# Tagged Questions

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1answer
275 views

### State of Matrix Product States

What is a good summary of the results about the correspondence between matrix product states (MPS) or projected entangled pair states (PEPS) and the ground states of local Hamiltonians? Specifically, ...
3answers
68 views

### Constructing a Hamiltonian (as a polynomial of $q_i$ and $p_i$) from its spectrum

For a countable sequence of positive numbers $S=\{\lambda_i\}_{i\in N}$ is there a construction producing a Hamiltonian with spectrum $S$ (or at least having the same eigenvalues for $i\leq s$ for ...
2answers
499 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 ...
1answer
86 views

### Hamilton operator in absence of causal order?

I hope, this question isn't too broad or vague. In a recent paper, Ognyan Oreshkov et al. worked out a theory of quantum correlations in absence of any causal order, dropping the assumptions of a ...
2answers
686 views

### How to construct the Hamiltonian matrix?

I'm trying to understand if there's a more systematic approach to build the matrix associated with the Hamiltonian in a quantum system of finite dimension. For example, I know that for the ammonia ...
2answers
391 views

### Expectation value of time-dependent Hamiltonian

I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ ...
3answers
152 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 ...
3answers
463 views

### What is the relationship between Schrödinger equation and Boltzmann equation?

The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics. ...
1answer
187 views

### Question concerning the Lindhard function

I'm having a question concerning the Lindhard function. The reference I'm using is the standard text "Quantum Theory of Solids" by Charles Kittel. I'm concerned with Chapter 6, subchapter "Method of ...
4answers
574 views

### How to calculate the quantum expectation of frequency of a particle?

I know how to calculate the expectation of < $\Psi$|A|$\Psi$ > where the operator A is the eigenfunction of energy, momentum or position, but I'm not sure how to perform this for a pure frequency. ...
2answers
215 views

### Can an Electromagnetic Gauge Transformation be Imaginary?

The Hamiltonian of a non-relativistic charged particle in a magnetic field is $$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$. Under a gauge transformation ...
1answer
614 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  ...
1answer
119 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 ...
2answers
126 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 ...
1answer
201 views

### Second quantization

In second quantization we use Hamiltonian in form: $$H=\int d^3x [ \psi^{\dagger}(x) h \psi(x)],$$ where $h$ is Hamiltonian density. The field operators have following form: \psi = \sum\limits _{i} ...
1answer
208 views

### Does the vacuum energy problem of quantum field theory only occur in the Hamiltonian approach, or also in the path integral approach and in AQFT?

In a standard QFT class, you're being indoctrinated that there is the "infinite vacuum energy density problem". (This is sometimes paraphrased as the "cosmological constant problem", which is in my ...
1answer
162 views

### center of mass Hamiltonian of a Hydrogen atom

I'm working through Mattuck's "A Guide to Feynman Diagrams in the Many-Body Problem", but I'm stuck on a bit which I feel should be trivial. In section 3.2 (p 43 in the Dover edition) he gives a ...
2answers
92 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 ...
1answer
128 views

### How does a state in quantum mechanics evolve?

I have a question about the time evolution of a state in quantum mechanics. The time-dependent Schrodinger equation is given as $$i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle$$ I am ...
1answer
154 views

### Conjugate Transpose of Hamiltonian Matrix

I read some notes saying, $$i\hbar \frac{dC_{i}(t)}{dt} = \sum_{j}^{} H_{ij}(t)C_{j}(t)\tag{1}$$ where $C_{i}(t) = \langle i|\psi(t)\rangle$ and $H_{ij}$ is hamiltonian matrix. However, what is ...
1answer
137 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 ...
1answer
93 views

### It seems to me that superpotentials can be defined in a theory with or without supersymmetry. Is this true?

I recently read "An Introduction to Supersymmetry in Quantum Mechanical Systems" by T. Wellman (amongst other sources) in an effort to find out what a superpotential actually is and how it relates to ...
1answer
492 views

### How to write the Fröhlich Hamiltonian in one dimension?

I am currently working on a (functional) analysis problem refining Pekar's Ansatz (or adiabatic approximation, as it is called in his beautiful 1961 manuscript "Research in Electron Theory of ...
3answers
230 views

### When Hamiltonian and the total energy are the same

In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
3answers
544 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 ...
2answers
108 views

### How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?

I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times: $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$. ...
1answer
157 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 ...
4answers
302 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, ...
1answer
103 views

### Perturbation method & eigenvalues

I have a problem but I don't understand the question. It says: "Show that, to first order in energy, the eigenvalues ​​are unchanged." What does it mean? It means that if the Hamiltonian has the ...
1answer
303 views

### Solving time dependent Schrodinger equation in matrix form

If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector $$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$ With Hamiltonian $H$ given by H=\hbar\omega ...
1answer
52 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 ...
1answer
179 views

### Computing a density of states of Hamiltonian $H=xp$

How could I compute the integral $$N(E)~=~ \int dx \int dp~ H(E-xp)$$ the 'Area' inside the Phase space is taken for $x \ge 0$ and $p\ge 0$? The result should be  N(E)~=~ ...
1answer
93 views

### Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
2answers
402 views

### Two expressions for expectation value of energy

I was looking up expectation value of energy for a free particle on the following webpage: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/expect.html It says that $E=\frac{p^2}{2m}$ and ...
1answer
132 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 ...
0answers
42 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 ...
1answer
47 views

### Hamiltonians, density of state, BECs

When working with Bose-Einstein condensates trapped in potentials, how can one tell what the density of state of a system of identical bosons given the Hamiltonian, $H$? (I have been told that it is ...
1answer
67 views

### What does $\psi_j(r_i)$ mean?

I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by $V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$ I ...
1answer
199 views

### Cyclic Coordinates in Hamiltonian Mechanics

I was reading up on Hamiltonian Mechanics and came across the following: If a generalized coordinate $q_j$ doesn't explicitly occur in the Hamiltonian, then $p_j$ is a constant of motion ...
3answers
120 views

### The notion of bounded states in quantum mechanics and their characterization with operators

Is there any case of potential $V$, such that the continuity of the operator $H=c\ \Delta+V$ is not spoiled? And I don't know any non-differnetial operator examples for continous spectra. I ...
1answer
129 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 ...
3answers
189 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 ...
1answer
148 views

### Symmetry and overlapping of ground states

In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: E_0 = ...
1answer
266 views

### The Hermiticity of the Laplacian (and other operators)

Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator? Alternatively: is the matrix representation of the Laplacian Hermitian? i.e. \langle \nabla^{2} x | y \rangle = \langle x | ...
1answer
306 views

### Commutation relation with Hamiltonian

How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?
3answers
262 views

### Propagators and Probabilities in the Heisenberg Picture

I'm trying to understand why $$\Bigl|\langle0|\phi(x)\phi(y)|0\rangle\Bigr|^2$$ is the probability for a particle created at $y$ to propagate to $x$ where $\phi$ is the Klein-Gordon field. What's ...
2answers
386 views

### Canonical transformations and conservation of energy

I have an important doubt about the nature of canonical transformations in hamiltonian mechanics. Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on ...
1answer
68 views

### Transform hamiltonian

I have got the following Quantum Hamiltonian: $$H=\frac{p^{2}}{2m}+k_{1}x^{2}+k_{2}x+k_{3}$$ Which transformation can I use to change this Hamiltonian into an harmonic oscillator hamiltonian? ...
2answers
216 views

### Confusion about Free Energy and the Hamiltonian

I'm probably making a relatively basic mistake here, but I'm a bit confused about the relation between the Hamiltonian and Helmholtz free energy. From what I can see, the free energy can be written ...
1answer
83 views

### Where can I find hamiltonians + lagrangians?

Where would you say I can start learning about Hamiltonians, Lagrangians ... Jacobians? and the like? I was trying to read Ibach and Luth - Solid State Physics, and suddenly (suddenly a Hamiltonian ...
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