The hamiltonian tag has no wiki summary.
19
votes
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, ...
8
votes
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 ...
8
votes
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 ...
8
votes
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 ...
6
votes
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 ...
5
votes
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 $F(t)=F_0 \sin(\Omega t)$ ...
5
votes
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 ...
5
votes
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.
...
5
votes
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 ...
4
votes
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.
...
4
votes
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 ...
4
votes
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
$$
...
4
votes
1answer
119 views
Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
Suppose we have an electron, mass $m$, charge $-e$, moving in a plane perpendicular to a uniform magnetic field $\vec{B}=(0,0,B)$. Let $\vec{x}=(x_1,x_2,0)$ be its position and $P_i,X_i$ be the ...
4
votes
2answers
126 views
Do asymptotically similar potentials yield similar energy levels asymptotically?
Let there be given two Hamiltonians
$$H_1~=~ p^{2}+f(x) \qquad \mathrm{and} \qquad H_2~=~ p^{2}+g(x). $$
Let's suppose that for big big $x$, the potentials are asymptotically similar in the sense ...
4
votes
1answer
201 views
Second quantization
In second quantization we use Hamiltonian in form:
$$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} ...
4
votes
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 ...
4
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 ...
3
votes
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 $H=T+V$?
3
votes
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 ...
2
votes
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
$W$.
...
2
votes
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 ...
2
votes
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, ...
2
votes
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 ...
2
votes
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
$$H=\hbar\omega
...
2
votes
1answer
52 views
Hamiltonian of polymer chain
I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic ...
2
votes
1answer
179 views
Computing a density of states of Hamiltonian $ H=xp$
How could I compute the integral
$$ 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)~=~ ...
2
votes
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 ...
2
votes
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 ...
2
votes
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
...
2
votes
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 ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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:
$$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
1
vote
1answer
148 views
Symmetry and overlapping of ground states
In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$:
$$E_0 = ...
1
vote
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.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
1
vote
1answer
306 views
Commutation relation with Hamiltonian
How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?
1
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3answers
262 views
Propagators and Probabilities in the Heisenberg Picture
I'm trying to understand why
$$\Bigl|\langle0|\phi(x)\phi(y)|0\rangle\Bigr|^2$$
is the probability for a particle created at $y$ to propagate to $x$ where $\phi$ is the Klein-Gordon field. What's ...
1
vote
2answers
386 views
Canonical transformations and conservation of energy
I have an important doubt about the nature of canonical transformations in hamiltonian mechanics.
Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on ...
1
vote
1answer
68 views
Transform hamiltonian
I have got the following Quantum Hamiltonian:
$$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?
...
1
vote
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 ...
1
vote
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 ...
