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
525 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
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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} ...
8
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1answer
576 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 ...
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59 views

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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146 views

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 - ...
5
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162 views

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 ...
5
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290 views

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 ...
4
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211 views

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 = -...
4
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631 views

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 ...
4
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170 views

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 ...
3
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45 views

Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#...
3
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1answer
76 views

Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$ i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...
3
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1answer
475 views

Is the Hamiltonian of a relativistic charged particle in an electromagnetic field only an approximation?

Consider a system of two relativistic charged point particles 1 and 2 which interact through their electric and magnetic fields. The equation of motion for the first particle is then given by the ...
3
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40 views

Is there any feature which distinguishes the Hamiltonian in the Poincare algebra?

The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ ...
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80 views

How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
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99 views

How to derive the simplest 1D Superpotential Hamiltonian?

In the superpotential wiki article there are definitions of two supersymmetric operators: $$Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW)b^\dagger\right] \\ Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW)b^\dagger\right] $$...
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62 views

Møller Opeartors and dressing relations: what picture?

Møller operators can be defined as (Urban, 2013;pg70): \[ \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\braket}[2]{\left<#1|#2\right>} \Omega_{\pm}=\underset{t\rightarrow \mp 0}{\...
3
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190 views

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 ...
3
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1answer
211 views

Momentum operator in effective mass approximation

When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: $$E(k) = \...
3
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551 views

Transfer from Heisenberg to Ising model

It is well know, that ferromagnets can be described using Hamiltonian $$ H = -\sum\limits_{i<j}J_{ij}\, (\mathbf{s}_i \cdot \mathbf{s}_j). $$ where (three dimensional) spins $\mathbf{s}_i$ ...
3
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1answer
100 views

Effective theories and unbounded operators

If you have two operators, one the true Hamiltonian $H$ and one we call an effective Hamiltonian $H_{eff}$ and say they agree on every eigenvector with eigenvalue up to $E_{eff}.$ Above that, they can ...
3
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1answer
309 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 ...
3
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556 views

What does it mean to expand a Hamiltonian using perturbation theory?

On UC Davis chemwiki website, the Hamiltonian for quadrupolar coupling in NMR is analyzed. (The details of this aren't important.) It is said in the analysis that: The expansion of the ...
2
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1answer
69 views

Why does the wave function of a non relativistic particle flatten out over time?

The Hamiltonian I used is the classical one with no potential energy: H=p^2/2m $$i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} $$ I want to gain ...
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97 views

Why doesn't Wigner's friend interact with the system?

So I was recently modelling something that turned out to be basically Wigner's friend. I saw there were some differences (in the Wiki page) in how it was modelled: Namely, that Wigner's friend ...
2
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30 views

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....
2
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42 views

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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59 views

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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108 views

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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77 views

Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...
2
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33 views

Looknig for resources on finding periodic orbit and stability on multidimensional Hamiltonian systems

I am looking for resources (books, papers, algorithms, codes) that explicitly explain the computation and analysis (using the monodromy matrix) of periodic orbits of multidimensional Hamiltonian ...
2
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1answer
317 views

What is the unitary matrix that diagonalizes the Hamiltonian?

If $H = H_0 + g H_1$ is our (free + interaction) Hamiltonian, and we assume that we have a basis of states $\{ | i \rangle \}$ under which $H_0$ is diagonal, then we may diagonalize $H$ by some ...
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157 views

How can I show that Second Quantization Hamiltonian is Hermitian?

How can I show that the non relativistic second quantization hamiltonian \begin{equation} \hat{H}=\lmoustache d^3x \hat{\psi}^\dagger_\alpha(x) T(x)\hat{\psi}^\dagger(x)+\frac{1}{2}\lmoustache d^3x\...
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72 views

What's the Hamiltonian of a spin subsystem?

Suppose I have 10 spins that are interacting through the Ising model with a random potential: $\mathcal{H} = \sum J S_i^x S_{i+1}^x + \sum S_i^z$ for a system size of L = n spins. Now I want to take a ...
2
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98 views

Solvable model with harmonic oscillator

I need help with a mathematical physics question. I have given the following system: A spin is coupled to a single harmonic oscillator mode with Hamiltonian $$H=(\epsilon/2) \sigma_{z} + \omega\, a^{...
2
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0answers
721 views

Parity of the ground state of a symmetric Hamiltonian

Is well know that if an Hamiltonian $\hat{H}$ commutes with the parity operator then exists a complete system of eigenstates with definite parity. So there will be even and odd states. I noticed that ...
2
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0answers
285 views

Exact solution of the Hubbard model in one-dimension

Does anyone know a full and well-explained account on how to solve the one-dimensional Hubbard model? I understand that the first solution is given here, but it's a little bit involved. Perhaps just ...
2
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0answers
140 views

What is a simple example of bound state in the continuum?

I am curious about the concept of bound state in the continuum. What is a simple example Hamiltonian which have this property and how can I derive this particular bound state? What are the ...
2
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1answer
77 views

Renormalization group to predict the ground state of a Hamiltonian

It is slightly difficult to phrase my question because I'm looking more of an entry point to something I know is a large field rather than a solution to a specific problem. I occasionally see papers ...
2
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0answers
45 views

Change in ground state after perturbing a hamiltonian

Lets consider a spin $\frac{1}{2}$ chain with $n$ spins and an associated local hamiltonian $H= \sum_i h_{i,i+1}$. We also assume that $\|h_{i,i+1}\|_{\infty} \leq 1$. In this question, we will be ...
2
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1answer
231 views

Scattering, 4 point correlator, #distinct Feynman diagrams

In order to compute the scattering probability that two particles of type 1 (associated to $\phi_1(x)$) which come from the far past with the momenta $p_1$ and $p_2$, to scatter and evolve into two ...
2
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116 views

Calculating the Closed String Hamiltonian (T-Duality)

Working with the bosonic string in a background space-time with one compact dimension, i.e.: $$ R^{1,24}\times S^1 $$ I have been able to calculate the mass-squared: $$ M^2 = \frac{n^2}{R^2} + \frac{m^...
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117 views

Holonomy of $\Lambda$-Configuration Hamiltonian

I'm currently reading through "Introduction to Topological Quantum Computing" By Jiannis Pachos in which the following is given as an exercise. "Consider a quantum system with three states in $\...
2
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0answers
282 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = \...
2
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300 views

A mysterious conserved quantity for a central potential

In teaching a course in classical mechanics and I have come across (from my predecessor) a to me mysterious conserved quantity. We are considering a gravitational (or electric) potential with the ...
2
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345 views

Fermion 1D Hubbard Model ground state in the U = 0 limit

I am trying to determine the ground state of the 1D fermionic Hubbard model at half-filling of $2L$ sites with $L$ electrons with spin-$\uparrow$ and $L$ electrons with spin-$\downarrow$ in the $U=0$ ...
2
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0answers
444 views

About the derivation of the Hamilton-Jacobi equation

It is an old question for me. In Goldstein's book, the H-J equation is derived in this way. We want to find a generating function $F(q,P,t)$ such that the transformed Hamiltonian vanishes identically, ...
2
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0answers
171 views

Adiabatic quantum evolution of single photon or biphoton system

The prerequisite for adiabatic quantum evolution of single photon or biphoton system is as follows. We have to prepare a single photon or biphoton quantum system which has a ground and a higher level ...
2
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0answers
81 views

Spin Transition Energies

I am reading a paper: http://arxiv.org/ftp/arxiv/papers/1305/1305.2445.pdf On p. 22, the following Hamiltonian is given: $$ H = \mu_B g \mathbf{B} \cdot \mathbf{S} + D(S_Z^2+\frac{1}{3}S(S+1)) + E(...
2
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133 views

Boundary condition Hamiltonian with point tinteractions

I`m studying the Hamiltonian with point interaction centered in $y$ in three dimensions. I know that the elements in the domain of the Hamiltonian are of the form $$\psi=\phi+qG^z(\cdot-y)$$ where $G^...