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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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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 ...
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Exact diagonalization by Bogoliubov transformation

I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, $$ H = \begin{pmatrix} \xi_\mathbf{k} &...
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Structure theorems of Bravyi-Vyalyi and zero conditional mutual information

A fundamental result in quantum information is that of Bravyi and Vyalyi (Lemma 8, https://arxiv.org/abs/quant-ph/0308021). Result 1: It states that for commuting hermitian operators $H_{ab}\otimes ...
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What is a singular continuous spectrum?

I read some answers about this and the wikipedia page that basically always say that a spectrum can be decomposed into: $$\mu = \mu_{ac} + \mu_{sc} + \mu_{pp}, $$ where $\mu_{ac}$ is absolutely ...
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The unitarity of the $\delta(x)$ potential

One of the common potentials to solve in quantum mechanics is when $$H=\frac{p^{2}}{2m}+\delta(x).$$ Is this Hamiltonian considered to produce unitary evolution? In particular, I'm not sure what is ...
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Derivation of the BCS Hamiltonian

$$ \hat{H}_{\mathrm{BCS}}=\sum_{k \sigma} \varepsilon_{k} c_{k \sigma}^{\dagger} c_{k \sigma}-\sum_{k k^{\prime}} G_{k k^{\prime}} c_{k \uparrow}^{\dagger} c_{-k \downarrow}^{\dagger} c_{-k^{\prime} \...
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Topological Quantum Field Theories

I've asked this on Math.SE, but with no avail. So, I decided to ask it here. I was wondering about the following after reading the Wikipedia article on TQFTs. It is said that TQFTs have vanishing ...
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How the supercharge operators act on superfields in quantum mechanics, and the adjoints of supercharges?

I'm watching this lecture on introductory Supersymmetry (Clay Cordova, 2019 TASI lecture 2 on Supersymmetry). My question relates to the first 20minutes or so. The lecturer is introducing Superfields ...
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How to explain the reason of harmonic approximation by Wilson RG?

It's a part of my homework. In many body physics, considering the hamiltonian of the ions, we often use harmonic approximation then the hamiltonian turns to $$ H=\sum_{k}\hbar\omega(b^\dagger b+\...
Alex Chen's user avatar
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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\...
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$S$-matrix commutation with Hamiltonian

I know from scattering theory that $S$-matrix and the free Hamiltonian $H_{0}$ commute due to energy conservation of incident and outgoing asymptotic states, but can the $S$-matrix and $H = H_{0} + V$,...
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Hamiltonian eigenvalues in a transformed reference frame

Under a time-dependent unitary transformation $V(t)$ of the state vectors $|{\psi}\rangle$ \begin{equation} |\psi'(t)\rangle = V(t) |\psi(t)\rangle \end{equation} The Hamiltonian $H(t)$ has to ...
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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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Steady state solution to density matrix

A density matrix follows the dynamics $$ \dot{\rho} = \mathcal{L}\rho, $$ where $\mathcal{L}$ is the Liouvillian super-operator. If put in Lindblad form, it can be written as $$ \mathcal{L}\rho = -...
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Many-body relativistic classical Hamiltonian

All classical mechanics textbooks I know only discuss the one-body Hamiltonian in an external field $$H = \sqrt{m^2c^4 + c^2(\mathbf{p}-e\mathbf{A}/c)^2} + e\phi$$ Jackson in his celebrated textbook ...
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CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

Statement of the problem: I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
Jonathan Rayner's user avatar
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Why does the Eigen-Energy-Shift for the AC-Stark-Shift, calculated in the "rotating frame", actually matter?

As is neatly described in this answer, the AC-Stark Shift is the observation that the energy-eigenvalues of stationary states in the "rotating frame" of a two-level-system behave as (Adam ...
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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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Imaginary Hamiltonian

The Hamiltonian for nuclear spin independent parity violation in atoms is given by: $$H_{PV} = Q_w\frac{G_F}{\sqrt{8}}\gamma_5\rho(r)$$ Here $Q_w$ is the weak charge of the nucleus (which is a scalar),...
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Mechanical systems with their configuration space being a Lie group

In Marsden, Ratiu - Introduction To Mechanics And Symmetry there is a certain focus on reducing cotangent bundles of Lie groups. More precisely, if $G$ is a Lie group, then $T^*G$ is naturally a ...
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Meaning of the concept of external parameters in Statistical Mechanics

I'm confused about the meaning of the concept of external parameter in Statistical Mechanics. According to my textbook, the Hamiltonian of a system is a function that depends on the generalized ...
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Physical meaning of gapped path between Hamiltonians in the same phase

I'm reading this famous paper about the classification of quantum phases, and I'm wondering about the physical meaning of the definition of phases the authors use. They say that two Hamiltonians $H_0$ ...
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Utility of the Magnus expansion (preserving symplectic form?)

There are (at least) two ways to perturbatively solve a matrix initial value problem: the Dyson expansion and the Magnus expansion To be explicit, suppose you're solving for a (interaction-picture) ...
QuantumEyedea's user avatar
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Is Schrieffer-Wolff transformation equivalent to Feynman diagram and Path integral?

In high energy community, people usually use path integral (or Feynman diagram) to derive effective action (or effective Hamiltonian). However, in condensed matter or AMO community, people usually use ...
Herman Chu's user avatar
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Ostrogradski’s theorem's proof

I am looking for a proof of the next theorem: "If the higher order time derivative Lagrangian is non-degenerate, there is at least one linear instability in the Hamiltonian of this system." Where ...
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Hamiltonian Operator for nonrenormalizable Effective Field Theories?

Assuming we have a Effective Field Theory, for example a Real Scalar Field Theory, defined through a Lagrangian density of the form $\mathcal{L}_{eff} = \frac{1}{2}\partial_\mu\phi \partial^\mu\phi - ...
Thomas's user avatar
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Non-Hermiticity when Fourier transforming onto a finite lattice

I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \sum \limits_{<ij>}a^\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to ...
Mathusalem's user avatar
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1 answer
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Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
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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 $H_0$...
Gao Minghao's user avatar
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0 answers
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Classical mechanics: Hamiltonian perturbation theory. What if the perturbing parameter is < 0?

In Hamiltonian Perturbation theory, we have a Hamiltonian of the form $$H(q,p) = H_0(q,p) + \lambda H_1(q,p).$$ One proceeds by expanding the equations of motion in powers of $\lambda$, assuming $\...
James Thiamin's user avatar
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143 views

What is the interpretation of the eigenvalues of $e^{-\beta (H-\mu N)}$?

In quantum statistical mechanics, the equilibrium state is characterized by a density matrix $\rho$. Let me focus on the grand canonical ensemble, although the question also holds for the canonical ...
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Second-order Trotter error involving an unbounded Hamiltonian

I have an Hamiltonian of this form: \begin{equation} H = \frac{p^2}{2m} + V(x), \end{equation} I would like to approximate the time evolution for a time $\tau$ of a known initial Gaussian state $|\...
Luke's user avatar
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Can a mixed state be a stationary state?

If we define a stationary state as a state in which the probability distributions for every observable are constant in time, is it fair to say that a mixed state can also be a stationary state? One ...
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Why must a Hamiltonian be gapped to have "local" excitations?

On page 4 of Kitaev's "Anyons in an Exactly Solved Model and Beyond" he states The notion of anyons assumes that the underlying state has an energy gap (at least for topologically ...
DeafIdiotGod's user avatar
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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 ...
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Which operator measures "energy", when unitary transformations don't change matrix elements, but they change time evolution?

This question is a more general (and shorter) version of this previous question of mine. We know that from any quantum-mechanical description of a system, we can go to an equivalent description by ...
Quantumwhisp's user avatar
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Is the QFT Hamiltonian on an eternal Schwarzschild black hole background unbounded below?

Consider the $t=0$ Cauchy slice of the maximally extended Schwarzschild black hole. Let the parts of the slice to the left and right of the bifurcation surface have Hilbert spaces $\mathcal{H}_L $ and ...
nodumbquestions's user avatar
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Factorization of 1d Ising model partition function

If I'm studying a 1-dimensional Ising model such that $\mathcal H = \sum_k J_k\sigma_k\sigma_{k+1}$, where $$J_k=\begin{cases}J&k \in2\mathbb N\\2J&k\in2\mathbb N+1 \end{cases}$$ can I ...
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0 answers
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Adiabatic elimination in the presence of single-photon transitions

I am trying to derive an effective Hamiltonian for a system that includes both periodic drives near resonance with certain transitions (that should lead to one-photon transitions) and pairs of drives ...
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Eigenkets of a two-state hamiltonian

I have a question related to this other question: Eigenenergies and eigenkets given the Hamiltonian. In it, OP is given the following hamiltonian: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\...
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1 answer
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How to construct a set of local Hamiltonians sharing the same ground state?

In the quantum approximate optimization algorithms, the superposition state $|+\rangle^{\otimes N}$ is usually prepared as the initial state, which is the ground state of driven Hamiltonian $H_M=\...
Yang Qian's user avatar
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0 answers
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Are all physically realistic Hamiltonians local?

My understanding of modern physics is that physicists think that, fundamentally, physical laws are local. For system A to interact with system B, they either need to be very close to each other or ...
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Time dependency in the Hamiltonian in Schwartz's book on Quantum Field Theory

On page 258 of his book on Quantum Field Theory and the Standard Model (in the comment between formulas 14.25 and 14.26) , Schwartz writes: "We will also write $\hat{H} (t) = \int d^3 x \hat{\...
OutrageousKangaroo's user avatar
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0 answers
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How to cast Frohlich's Hamiltonian from phonon creation and annihilation operators to phonon position and momentum operators?

I have had trouble casting Fröhlich's Hamiltonian (Eqn 2.33 in [1]) from its form in terms of phonon creation $\hat{b}^\dagger_{\mathbf{k}}$ and annihilation operators $\hat{b}_{\mathbf{k}}$ to its ...
Bradley Martin's user avatar
3 votes
0 answers
400 views

Non-Hermitian Hamiltonian diagonalization

Edit : 2x2 simple system instead, simplification of the question. I would like to study a system into its diagonal form, but this system is represented by a non-Hermitian Hamiltonian $\tilde{\mathcal{...
Koyot's user avatar
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1 answer
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Finite temperature greens function in grand canonical ensemble

I see this question was asked several times before but I don't think any answer can explain the issue perfectly. I am studying many body theory and encounters finite temperature Green's function. At ...
Wong Harry's user avatar
3 votes
0 answers
106 views

Eliminating an eigenvalue from the Hamiltonian

I have a momentum space Hamiltonian $H(\vec k)$ for a Kagome lattice and I want to find its eigenvalues which may be dependent on $\vec k$. Now, I'm told that one of the eigenvalues for such ...
hhsomething69's user avatar
3 votes
1 answer
98 views

Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?

I want to arrive to Hamilton-Jacobi equation using the Riemannian geometry. So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
Jagoba Barata's user avatar
3 votes
0 answers
168 views

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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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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