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

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15 views

How to include Berry Connection in Hamiltonian?

When we calculate Berry connection, $A(R)=i<\psi(x,y)|\frac{d}{dR}|\psi(x,y)>\hat{R}$ corresponding to the Berry phase of any system, the gauge potential is related to the $R$ of the parameter ...
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3answers
95 views

Why does Hamiltonian follow the property $H^*_{ij} = H_{ji} $?

I was reading Feynman's Lectures III's Hamiltonian Matrix. There I found this property of Hamiltonian Matrix: The Hamiltonian has one property that can be deduced right away, namely, that ...
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6answers
239 views

Why does time evolution operator have the form $U(t) = e^{-itH}$?

Let's denote by $|\psi(t)\rangle$ some wavefunction at time $t$. Then let's define the time evolution operator $U(t_1,t_2)$ through $$ U(t_2,t_1) |\psi(t_1)\rangle = |\psi(t_2)\rangle \tag{1}$$ and ...
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1answer
31 views

Complex scalar theory: annihilation and creation operators give wrong commutators with Hamiltonian

The theory of a real (hermitian) scalar field can be found in many books and everywhere online. On the other hand, if we take the field non-hermitian, then I can only find notes on path integrals. I ...
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65 views

How to get Heisenberg Equation of Motion?

If a system Hamiltonian is given by, $$ ...
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1answer
101 views

Example where Hamiltonian $H \neq T+V=E$, but $E=T+V$ is conserved

I'm looking for an example of a Hamiltonian $H$, where $H\neq T+V$, but the total energy in the system, $E=T+V$, is still conserved. While I'm at it, I might as well add that I'd be most interested ...
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8 views

Trapped ion Hamiltonian: simultaneous blue and red sidebands

In the paper "Spin-motion entanglement and state diagnosis..." (Home et al., Nature, 2015), the authors talk about a single trapped ion, optically pumped, simultaneously driving the red and blue ...
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1answer
93 views

Is the Klein-Gordon Hamiltonian unbounded below?

This question is about the Klein-Gordon Hamiltonian for simplicity, but the problem seems to remain when dealing with other fields (e.g. Dirac, photon...). One usually writes the Hamiltonian ...
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1answer
72 views

A trace formula of two noncommutative operators

In many cases of quantum many-body problems, the Hamiltonian $H$ can always be divided into two parts, i.e. $H_0$ and $H'$. In this occasion, one can systemically calculate the partition function ...
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1answer
35 views

What is the form of the kinetic energy operator on a one dimensional (real space) lattice? (In second quantization)

I'm trying to figure out how one would write down the hamiltonian of a free fermion system (eventually in second quantization) on a one dimensional lattice and I'm having trouble both coming up with ...
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1answer
59 views

Any three-body Hamiltonian? [closed]

Is there an extension of spin interactions into three-body interactions such as $$H\sim \sum J \sigma_1\otimes \sigma_2 \otimes \sigma_3$$
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1answer
111 views

Quantum mechanic particle

In non relativistic quantum mechanic, we are dealing with a problem involving a particle in one dimensional space, and it has been given the potential and it reads: ...
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49 views

Constructing a dispersion relation from the Hamiltonian

I'll begin by saying that I'm not entirely clear on if this is possible. I have a Hamiltonian of the form $$ \left( \begin{array}{cccc} \text{$\omega $1} & \text{J12} & 0 & \text{J14} \\ ...
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1answer
87 views

Why do $H$ and $L^2$ commute in spherically symmetric potential?

In this PDF document (a lecture by Shivaly Reddy, page 13), he says that $L^2$ is independent of $r$; therefore it commutes with any function of $r$. This seems related to a problem in ...
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2answers
98 views

Would $[\hat{Q},\hat{H}]$ correspond to an observable? [closed]

Would $[\hat{Q},\hat{H}]$ correspond to an observable? Where $\hat{Q}$ is an observable and $\hat{H}$ is the Hamiltonian. Surely that would just mean that $[\hat{Q},\hat{H}]$ would commute i.e. = 0?: ...
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1answer
55 views

How does the Hamiltonian change when going to a moving frame?

The Hamiltonian of a free particle in a rotating frame is given by $$ H = H_0 - \omega \cdot J, $$ where $H_0$ is the Hamiltonian in the non-rotating frame, $\omega$ is the angular velocity of the ...
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1answer
55 views

Changing the zero-point energy

I have the following Hamiltonian $$\mathcal{H}(\{x_i,y_i \})=-l\sqrt{2}\sum_{i=1}^N \mathbf{f}_i \cdot \hat{\mathbf{b}}_i+E_0$$ For calculating things like the partition function it would be ...
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1answer
73 views

Jacobi energy function $h$ and the Hamilton $H$ and the Hamilton-Jacobi equation

My understanding of the Jacobi energy function $h$ as defined in Goldstein is that it is the total energy $T+V$ expressed as, \begin{equation} h(q,\dot q,t)=\sum \frac{\partial L}{\partial \dot q}\dot ...
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1answer
282 views

Mean field theory Weiss Approximation for the Isling Model of a Protein

A model for protein in 2D can be formed by adding bonds of fixed length $l\sqrt{2}$ on a square lattice along the diagonal, ie $\hat{\mathbf{b}}_i=\frac{1}{\sqrt{2}}(\pm \hat{\mathbf{x}}\pm ...
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1answer
52 views

What does it mean for a quantum particle to have energy $E_n$? And what is its general normalised state?

In this particular case, I have found the energy to be quantised with energy levels $\frac{h^2n^2}{2m} >0 $ where $n$ is an integer. Suppose a particle has energy $E=\frac{4h^2}{2m}$, then this ...
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34 views

Does the time Evolution operator commute with the both free and interaction Hamiltonian?

Consider a quantum system (finite dimensional) which has overall Hamiltonian: $H_s = H_0 + w(s)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of ...
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1answer
105 views

How do we find the phase space density from the Hamiltonian?

How do we find the phase space density from the Hamiltonian? For example: Consider a classical gas made of N identical non-interacting particles in 1d. Each molecule is characterised by centre mass ...
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1answer
43 views

Trick for reformulating in terms of centre of mass and relative variables

I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian $$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$ I want to express the ...
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1answer
56 views

The grand partition function of non interacting hamiltonians

In the case of non interacting particles I know we can write the Hamiltonian as $$H(\mathbf{q}_1,\dots,\mathbf{p}_1,\dots)=\sum_{i=1}^N h(\mathbf{q}_i,\mathbf{p}_i)$$ but I am having trouble ...
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42 views

Hamiltonian flow?

I was wondering what the Hamiltonian flow actually is? Here is my idea, I just wanted to know if I am correct about this. So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and ...
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1answer
113 views

What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
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0answers
46 views

Time evolution of interaction Hamiltonian in the Heisenberg picture

How does the interaction Hamiltonian of a (finite dim) quantum system with Hamiltonian: $H(t) = H_0 + w(t) H_I$ evolve in time in the Heisenberg picture. Is there anything special about the way ...
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1answer
98 views

Commutator and Hamiltonian [closed]

Assume that $[\hat{A},\hat{H}]_-=0$ and $[\hat{B},\hat{H}]_-=0$ but we know that $[\hat{A},\hat{B}]_-\neq 0$. Then there exists degenerate stationary states of $H$. How to prove it?
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62 views

Unitary evolution operator

Assume we have a system in a state $\psi$ that is a superposition of eigenvectors of some observable $A$. Now we make a measurement of the observable $A$; the state after the measurement $\phi$ is a ...
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2answers
106 views

Time dependent Hamiltonian and Gauge invariance

In general, in quantum mechanics we can prove probability current or the Schrodinger equation and other quantities are gauge invariant. However, the Hamiltonian isn't gauge invariant. Under a gauge ...
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2answers
152 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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53 views

Computing the probability density of wavefunctions

Suppose I am given a Hamiltonian operator $\hat{H}$ that satisfies the time-independent Schrödinger equation $$\hat{H} \psi = E\psi$$ I can compute energy eigenvalues and their associated ...
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37 views

How geometry, and hence, a tight-binding Hamiltonian dictates the eigenvalues?

Considering an 'N' atom system, how should we understand the geometric dependence on the calculated eigenvalue spectrum by solving the nearest neighbor tight-binding Hamiltonian? A simple example ...
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2answers
82 views

Energy conservation $\iff \frac{dE}{dt} = 0\ $?

If I'm asked to prove that a system is/ isn't conservative and compare it to whether or not the Hamiltonian is conserved, does that mean I need to compute the time derivative of energy $(T+U)$? Doing ...
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3answers
151 views

Does Peskin & Schroeder Eq. (4.26), $U(t_1,t_2)U(t_2,t_3) = U(t_1,t_3)$ imply $[H_0,H_{int}] = 0$?

Peskin & Schroeder equation (4.17) define the operator, \begin{equation} U(t,t_{0})~=~e^{i(t-t_{0})H_{0}}e^{-i(t-t_{0})H} \tag{4.17} \end{equation} where $$H~=~H_0+H_{\text{int}}\tag{4.12}$$ is ...
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1answer
63 views

Commuting with time evolution operator implies commuting with Hamiltonian

Consider a quantum system (finite dimensional) has overall Hamiltonian: $H_t = H_0 + w(t)H_c$ with $H_0, H_c$ constant in time and traceless and $w(t)$ a, not too badly behaved, function of time. ...
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1answer
58 views

Formulating a symplectic integrator for a non-local Hamiltonian

I recently asked two questions, Q. [1] and Q. [2], regarding reformulating non-local Lagrangians as Hamiltonians. In these questions, the Hamiltonian is formulated as an integral because of it's ...
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30 views

CSCO and Hamiltonian

Is there a relationship between the Hamiltonian and the number of observables needed in a CSCO to fully describe a quantum system? I am thinking that a new observable is needed each time there is a ...
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0answers
34 views

Why doesn't a quantum pairwise Hamiltonian couple states in which more than one interaction occurs?

This question is about the standard quantum mechanical pairwise interaction Hamiltonian. I'll phrase it in terms of an example using Rydberg atoms, but you could just as well imagine spins (for ...
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1answer
53 views

Average Energy of a coherent state

The question is relating to a previous problem concerning the harmonic oscillator. Determine the average energy < E > in a coherent state |alpha>. From my understanding the expectation of the ...
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60 views

Does the additivity property of Integrals of motion and Lagrangians valid in all situations?

I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
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3answers
208 views

Quantum Mechanincs - Dirac notation and solving time dependant schrodinger [closed]

The $\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}$ obviously correlate to $x,y,z$ components of the operators. Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a ...
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1answer
91 views

How much information does the Hamiltonian contain in quantum mechanics? [closed]

Given a Hamiltonian, let's say of a many-body system, through the Schrodinger equation,in principle we can find the eigenfunctions and their corresponding eigenvalues (spectrum). Now given an ...
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47 views

Hamiltonian for semiconductor

I was wondering which terms we need in a semiconductor Hamiltonian where no transition between the valence and conduction band occur? First we would have a term describing the energy of the full ...
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2answers
43 views

Allowed system energies from quantized Hamiltonian spectra

To find the allowed energies for a system, I can find the spectrum of the Hamiltonian $\hat{H}_{\psi}$ given a wavefunction $\psi$ representing the state of the system. 3 cases might happen: either ...
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1answer
98 views

Relation between scattering matrix and an effective Hamiltonian

Could somebody provide the proof (or reference to some accessible literature) of relation $$S(E) = 1 + 2πiW^{†} (H_M − E − iπW W^{†} )^{−1} W \tag{2}$$ of arXiv:0806.4889, which relates $S$-matrix to ...
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1answer
167 views

How do I show that a given Hamiltonian does not affect the overall number of particles in a given state?

I'm struggling with the following problem: Consider a system of an arbitrary number of indistinguishable bosonic particles. The system has two sites and $a_i^{\dagger}$ and $a_i$ are the ...
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0answers
40 views

Alternative formulations of Lagrangians and Hamiltonian? [closed]

We have the Hamiltonian, a concept that was based on trajectories being used extensively in General Relativity, Electromagnetism, Quantum Mechanics, Classical Physics and lot more. Where we use the ...
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1answer
58 views

Electron-Hole Spin Exchange Interaction

I am stuck with this seemingly "simple" Hamiltonian. I am dealing with an exchange term of a Hamiltonian for two different spin species: $$H_\text{exchange} = - \lambda J \cdot S = ...
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59 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...