaces depend on the integrator

First question on StackOverflow so go easy on me. I have a Hamiltonian system that consists on the following Hamiltonian: $H(p,x;\textbf{P,X})=\frac{p^2}{2m}-a\frac{x^2}{2}+b\frac{x^4}{4}+x\sum_{n=1}^...
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Poisson bracket of Hamiltonian with Hamiltonian always vanishes

Since Poisson bracket of Hamiltonian with Hamiltonian always vanishes then in case of explicit time dependence of Hamiltonian, how does Poisson bracket gives correct result?
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Validity of Linear Response Theory

Suppose the perturbation of the hamiltonian is some multiple of the free hamiltonian, that is $$H=H_0+H_1=H_0+\lambda H_0=(1+\lambda)H_0.$$ Here, certain operators apparently have no response due to ...
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How to generate Hubbard Hamiltonian?

I am a beginner with Hubbard Hamiltonian and my question is very basic: how can I generate the matrix form of the Hubbard Hamiltonian? I know the theory but I don't know how to put it numerically. $$\...
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Balance the units of the following hamiltonian

The following image is taken from an article and shows the hamiltonian of a spin chain model. I knew that the dimensional units in an equation must balance. To ensure this, the author took a procedure ...
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What happens to the configuration manifold when one quantizes the Hamiltonian?

A system in classical mechanics can be described by a configuration manifold $Q$ and a Lagrangian \begin{equation} L:TQ\rightarrow \mathbb{R} \end{equation} where $TQ$ is the tangent bundle or a ...
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How to check whether Weyl field Hamiltonian is bounded below?

When constructing the Lagrangian for a two-component left-handed Weyl field $\psi$, in e.g. Srednicki, one rejects the choice of $\partial^\mu \psi \partial_\mu \psi+\partial^\mu \psi^\dagger \...
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“Unnatural” Hamiltonian systems from a statistician's perspective

I would like to learn more about "unnatural" Hamiltonian systems, that is, systems whose energies cannot be written as $$H(p,q) = K(q) + U(p).$$ I have seen the term "natural" applied to systems ...
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Are the classical hamiltonian and quantum hamiltonian different types of objects?

Context: I'm not a physicist. I've come across the Hamiltonian in classical physics and in quantum physics, and I can't recognise why they have the same name. They seem very different. So I probably ...
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A non-Hermitian system whose “Hamiltonian” is the annihilation operator

Consider a notional quantum system whose "Hamiltonian" is the annihilation operator, $$H=a .$$ Its initial state $|ψ(0)\rangle$ is $$|\psi(0)\rangle=\sum_{n=0}^{\infty} c_{n}| n\...
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Why is the Hamiltonian of a photon = 0?

I'm studying the motion of light near Schwarzschild black holes, and I was wondering why the Hamiltonian of the Schwarzschild metric $$H = - \left( 1-\frac{2M}{r} \right)^{-1} \frac{p_{t}^2}{2}+\left(...
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Can any sum of infinitesimal canonical transforms on phase space be obtained from evolution under a static Hamiltonian?

Suppose I have a canonical transformation on phase space, which is obtained by evolving a classical Hamiltonian system from time $t=0$ to $t=T$, with some arbitrary time-dependent Hamiltonian $H(t)$. ...
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Eigenfunctions of a quantum plane rotor?

I'm trying to determine the energetic levels of a system with Hamiltonian $$H=-\frac{h^2}{2m}\frac{\partial^2}{\partial \phi^2}$$ And the border condition $$\psi(0)=\psi(2\pi)$$ The eigenvalues ...
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Clarification of a question in Analytical Mechanics - Hand & Fitch

In problem 11 in the first chapter they tell you to give an example of Hamiltonian that is not conserved but equals to $E$, however if $H = E$ it follows that: $$\frac{dH}{dt} = \frac{dE}{dt} = 0$$ ...
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How is the Hamiltonian changed in Adiabatic Quantum computing?

In adiabatic quantum computing, we define a time-dependent Hamiltonian, which is then changed gradually to obtain a solution to an optimization problem. Is the Hamiltonian changed automatically, or we ...
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Hamiltonian energy density, energy flux, and Poisson brackets

I came across an old paper (Hardy, Energy Flux Operator for a Lattice. Physical Review. Volume 132 Number 1, 1963) that states a relation between the time rate of energy density, the "energy density ...
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How to obtain the operator in time domain using multi-mode Hamiltonian of the optical cavity system?

I'm trying to get the average photon number of each mode in the optical cavity system with the hamiltonians: $\hat H_{Free\ optical\ wave} = \hbar w \hat a_i^† \hat a_i$ $\hat H_{Free\ cavity} = \...
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How do I simulate an atom?

Let us assume I wish to simulate a Helium atom, since there does not exist a closed-form solution. However, I presume I would need to simulate the time-dependent Schrodinger wave equation. I would ...
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Spin-Spin Interaction: Ground State Degeneracy

I'm given the hamiltonian $$\hat{H}=\sum_{i=1}^{L-1}\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}_{i+1}$$ (reminiscent of a para- or ferromagnetism situation?) where $\hat{\mathbf{S}}_i\cdot\hat{\mathbf{S}}...
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When solving Schrödinger's equation by separation of variables, why is the separation constant taken as the energy?

For simplicity, let's take the 1D Schrödinger's equation for a single non-relativistic particle: $$-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x) \Psi(x,t) = i\hbar \frac{\...
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Why does the quantum expectation value depend only on the diagonals of the Hamiltonian in the long time limit?

I'm trying to understand the eigenstate thermalization hypothesis more and I keep coming across a limit I don't understand. If the initial state of a system is in it's energy eigenstate basis as $|\...
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Why does $\langle \psi_1 \psi_1 |H_1 + H_2 |\psi_1 \psi_1 \rangle = 2E_1$ (and not simply $E_1$)? [closed]

This is "somewhat" related to Why does $\langle \psi_1 \psi_2 | H_1|\psi_1 \psi_2 \rangle= \langle \psi_1 | H_1|\psi_1 \rangle \langle \psi_2 | \psi_2 \rangle$? but I'm asking about something ...
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Coupled position-momenta term in hamiltonian

I am studying a hamiltonian system of the form: $$H=H_0+b(x^ix_i)^{1/2}(p^ip_i)^{1/2}$$ In which $H_0$ is the hamiltonian of a completely integrable system and $b$ a real constant. I have tried ...
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Getting rid of double delta-functions in Kubo formula?

I am trying to understand the Kubo formula, written, for example as $$ \sigma_H \sim \sum_{E_\alpha<E_F<E_\beta} \frac{\epsilon_{ab}}{(E_\alpha-E_\beta)^2} \langle\alpha|[\hat H,\hat x^a]|\...
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Wavefunction of a particle on a polar potential

Suppose you have the following hamiltonian: $$ H=-\frac{\hbar^2}{2mR^2}\frac{d^2}{d\phi^2} \tag{1} $$ which we can recognize as $\hat H=\frac{\hat L_z^2}{2mR^2}$. But working with (1) we see that: $...
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Heisenberg Hamiltonian: Energy per site on the triangular lattice

I want to find the energy per site for the Heisenberg Model with 3D spin-vectors $\bf S_i$ on a 2D triangular lattice and nearest neighbor interactions. I have a Monte-Carlo Simulation (green +) and I'...
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Calculating the Hamiltonian of complex scalar field

I am working on the Peskin's QFT problems, finding difficulties in calculating the Hamiltonian of complex scalar field. The Hamiltonian read: $$H = \int d^3x (\pi^*\pi+\nabla\phi^*\nabla\phi+m^2\phi^*\...

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