Questions tagged [hamiltonian]

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

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1answer
71 views

Can we craft a Hamiltonian such that the measurement is consistent with the discrete measurement taught in Quantum physics?

So the way I understand this, the way measurement is taught is that you have a wave function $\Psi(t)$. It's evolution over time is : $$i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H(t)\vert\Psi(t)...
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213 views

Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
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33 views

Generically, why do we want to evolve states with unitary operators? [duplicate]

Why is it so important that operators that evolve states are unitary?
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Quantum field theory , Schrödinger wavefunction

$\psi$ is a state that given $|\psi\rangle=\int d^3x\psi(x)|x\rangle$ How does the wave function change in time? The Hamiltonian can be written as $$H=\int d^3x\frac{1}{2m}\nabla\psi^*\nabla\psi=\int ...
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How to know the symmetries of a coupling in the Hamiltonian

Suppose you have an interaction term in your hamiltonian that looks like \begin{equation} H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \end{equation} where $U$ is the coupling and $c$, $c^\...
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1answer
37 views

Can we automatically find the Hamiltonian from knowledge for multiple wave functions?

Say we had a set of wave functions $\psi(x,t)$ that we new the values of for all $x$ and $t=t_0..t_1$. Say we had $N$ of these wavefunctions, perhaps $N=10$. All these wave functions start off at ...
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Fisher Information in Statistical Mechanics

I am studying the canonical ensemble and it seems to me there is an analogy between derivatives of the partition function, which can extract energy momenta for the system and Fisher score /information....
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2answers
66 views

Manipulating Dirac Notation

I have trouble getting my head around manipulating Dirac notation, it's still new to me and I'm not used to it. I'm following the rotating wave approximation derivation for Rabi oscillations and light ...
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1answer
133 views

Need verification if this simple derivation of the Schrödinger equation is valid

By 1924 it was well observed that matter (as well as light) has wave-particle duality (later named quantum), and the wavelength-momentum-energy relation of quanta $$\lambda=\frac{h}{p}\;\;\...
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1answer
72 views

Born-Markov Approximation: Why is $\rho_{I}(s) \to \rho_{I}(t)$ taken, and not $\rho(s) \to \rho(t)$?

I am following along Chapter 3 of Breuer and Petruccione's book. For a Hilbert space $\mathcal{H}_{S} \otimes \mathcal{H}_{R}$ and Hamiltonian $$ H = H_{S} \otimes \mathbb{I}_{R} + \mathbb{I}_{S} \...
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Antisymmetry of an approximate wavefunction in a finite-basis calculation

Let's say I have a molecule and instead of using the Hartree-Fock procedure to find a Slater determinant wavefunction in a basis of atom-centered orbitals, I want to approximate the ground-state ...
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1answer
62 views

Eigenvectors of spin-spin coupling Hamiltonian

We want to find the eigenvectors and eigenvalues of the Hamiltonian, $H = \vec{\sigma_1}.\vec{\sigma_2}$ , where the subscript indicates the particle number. The usual way to go about it is to find ...
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1answer
76 views

Non-Hermitian Hamiltonian for electron conductance in electric field?

Electron conductance in a solid state is usually driven by electric field - making some direction of jumps more likely. It makes (e.g. Hubbard's) Hamiltonian no longer self-adjoint, how to simulate ...
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2answers
88 views

What are the fermions in the SYK model doing?

The Hamiltonian of the SYK model is \begin{equation} H = \mathcal{N}\sum_{ijkl}^N J^{ijkl} \chi_i \chi_j \chi _k \chi _l \end{equation} where $\mathcal{N}$ is some normalization to make the energy ...
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Determining Whether a Given Hamiltonian is Conserved

in simple terms I'm looking to understand how we can tell whether a given Hamiltonian (or one that we've deduced) is conserved or not? I've tried looking at other similar questions but am not sure I ...
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1answer
157 views

The WKB approximation and the Cotangent bundle

When we say (see pag. 9 of Lectures on the Geometry of Quantization) that the image of the differential of the phase function lies in the level set of the classical Hamiltonian is it simply ...
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3answers
344 views

Why symmetry transformations have to commute with Hamiltonian?

Let us consider a unitary or antiunitary operator $\hat{U}$, that associates with each quantum state $| \psi \rangle$ another state $\hat{U} | \psi \rangle$. I have read that to $\hat{U}$ be a ...
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1answer
56 views

Problem with understanding Time Evolution of a Quantum State [closed]

I was given the following task and I'm having some troubles with understanding a few things about it: There is given a system with Orthonormal basis $ |u_1 \rangle , |u_2 \rangle, |u_3 \rangle$ ...
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54 views

Examples of Simple Hamiltonians Giving Different Phases?

I am trying to study about simple condensed matter models that I can simulate numerically and use to calculate some topological invariant that defines a (topological) phase. My interest is in ...
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81 views

Solution to two non-coupled quantum harmonic oscillators

Given the following Hamiltonian: $$\hat H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_1^2} -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_2^2} + \frac{1}{2}m\omega_1^2x_1^2 + \frac{1}{2}m\...
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1answer
32 views

Galilei group and Constrained QM

Let's assume spin-0 for simplicity. So far as I understand the issue, the Galilei simmetries constraints the possible hamiltonians of a quantum systems so that the only possible interactions of a ...
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1answer
60 views

How to derive the analytical expression for the retarded Green's function with quadratic Hamiltonian?

For two operators, $A(t)$ and $B(t)$ the retarded Green’s function is defined as \begin{equation} G^R(t,t') \equiv \langle \langle A(t)|B(t) \rangle \rangle^R = -i\theta(t-t')\langle \{A(t),B(t')\} \...
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1answer
153 views

Symmetry transformations that are self-inverse and global symmetries of the Hamiltonian

I have the simplified Ising model. The Hamiltonian is given by $$ \mathcal{H} = -\mathrm{J}\sum_{<ij,i' j'>} \sigma_{ij} \sigma_{i'j'}. $$ Where the sum over $<ij,i'j'>$ means just the ...
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1answer
57 views

Domain of the infinite square well hamiltonian

I am reading the book by Gitman et al. 'self-adjoint extensions in quantum mechanics'. In the book, they give a precise definition of the domain of the hamiltonian of an infinite square well. For ...
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2answers
42 views

Wave function of a particle under $V(x)$ (QM)

Suppose I have a particle with mass $m$ and it's under potential of a certain $V(x)$. (NOT an infinite or finite potential well) Also given is the wave function at time $t=0$, $\psi(x,0)$. What is ...
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Symmetry in quantum mechanics

My professor told us that in quantum mechanics a transformation is a symmetry transformation if $$ UH(\psi) = HU(\psi) $$ Can you give me an easy explanation for this definition?
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Virtual terms in the Dyson series (time dependent perturbation theory)

Let the interaction evolution operator in the interaction picture be $$U_I(t,t_0)=T \exp \Big( -i \int_{t_0}^t dt_1 H_I(t_1) \Big) ,$$ where $T$ is the time order operator and $H_I=H-H_0$ is the ...
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2answers
368 views

Hamiltonian with one constant of motion (besides the Hamiltonian itself)

The background of my question is a well known fact: a Hamiltonian system with $n$ degrees of freedom with $n$ constants of motion is integrable. My question is about the case in which there are only ...
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1answer
55 views

Two-level system Hamiltonian from electric-dipole approx

After making the electric-dipole approx., I can express the interaction of a monochromatic field with angular frequency $\omega$ and a dipole moment ${\bf \mu(x)}$ as $V({\bf x},t) = - {\bf \mu(x)} \...
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A particle on a ring: orthogonality of eigenstates

Let us consider the quantum-mechanical problem---a particle on a ring of a circumference as $2\pi$ with a magnetic flux $A$ inserted through it: \begin{eqnarray} H=(-i\partial_\phi-A)^2, \end{...
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Hamiltonian directly expressed in $(q,\dot{q})$ : how to find what is $p$?

I am reading a book about non relativistic quantization of E.M field. But first we do classical field theory. We directly wrote the Hamiltonian of our study, and a part of our Hamiltonian is the ...
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1answer
62 views

Quantum mechanics on operator [closed]

If any operator is commute with Hamilton then they are labelled such a way that the energy eigenstate are equal and we also know it is a constant of motion. I don't related constant of motion with ...
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2answers
214 views

Lagrange multiplier in spin liquid mean-field theory (Paper by X.G. Wen)

My question is about a step in this paper: PhysRevB.65.165113 (X.G. Wen) page 6 Or alternatively: PhysRevB.90.174417 page 3. All the papers concerning spin liquids and the projective symmetry group ...
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Spontanous emission Hamiltonian model

I am looking for a clear (and not too long) model of spontaneous emission, for an atom modeled by a two level system in a cavity where the field is multimode I am looking for model bases on ...
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2answers
190 views

Baker-Campbell-Hausdorff (BCH) Formula for the Time Evolution Operator

In following Prof. Toyer's Computational Quantum Physics lecture notes, I came across the following: In computing the Schrödinger equation in real space, one can make a "split operator" Ansatz, for ...
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What does “substantial changes” in material properties mean in geometric optics?

We know that (I read it from Kip Throne's Modern Classic Physics) if a wave's wavelength is smaller than the length scale over which "substantial changes" of material properties occur, then the wave ...
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1answer
104 views

Hamiltonian in a Master Equation

I am going through this paper on the complete positive map with memory. The bath operators $\Gamma_k (t)$ are told to satisfy the correlation $\langle \Gamma_j(t) \Gamma_k(t^\prime) \rangle = a_k^2 e^{...
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1answer
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Can I express the Hamiltonian in terms of $L_z$ operator only, not $L_x$ and $L_y$? Is it generally true, that $-\omega L_z = H$?

I encountered the relation in the Solution of Problem 5.1 in the book by Kyriakos Tamvakis titled "Problems and solutions in quantum mechanics": $$\frac{i}{h}[H,\textbf{L}]=-\frac{i \omega}{h}[L_z, \...
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2answers
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Can the Hamiltonian operator act on a bra, if it was once acting on a ket?

I was watching a MIT Quantum Physics III class when I got a doubt about a specific bra-ket manipulation. My doubt is about the step from the expression $(3.7)$ to the expression $(3.8)$ of the lecture ...
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Solving time evolution equations in Hamiltonian formalism

I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential ...
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1answer
44 views

Hamiltonian-commutation, hermiticity and non-hermiticity (QM)

When we have a QM system in an energy eigenstate (say after a measurement of energy) then we can measure any time another quantity that is described by an hermitian operator that commutes with the ...
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2answers
72 views

What is the simplest possible Hamiltonian that yields an Antisymmetric Wavefunction?

I am using a Split-Operator Fourier Transform (SOFT) technique to solve the time-dependent electronic Schrödinger Equation (TDSE) for a Hydrogen molecule under the Born-Oppenheimer approximation. So I ...
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2answers
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In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?

When we go from the classical many-body hamiltonian $$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{...
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1answer
151 views

What does $|$ mean in the Schrödinger Equation?

I saw the $|$ symbol in the Schrödinger Equation $$i\hbar\frac{\partial}{\partial{t}}|\Psi(r,t)\rangle=\hat{H}|\Psi(r,t)\rangle$$ But I don't know what the $|$ means. What does $|$ mean in the ...
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1answer
104 views

Matrix of Hamiltonian $H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$ [closed]

I have a second order system, with a Hamiltonian $$H=\frac{ħω}{2}(|1〉〈1|−|2〉〈2|)+\frac{iħχ}{2}(|1〉〈2|−|2〉〈1|)$$ where $|1〉,|2〉$ form a complete basis for the system. I'm trying to get the matrix that ...
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1answer
46 views

Why do we consider only one mass when solving linear harmonic oscillators in quantum physics?

While solving the Hamiltonian, books concentrate on the horizontal flow with only one mass attached to the string. Isn't there any consequences if we add more masses and why is friction always ignored?...
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2answers
92 views

Why does the $\phi$-cubed theory have no ground state?

In the book of Sredinicki's, he claimed that the $\phi^3$ theory has no ground state, hence this is not a physical theory. My question is that I can't see why this system has no ground state. And I ...
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1answer
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Does adiabatic quantum computation require the initial and final ground states to be non-orthogonal?

Background At a recent talk, I was told by the speaker that it is not possible to adiabatically transfer from one ground state $|\psi_0 \rangle$ to another $|\psi_1 \rangle$ if these states are ...
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2answers
117 views

Is a quantum harmonic oscillator always infinite dimensional?

Let us assume we have a quantum particle in a harmonic potential with the Hamiltonian $$H = \sum_n n \omega |n\rangle\langle n|$$ If I am not mistaken. Now when talking about harmonic oscillators ...
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40 views

Time units in simulation

I am hoping that someone could give me some insight about my problem. Currently I want to simulate the evolution of a gaussian wavepacket in a time-dependent potential using the Crank-Nicolson scheme. ...