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

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187 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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75 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
127 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: $$V(x)~=~A'(x)^2-\frac{\hbar}{\...
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124 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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166 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 Schaum'...
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114 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
72 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
67 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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323 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 \mathbf{y})...
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1answer
73 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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55 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
709 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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72 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
98 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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1answer
70 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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344 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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1answer
150 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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293 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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299 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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66 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 eigenvectors....
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62 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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89 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
204 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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73 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. ...
1
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1answer
68 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 non-...
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55 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
101 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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0answers
78 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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326 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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108 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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2answers
55 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
153 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
173 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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1answer
89 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 = -\sum_{i=x,y,z}\...
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64 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 ...
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1answer
301 views

Eigenvalues of a nearest-neighbour tight-binding Hamiltonian in (Mahan, 2003)

In this paper by G. D. Mahan, he obtains the following electron Hamiltonian in a nearest-neighbour tight binding scheme: (page 2 of the paper, top of the right column) \begin{align} H_0 &= J_0 \...
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1answer
53 views

Eigenvalue for interacting Hamiltonian [closed]

Consider the Hamiltonian $$H=\omega_{1} a_{1}^\dagger a_{1}+\omega_{2}a_{2}^\dagger a_{2}+\alpha a_{3}^\dagger a_{3}(a_{1}^\dagger a_{2}+a_{2}^\dagger a_{1})$$ with $$ [a_\alpha^\dagger,a_\...
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0answers
70 views

Hamiltonian System Outside Physics [closed]

What are good examples of Hamiltonian systems outside physics? I heard there are financial systems that can be described by a Lagrangian, and was interested to see some examples
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1answer
539 views

Dealing with tensor products in an exponent

I am looking at the following problem and I am struggling to follow the steps involved. Consider the non-interacting Hamiltonian $$H_{AB}=H_A\otimes I_B+I_A\otimes H_B$$ So I'm trying to prove that ...
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2answers
205 views

Calculation of the $\langle H \rangle$ for a particle in a box

I am working through a problem in which a particle is in an infinite potential well of length $L$ at $t=0$ before the spontaneous change of the box being expanded to length $2L$. I have calculated the ...
0
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1answer
93 views

Two site Hamiltonian [closed]

So my question is I have this Hamiltonian: $$ H = \sum_i \epsilon_i \sigma^+_i\sigma^-_i + \sum_{i\neq j} V_{ij} \sigma^+_i \sigma^-_j, $$ and I want to write it out for two site. Is this correct?: $$\...
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119 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
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0answers
71 views

Exercise about Bethe Ansatz for $N=3$ particles on a ring of length $L$

Suppose there are $3$ bosons living on a 1-dimensional ring of length $L$. The Hamiltonian is given by $$H=-\sum_{i=1}^3\frac{\partial^2}{\partial x_i^2}+\sum_{1\leq j<k\leq 3}2c\delta(x_j-x_k).$$...
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1answer
65 views

Trouble getting the matrix representation of a 4-state Hamiltonian

$\newcommand{\bra}[1]{\left\langle #1 \right|} \newcommand{\ket}[1]{\left| #1 \right\rangle}$I have a simple 4-state Hamiltonian and am trying to find the matrix representation (in order to determine ...
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96 views

Basis spin states

We are given a system of $N$ spin states and the following (non-hermitian) Hamiltonian $$H = \frac{N \hbar \nu}{2M} \sin(\alpha)+ \sum_{i=1}^N \frac{\hbar \omega_i }{2} \sigma_{z,i} + \frac{\hbar \nu ...
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1answer
76 views

How can you tell if a GTO function is an eigenfunction of hamiltionan H?

How can you tell if a Gauss-type orbital is an eigenfunction of Hamiltionan $H$? For example: $$GTO = N z^2 \exp\left(-\alpha r^2\right)$$ I know it is and eigenfunction of $L_z$ and not $L_x$ and $...
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0answers
103 views

What happens to the Hamiltonian of the wave function after measurement?

As I understand it, the Hamiltonian is the kinetic plus the potential energy of the wave function. When a measurement is done what happens to the kinetic and potential energy? Does it dissipate? Is ...
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1answer
1k views

Commutation of Hamiltonian with momentum

In which case does the Hamiltonian $H$ commutes with the momentum $P$? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ...
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60 views

What is the physical significance of the two integration constants that appear in the ladder operator decomposition of the Quantum Hamiltonian?

If I have a simple one dimensional Hamiltonian of the form \begin{align} H = V - \partial_x^2 \end{align} and if I know one zero energy state solution $H\psi_0=0$ then I can use the Wronskian to ...
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0answers
120 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 $U=0$ ...