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Questions tagged [hamiltonian]

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

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How to get standard symplectic form so that one can get a vector field from a Hamiltonian function? [on hold]

In https://en.m.wikipedia.org/wiki/Hamiltonian_vector_field under 'Examples' a standard symplectic form w=(summation)dq(i)^dp(i) where p and q are the momentum and position of a particle. My question ...
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Symplectic form on a Hamiltonian

What are the Hamiltonian equations with respect to standard symplectic form? What is the standard symplectic form?
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25 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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+50

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
46 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
39 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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991 views

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
113 views

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

It is well known that 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 two constants of motion, one ...
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1answer
34 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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35 views

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
61 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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192 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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49 views

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
101 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
89 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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30 views

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
37 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
58 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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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
141 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
99 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
45 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
81 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
23 views

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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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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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. ...
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1answer
384 views

General derivative of the exponential operator w.r.t. a parameter

I am interested in the calculation of the general $N$th derivative w.r.t. a parameter $\lambda$ of a quantum mechanical exponential operator with the following structure: \begin{equation*} \frac{\...
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Interaction picture: why the Hamiltonian describing the dynamic doesn't change with the same law as other observables?

First: what happens in a general change of picture? If I have the following equation: $$ A | x \rangle = | y \rangle .$$ To do a change of picture is to apply a unitary $U$ on all vectors of the ...
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1answer
58 views

Discretization of Hamiltonian with first derivative

In a particular 1D system, the Hamiltonian can be writen as $$H=\mathrm{i}\left(f(r)\frac{\partial}{\partial r}+\frac{1}{2}f'(r)\right)\; ,$$ wher $\mathrm{i}$ is the imaginary unit, and $f(r)$ is a ...
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How to interpret overlap in Hamiltonian if it is not a degeneracy?

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
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1answer
79 views

Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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1answer
43 views

Is the tight binding model an effective free fermion model?

The tight-binding Hamiltionian has the form $$H=-t\sum_i\left(c_i^\dagger c_{i+1} + c_{i}c_{i+1}^\dagger\right)$$ But does this mean that it can be represented in the form of free fermion modes?
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114 views

How does a Hamiltonian 'generate' a unitary?

I know that the unitary (propagator) is given by $$U=e^{iHt}\tag{1}.$$ But I actually never saw how a Hamiltonian translates into a unitary. For example when I consider a two-level rotation in a ...
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1answer
53 views

Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$ I was hoping to understand this equation better. This suggests that ...
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25 views

What is the 4x4 matrix for the charge inversion operator and how do you construct it?

I have a 4x4 Hamiltonian describing a part of my system. To get a holistic view of the situation I need to do a charge inversion on the matrix. What is the 4x4 charge inversion operator? And what is ...
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2answers
63 views

Wilson Sommerfeld Method to solve for Energy

I have an example in my notes to find the quantum energy levels when the Hamiltonian is $H(p,q)={p^2}/{2m}+(mw^2q^2)/2$. However when given the Hamiltonian $H(p,q)={p^2}/{2m}$, I'm having ...
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Mathematical representation of Symmetry Transformation

Consider a general Hamiltonian that is made up of three terms $\mathcal{H}$ = term I + term II + term III . Suppose the combination of charge conjugation and parity (CP) is a symmetry of this ...
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1answer
61 views

What people mean by “state evolving with the interacting/free theory”?

This is a quite basic question but I confess it is something I didn't get up to this point. When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
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1answer
62 views

Potential must be real for Hamiltonian to be Hermitian?

I have seen a few proofs specify for finite wells, step functions, and harmonic oscillators, that $V$ must be real for $H$ to be Hermitian. Why is that? If we're solving the Schrodinger equation, we ...
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2answers
58 views

Good /not good quantum numbers in spin-orbit coupling

Given that the Hamiltonian associated with the spin-orbit interaction can be expressed in terms of the total orbital angular momentum and total spin operators as: $$ H_{SO} = -\frac{e}{2m_e ^2 c^2} \...
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1answer
44 views

Time reversal of a QM Hamiltonian

I'm interested in the time reversal properties of a term in the non-relativistic QM Hamiltonian proportional (up to a true scalar) to $$ H \propto (\vec S_1 \times \vec S_2) \cdot \vec L $$ The ...
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1answer
82 views

Quantum walk and the Hamiltonian operator

The Hamiltonian operator is defined on the graph $G$ as $H_{A}(t) = \exp(itA)$ where $A$ is the adjacency matrix of the graph $G$. It is said that this operator is a transition matrix and represents ...
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3answers
113 views

Time translation invariance of Hamiltonian

I am learning about the time translation invariance of the Hamiltonian. I read that the time translation invariance is already manifest in the fact that our Hamiltonian is chosen an ...
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1answer
98 views

‘Supersymmetrizing’ an arbitrary quantum-mechanical potential

To my understanding, it is not possible to $``\text{supersymmetrize}"$ an arbitrary quantum-mechanical system unless one knows how to represent the corresponding Hamiltonian in the form $$ H = A^\...
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1answer
41 views

Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...