Questions tagged [hamiltonian]

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

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Conditions of separation of variable of wave function

In quantum mechanics as far as I know when the hamiltonian operator can be written as $$H = H_1 + H_2$$ then the wave function can be written as $$\psi_1 \cdot \psi_2$$ but as we get further in ...
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Calculation of effective Hamiltonian

I'm stuck with calculating the effective Hamiltonian of a model system. The energy-dependence of the effective Hamiltonian is confusing me a lot. My question is how to treat this dependence ...
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Can energy eigenstates be in a superposition of quantum numbers?

I know that given say 4 quantum numbers $J^2$, $J_z$, $J_1^2$, $J_2^2$ (e.g. for the Hamiltonian $H=\lambda J_{1}.J{_2}$), the state |$J$,$J_z$,$J_1^2$,$J_2^2$>=|2,2,1,1> will be an energy eigenstate (...
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Energy degeneracy given a rigid-rotor Hamiltonian

I'm trying to work out the degeneracy of some energy levels in a Hamiltonian given by $H = \frac{1}{2a} (L_x^2 + L_y^2) + \frac{1}{2b} L_z^2$. Looking for a common base of eigenstates $Y_l^m$ ...
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Numerical calculation of a quantum field's observables

Okay so QFT is definitely beautiful and elegant theory, its mathematics is rich and ingenious, but there is so much one can do with symbolic manipulations of mathematical objects only, how can I ...
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Unitary time evolution operator for hamiltonian

I have homework, which I've seen solution with which I have some problem. The homework is to find time evolution of state $|\psi(t)\rangle$, if the hamiltonian of the state is $\hat{\textbf{H}}=\...
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Hamiltonian for the spin-orbit coupling in spinor-notation

After some research I can't find a good source which explains how to write the Hamiltonian of a spin-orbit coupling in spinor notation. The following notation is commonly used $\hat H =\xi \hat L.\...
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Why does Hamiltonian no longer commutes with $\mathbf L$ and $\mathbf S$ in presence of spin orbit coupling?

Is this something due to change in Hamiltonian due to relativistic correction. i.e. $H$ being $$(\frac{e^2}{4\pi {\epsilon _0}}) \mathbf{S \cdot L}$$ instead of $$(\frac{e^2}{8\pi {\epsilon _0}}) \...
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In the derivation of Kubo Formula for electrical conductivity, how we can calculate the perturbation term which is due to the total electric field?

The perturbed Hamiltonian for the system can be taken as $H + V$, where $V$ is interaction between the total electric field and the particles of the system. In my calculation I need to find a form of ...
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Functional Analytic Square Root of Hamiltonian Alternative to Dirac

I was thinking about the history of the Dirac equation and asked myself, what happens if one simply considers the Schrödinger equation $i\hbar\frac{\partial\phi}{\partial t}=\sqrt{-c^2\hbar^2\Delta+m^...
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Hamiltonian for a system of 3 interacting particles and the meaning of potential

I'd like to discuss some physics basic. assume we have 3 particles $\{\vec{q_i},\vec{p_i}\}, \quad i=1,2,3$ whereas $\vec{q_i}$ is the position and $\vec{p_i}$ is the momentum. We also have a "pair-...
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Projection operator (relative angular momentum) in FQHE Toy hamiltonian

I am working on Fractional Quantum Hall Effect and reading these lecture notes http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf. As all others sources I have found, none of them precisely define the ...
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Is it possible for a quantum system to evolve out of a determinate state of some observable before measurement is made?

On page 96 of his book, Griffiths explains that determinate states of some observable $Q$ are eigenfunctions of that operator. So if a particle starts out in that state it will continue to be in that ...
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In context of defining a hamiltonian of a system, can someone please explain the meaning of zero-point oscillations/vibrations?

I came across this term while studying electron delocalization in amorphous semiconductors. The delocalization of the electronics wavefunction is explained in terms of zero-point oscillations.
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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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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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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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