The hamiltonian tag has no wiki summary.
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1answer
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Hamiltonion in 2-dimensions?
I am trying to construct a Hamiltonian for a system in 2 dimensions using Matlab.
I am not sure how this Hamiltonian will look like in matrix form.
If somebody can help me visualize this matrix that ...
-1
votes
1answer
65 views
Proof $\left[ {\hat H,{{\hat p}_i}} \right] = - \frac{\hbar }{i}\frac{{\partial \hat H}}{{\partial {{\hat q}_i}}}$ [closed]
I have a problem with the Hamiltonian, I don't think anything to solve it!!
So could you give me some hints!
Knowing that:
$$\left[ {{{\hat p}_i},{{\hat q}_k}} \right] = \frac{\hbar }{i}{\delta ...
2
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2answers
112 views
How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?
I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times:
$\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue
$W$.
...
2
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0answers
42 views
Boundary condition Hamiltonian with point tinteractions
I`m studying the Hamiltonian with point interaction centered in $y$ in three dimensions.
I know that the elements in the domain of the Hamiltonian are of the form
$$\psi=\phi+qG^z(\cdot-y)$$
where ...
4
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1answer
201 views
Second quantization
In second quantization we use Hamiltonian in form:
$$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} ...
3
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2answers
93 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 ...
5
votes
3answers
159 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 ...
3
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1answer
133 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 ...
0
votes
0answers
59 views
Quantum harmonic oscillator. Finding operators
Problem:
I'm trying to verify that $p_H(T)$ and $x_H(T)$ satisfy the following equations, (by solving the Heisenberg equation):
$x_H(t)=x_H(0)cos(\omega t)+(1/m\omega)p_H(0)sin(\omega t)$
...
2
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1answer
49 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
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1answer
134 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
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1answer
53 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
94 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
1answer
158 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 ...
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vote
3answers
190 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:
$$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
0
votes
2answers
136 views
Hamiltonian and non conservative force
I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$.
I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
1
vote
1answer
206 views
Cyclic Coordinates in Hamiltonian Mechanics
I was reading up on Hamiltonian Mechanics and came across the following:
If a generalized coordinate $q_j$ doesn't explicitly occur in the
Hamiltonian, then $p_j$ is a constant of motion ...
6
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2answers
711 views
How to construct the Hamiltonian matrix?
I'm trying to understand if there's a more systematic approach to build the matrix associated with the Hamiltonian in a quantum system of finite dimension. For example, I know that for the ammonia ...
3
votes
1answer
144 views
Alkali atom in oscilating electromagnetic field
I am trying to calculate atom - light (EM field) interaction Hamiltonian, and the results I get seem to me rather unphysical - I get some nonzero matrix elements which should not be there. Please, can ...
1
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0answers
86 views
Measurement in Quantum mechanics
I have got a quantum conservative system whose Hamiltonian is $H$. I consider an selfandjoint operator $O$ whose eigenvalues and eigenvectors are: $$O|\psi _{n}\rangle = \lambda _{n}|\psi ...
1
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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?
...
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0answers
39 views
Length of the orbit (semiclassical orbits)
The Gutzwiller trace is about
$$ d(E)=d_{0} (E)+ \sum_{p.o}A_{p}Cos(S(E)l_{p}) $$
and $ l_{p} $ are the length of the orbit
However my question is, how can one derive the length of the orbit from ...
2
votes
2answers
422 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 ...
5
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2answers
403 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 $F(t)=F_0 \sin(\Omega t)$ ...
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1answer
87 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 ...
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 ...
2
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4answers
303 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, ...
4
votes
1answer
633 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
$$
...
1
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0answers
84 views
Hubbard Model Hamitonian
$H = -\sum\limits_{i,j} A_{ij} c_i^{\dagger} c_j + \frac{U}{2} \sum\limits_i(c_i^\dagger c_i)(c_i^\dagger c_i -1)$ is defined to be a Hamiltonian for modeling quantum random walk of identical ...
4
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1answer
120 views
Finding the energy levels of an electron in a plane perpendicular to a uniform magnetic field
Suppose we have an electron, mass $m$, charge $-e$, moving in a plane perpendicular to a uniform magnetic field $\vec{B}=(0,0,B)$. Let $\vec{x}=(x_1,x_2,0)$ be its position and $P_i,X_i$ be the ...
2
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1answer
103 views
Perturbation method & eigenvalues
I have a problem but I don't understand the question. It says:
"Show that, to first order in energy, the eigenvalues are unchanged."
What does it mean?
It means that if the Hamiltonian has the ...
2
votes
1answer
306 views
Solving time dependent Schrodinger equation in matrix form
If we have a Hilbert space of $\mathbb{C}^3$ so that a wave function is a 3-component column vector
$$\psi_t=(\psi_1(t),\psi_2(t),\psi_3(t))$$
With Hamiltonian $H$ given by
$$H=\hbar\omega
...
1
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1answer
308 views
Commutation relation with Hamiltonian
How do we get $[\beta , L] = 0$ , where $L$= orbital angular momentum and $\beta$= matrix from Dirac equation?
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1answer
151 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 $H_0$:
$$E_0 = ...
3
votes
3answers
232 views
When Hamiltonian and the total energy are the same
In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
1
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3answers
267 views
Propagators and Probabilities in the Heisenberg Picture
I'm trying to understand why
$$\Bigl|\langle0|\phi(x)\phi(y)|0\rangle\Bigr|^2$$
is the probability for a particle created at $y$ to propagate to $x$ where $\phi$ is the Klein-Gordon field. What's ...
1
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2answers
222 views
Confusion about Free Energy and the Hamiltonian
I'm probably making a relatively basic mistake here, but I'm a bit confused about the relation between the Hamiltonian and Helmholtz free energy.
From what I can see, the free energy can be written ...
2
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1answer
68 views
What does $\psi_j(r_i)$ mean?
I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by
$V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$
I ...
1
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1answer
83 views
Where can I find hamiltonians + lagrangians?
Where would you say I can start learning about Hamiltonians, Lagrangians ... Jacobians? and the like?
I was trying to read Ibach and Luth - Solid State Physics, and suddenly
(suddenly a Hamiltonian ...
1
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1answer
273 views
The Hermiticity of the Laplacian (and other operators)
Is the Laplacian operator, $\nabla^{2}$, a Hermitian operator?
Alternatively: is the matrix representation of the Laplacian Hermitian?
i.e.
$$\langle \nabla^{2} x | y \rangle = \langle x | ...
0
votes
1answer
235 views
Base states with hamiltonian matrix
It is better for you to have studied "Feynman lectures on Physics Vol.3", because I cannot distinguish whether the words or expressions are what Feynman uses only or not and in order to summarize my ...
0
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1answer
121 views
Stationary states with a pair of hamiltonian equations
I read some derivation related with probability amplitudes and hamiltonian matrix in some book, and have a few questions.
Here what the book says is.
We want the general solution of the pair of ...
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3answers
121 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 ...
3
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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 ...
4
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1answer
164 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 ...
4
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1answer
213 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 ...
2
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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
$$ N(E)~=~ ...
4
votes
2answers
216 views
Can an Electromagnetic Gauge Transformation be Imaginary?
The Hamiltonian of a non-relativistic charged particle in a magnetic field is
$$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$.
Under a gauge transformation ...
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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 ...
8
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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 ...
