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

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5answers
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What does it mean for a Hamiltonian or system to be gapped or gapless?

I've read some papers recently that talk about gapped Hamiltonians or gapless systems, but what does it mean? Edit: Is an XX spin chain in a magnetic field gapped? Why or why not?
2
votes
3answers
1k views

Is there a valid Lagrangian formulation for all classical systems?

Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths? On the wikipedia page of Lagrangian mechanics, there is an ...
5
votes
1answer
400 views

Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
8
votes
1answer
140 views

How to prove that a hamiltonian system is not integrable?

To show that a system is integrable, we just need to find $N$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For example, ...
0
votes
3answers
494 views

Why can the Schroedinger equation be used with a time-dependent Hamiltonian?

I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems. However, one of the axioms in quantum mechanics postulates that ...
7
votes
1answer
4k views

Evolution operator for time-dependent Hamiltonian

When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
2
votes
1answer
185 views

Quantum Stat-Mech Proof of an Inequality for the Partition Function

I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been ...
15
votes
2answers
798 views

Is the Ground State in QM Always Unique? Why?

I've seen a few references that say that in quantum mechanics of finite degrees of freedom, there is always a unique (i.e. nondegenerate) ground state, or in other words, that there is only one state ...
8
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2answers
573 views

Regularisation of infinite-dimensional determinants

Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite ...
5
votes
1answer
323 views

Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
3
votes
2answers
196 views

What is a symmetry of a physical system?

If I understand correctly, in many context in physics (quantum mechanics?), a physical system is specified by giving its Hamiltonian. I also hear that symmetries are rather essential. As far as the ...
2
votes
0answers
110 views

Number theoretical function applied in physics? [closed]

I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does ...
2
votes
4answers
602 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
1
vote
3answers
1k views

Energy Spectrum of pair of spin-1/2 particles with general Hamiltonian

I found this problem, and so far I am stumped. I was wondering if anyone wanted to solve it with me, or help me calculate eigenvectors, or just give insight on my questions. Consider a system of ...
7
votes
2answers
3k views

How to express a Hamiltonian operator as a matrix

Suppose we have Hamiltonian on $\mathbb{C}^2$ $$H=\hbar(W+\sqrt2(A^{\dagger}+A))$$ We also know $AA^{\dagger}=A^{\dagger}A-1$ and $A^2=0$, letting $W=A^{\dagger}A$ How can we express $H$ as $H=\hbar ...
4
votes
1answer
534 views

Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
4
votes
1answer
175 views

Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field

Suppose we have an electron, mass $m$, charge $-e$, moving in a plane perpendicular to a uniform magnetic field $\vec{B}=(0,0,B)$. Let $\vec{x}=(x_1,x_2,0)$ be its position and $P_i,X_i$ be the ...
3
votes
1answer
599 views

Alkali atom in oscilating electromagnetic field

I am trying to calculate atom - light (EM field) interaction Hamiltonian, and the results I get seem to me rather unphysical - I get some nonzero matrix elements which should not be there. Please, can ...
2
votes
4answers
316 views

Bogoliubov transformation with a slight twist

Given a Hamiltonian of the form $H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} ...
1
vote
3answers
286 views

Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field

I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
5
votes
2answers
152 views

Do asymptotically similar potentials yield similar energy levels asymptotically?

Let there be given two Hamiltonians $$H_1~=~ p^{2}+f(x) \qquad \mathrm{and} \qquad H_2~=~ p^{2}+g(x). $$ Let's suppose that for big big $x$, the potentials are asymptotically similar in the sense ...
3
votes
1answer
619 views

Hamiltonian matrix off diagonal elements?

I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I ...
2
votes
1answer
125 views

Hamiltonian form of Noether's Theorem

I understand that Noether's Theorem has a Hamiltonian form, whereby {X, H} = 0 iff {H, X} = 0. The proof of this is trivial, as it follows from the antisymmetry of the Poisson Brackets. First ...
2
votes
0answers
151 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
votes
1answer
316 views

The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if: Lagrangian be time-independent, potential be independent of velocity, coordinate be time independent. Otherwise ...
2
votes
1answer
191 views

Hamiltonian of polymer chain

I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic ...