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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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5 votes
1 answer
240 views

Details in the derivation of the Lippmann-Schwinger equation

So the argument goes that for a slightly perturbed Hamiltonian $$ H = H_0 + V, $$ there will be some exactly known states, $\left|\phi\right>$, solving $$ H_0\left|\phi\right> = E\left|\phi\...
8 votes
4 answers
1k views

Examples of systems with infinite dimensional Hilbert space, whose energy is bounded from above

We often encounter (and love to!) deal with systems whose energy is bounded from below, for good reasons like stability, etc. But what about systems whose energy is bounded from above? In finite ...
1 vote
1 answer
362 views

Hamiltonian operator vs zeroth component of the momentum four-vector

The zeroth component of a particle's four-momentum is the energy divided by the speed of light. For a free particle of mass $m$, that is $$p_0 = \frac{E_{p}}{c} = \sqrt{\vec{\bf p}^2 + m^2c^2}.$$ Now, ...
1 vote
1 answer
123 views

Crystal field Hamiltonian using Stevens operators

I am trying to explicitly find (or understand how to find at first) Hamilton operators for crystal fields in different symmetries, e.g. $T_d$, using Stevens operators. The Hamiltonian is then of the ...
2 votes
1 answer
775 views

Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
6 votes
1 answer
173 views

Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?

I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
0 votes
1 answer
588 views

Calculating the expectation value of a spin operator in a uniform magnetic field

I'm trying Usually for these types of questions, I'm used to the field being in a specific direction. For example, if the field was in the $z$ direction, I could find this solution by checking $|\...
1 vote
2 answers
45 views

In degenerate perturbation theory why can we assume that matrix elements above and below the degenerate subspace disappear?

The picture shows some original Hamiltonian H which has some degeneracies. Suppose I have some perturbation V to the system and I want to find the new energies and eigenstates of the system. Then from ...
4 votes
1 answer
73 views

Solving for unitary operation using perturbation theory

Let the time-dependent Hamiltonian be \begin{equation} H(t) = H_0(t) + \lambda H_1(t), \end{equation} where $\lambda$ is a small parameter. In the interaction picture (i.e. rotating frame w.r.t ...
1 vote
1 answer
319 views

Symmetry relations of a Hamiltonian (Mirror, Chiral, Inversion, Rotation)

I'm reading this paper and I have the following questions regarding Eqs. (4)~(6): Eq. (5) shows a mirror symmetry relation $M_z H \left( k_x , k_y , k_z \right) M_z ^{-1} = H \left(k_x , k_y, -k_z \...
5 votes
1 answer
279 views

Are "good" states in perturbation theory eigenstates of both the unperturbed and perturbed Hamiltonian?

In my quantum course, my professor asked us the true/false question: "Are 'good' states in degenerate perturbation theory eigenstates of the perturbed Hamiltonian, $H_0 + H'$?" I was ...
0 votes
0 answers
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Mean energy measurement in an arbitrary quantum state

I've gone through many papers looking for a way to measure a mean energy in an arbitrary state $\langle \psi | H | \psi \rangle$. I am interested in a theoretical protocol or an exemplary experimental ...
0 votes
0 answers
41 views

What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?

The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
2 votes
1 answer
102 views

Derivation of Dirac Hamiltonian

In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads $$ L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi. ...
0 votes
2 answers
83 views

Energy and momentum operators using Hamilton's equations

The energy operator is: $${\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}\tag1$$ and the momentum operator is $${\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}.\...
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0 answers
49 views

Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?

I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12: $$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
2 votes
1 answer
246 views

Time evolution operator in quantum mechanics

One of the postulates of quantum mechanics is that, given a quantum state $\psi_{0}$ at time $t=0$, the state of the system at a posterior time $t > 0$ is given by $\psi_{t} = e^{-iHt}\psi_{0}$, ...
3 votes
1 answer
202 views

Line integral in Peierls substitution

I'm trying to understand the reasoning behind Peierls substitution. The final result seems to be simply replacing the hopping elements $$t_{ij} \to t_{ij} e^{i \frac{q}{\hbar} \int_i^j \vec{A} \cdot d ...
0 votes
1 answer
31 views

Eigenstates of the Laplacian and boundary conditions

Consider the following setting. I have a box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$, for some $L> 0$. In physics, this is usually the case in statistical mechanics or some problems in quantum ...
10 votes
2 answers
763 views

Why does time reversal symmetry requires a real Hamiltonian?

I have some problems understanding the consequences of time reversal symmetry. If Hamiltonian $H$ is symmetric under time reversal, it satisfies: $$ \mathcal{T} H \mathcal{T}^{-1} = H \quad \mathrm{...
0 votes
0 answers
42 views

Math in Hamiltonian of the hyperquantization of EM field

1. Background: I encounter this when looking into the hyperquantization of EM field. We have the secondly quantized field as below: $$\hat{E}^{(+)}(t)=\mathscr{E} e^{-iwt+i\vec{k}\cdot\vec{r}}\hat{a}=\...
0 votes
0 answers
11 views

AC Stark shift in the non-perturbative regime

I am trying to simulate the following situation. I have a 2 level system, with the energy spacing $\omega_0$. I have a laser, with Rabi frequency $\Omega_1$ and frequency $\omega_1$, which I can scan ...
0 votes
0 answers
33 views

Hamiltonian in Non-Linear Optics

I want to know why we add an additional term known as hermitian conjugate in the hamiltonian of many non-linear optical processes like SPDC. For example the in the equation below,
9 votes
1 answer
888 views

Temperature in the Hamiltonian limit

There is a well known connection between statistical mechanics in D spatial dimensions and quantum field theory in D-1 spatial dimensions. Changing the temperature in statistical mechanics corresponds ...
3 votes
0 answers
50 views

Existence of eigenstates in the context of continuous energies in the Lippmann-Schwinger equation

In the book QFT by Schwartz, in section 4.1 "Lippmann-Schwinger equation", he says that: If we write Hamiltonian as $H=H_0+V$ and the energies are continuous, and we have eigenstate of $H_0$...
6 votes
1 answer
75 views

How do I formulate a quantum version of Hamiltonian flow/symplectomorphisms in phase space to have a "geometric", quantum version of Noether's theorem

I'm currently exploring how Noether's theorem is formulated in the Hamiltonian formalism. I've found that canonical transformations which conserve volumes in phase space, these isometric deformations ...
10 votes
3 answers
1k views

Quantum harmonic oscillator meaning

Imagine we want to solve the equations $$ i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right> $$ where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
1 vote
2 answers
82 views

Why is the time derivative of the wavefunction proportional to a linear operator on it? [closed]

I am currently trying to self-study quantum mechanics. From what I have read, it is said that knowing the wave function at some instant determines its behavior at all feature instants, I came across ...
2 votes
1 answer
36 views

Why is the "decision" version of the local Hamiltonian problem promised to have a positive gap?

The Wikipedia article on the local Hamiltonian problem is ungrammatical and unclearly written. I think that this is what it is supposed to say: The decision version of the $k$-local Hamiltonian ...
2 votes
1 answer
355 views

Discretized derivation of Majorana path integral

Shankar's QFT book gives an overview for deriving a path integral representation for Majorana fermions. In the derivation, he works directly in continuous imaginary time, sweeping issues of ...
0 votes
1 answer
232 views

The most general $SU(2)$ invariant spin-$1/2$ Hamiltonian on 5 sites

I have periodic chain of spins $s=1/2$. I want to know what is the most general $SU(2)$ invariant and translation-invariant Hamiltonian. My guess is: $$\sum_i (j_1 S_i \cdot S_{i+1}+j_2 S_i \cdot S_{i+...
0 votes
0 answers
50 views

How Can I find Free Hamiltonian for this Problem?

I have got an Open Quantum System in which two two level atoms (two identical qubits) interact separately with two independent environments in the presence of the ...
5 votes
2 answers
2k views

Higher orders in non-degenerated time-independent perturbation theory

I would like to compute an energy level up to many orders in perturbation theory. My difficulty right now is not in the calculation itself but in understanding the algebraic structure of the higher ...
0 votes
1 answer
55 views

Discrete to continuous quantum operator

Let's say that we have a discrete lattice with $N$ sites. Let's label the site by the index $i$. Let's say that we have the operators $a_i$ and $a_i^\dagger$ which correspond to the creation and ...
3 votes
1 answer
281 views

Systematically going beyond minimal coupling?

Minimal coupling is a fairly standard procedure for describing the coupling of a charged particle with the electromagnetic field, and is often given by the following substitution in a Hamiltonian: $$p ...
3 votes
2 answers
179 views

Form of the Hamiltonian at Half-filling

I am trying to understand why chemical potential $= U/2$ is considered to be at half-filling in the case of the Hubbard Model Hamiltonian. So when I substitute this in its Hamiltonian, this is the ...
0 votes
0 answers
18 views

Derivation of the number operator in the energy basis of a qubit

I am trying to model the capacitive coupling of two transmon qubits. I would like to write the number operator in the energy basis, currently, I am working on using $$ \hat H = \hat H_1 + \hat H_2 + \...
2 votes
2 answers
1k 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} =...
0 votes
2 answers
61 views

Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
1 vote
1 answer
76 views

How to deal with explicit time dependence in the Heisenberg picture?

I am studying for my test in Quantum Mechanics, and there is something I don't quite understand about the Heisenberg picture and Heisenberg's equation of motion. In the lecture, we derived Heisenberg'...
3 votes
2 answers
850 views

How to find ladder operators that diagonalize a Hamiltonian in QFT?

I have some trouble understanding how one can, in the context of QFT, diagonalize a Hamiltonian $H$ by the introduction of ladder operators $a$ and $a^\dagger$ (I have trouble understanding how one is ...
0 votes
1 answer
525 views

Hamiltonian Simulation in Qiskit

I would like to simulate the time evolution of a quantum system Qiskit. Qiskit, however, only supports Hamiltonians that are a sum of hermitian matrices and that can be expanded into tensor products ...
1 vote
1 answer
614 views

One-dimensional Ising Model in a three spin chain

I have a system of three aligned spins with $S=\frac{1}{2}$. There are interactions between nearest neighbors, and each spin has a magnetic moment. The Hamiltonian of the system is: $$H=J[S_z(1)S_z(2) ...
0 votes
2 answers
69 views

Where does the complex conjugate term generally come from in a Hamiltonian?

I find myself stumbling across Hamiltonians which go like $$ \hat{H}\sim\alpha\hat{a}+\alpha^*\hat{a}^\dagger $$ How does this form of Hamiltonian actually come about? To my knowledge, the Hamiltonian ...
2 votes
0 answers
39 views

Is the overall (distinguishble-particle) ground state for a many-body identical particle Hamiltonian also immediately the bosonic ground state?

Consider the following many-body Hamiltonian of $N$ particles in an external trapping potential with inter-particle interactions: \begin{align} \hat{H}= \sum_{i=1}^{N} \left[-\frac{\hbar^2}{2m} \...
0 votes
0 answers
25 views

Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian

I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
0 votes
1 answer
53 views

Does the Hamiltonian always commute with the Time Evolution Operator?

The time evolution operator $U(t, t_0)$ is given as the solution of the equation $$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$ whether or not the system is conservative. When the ...
1 vote
1 answer
130 views

Understanding equation for eigenvalues of a Hamiltonian

I'm reading the paper Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics. I'm not understanding a claim they make about the eigenvalues of a certain Hamiltonian. In particular, how eqn 3....
0 votes
0 answers
60 views

Going to momentum space from Hamiltonian equation 17.4 chapter 17 in Schwartz

I'm reading the chapter 17 on the anomalous magnetic moment in QFT and the SM (Schwartz). In the section "17.1 Extracting the moment" he says "Going to momentum space,the Dirac equation ...
4 votes
2 answers
541 views

How do we justify the chemical potential term in a Hamiltonian of interacting fermions?

Consider a noninteracting fermi gas of electrons. If we know the chemical potential it makes sense that the Hamiltonian is $$\sum_{|k| > k_f} E_kc_k^{\dagger}c_k +\sum_{|k| < k_f}E_k c_kc_k^{\...

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