All Questions
Tagged with hamiltonian hilbert-space
219
questions
8
votes
4
answers
1k
views
Examples of systems with infinite dimensional Hilbert space, whose energy is bounded from above
We often encounter (and love to!) deal with systems whose energy is bounded from below, for good reasons like stability, etc. But what about systems whose energy is bounded from above? In finite ...
- 785
1
vote
2
answers
49
views
In degenerate perturbation theory why can we assume that matrix elements above and below the degenerate subspace disappear?
The picture shows some original Hamiltonian H which has some degeneracies. Suppose I have some perturbation V to the system and I want to find the new energies and eigenstates of the system. Then from ...
- 103
5
votes
1
answer
286
views
Are "good" states in perturbation theory eigenstates of both the unperturbed and perturbed Hamiltonian?
In my quantum course, my professor asked us the true/false question:
"Are 'good' states in degenerate perturbation theory eigenstates of the perturbed Hamiltonian, $H_0 + H'$?"
I was ...
- 63
0
votes
0
answers
51
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What are the similarities and differences between the Magnus expansion and the Schrieffer-Wolff transformation?
The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates ...
- 19
10
votes
3
answers
1k
views
Quantum harmonic oscillator meaning
Imagine we want to solve the equations
$$
i \hbar \frac{\partial}{\partial t} \left| \Psi \right> = \hat{H}\left| \Psi \right>
$$
where $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial ...
- 221
3
votes
2
answers
203
views
Form of the Hamiltonian at Half-filling
I am trying to understand why chemical potential $= U/2$ is considered to be at half-filling in the case of the Hubbard Model Hamiltonian. So when I substitute this in its Hamiltonian, this is the ...
- 53
0
votes
0
answers
32
views
Time evolution using non-Hermitian (not a PT symmetric) Hamiltonian
I am currently dealing with non-Hermitian hamiltonian and dynamics using it. In general the diagonalizable non-Hermitian matrix might have complex eigenvalues and the eigenvectors may not be ...
- 711
1
vote
1
answer
82
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Exercise on self-adjointness of Hamiltonian [closed]
I am struggling with some exercise I have to solve for my quantum mechanics class.
PROBLEM:
Suppose $|\psi\rangle, |\phi\rangle$ are normalised and linearly independent (but not necessarily ...
- 743
0
votes
0
answers
62
views
Wigner's formula for the kinetic energy density in QM
In the Schroedinger equation the kinetic energy is represented by the operator $T = -\frac {\hbar^2} {2m} \Delta$ which acts on a wavefunction $\Psi$. If we multiply this by the complex conjugate of ...
- 743
0
votes
2
answers
113
views
Is the initial state the eigenstate of a Hamiltonian?
Solutions to the Schrödinger equation can take the form $ \psi(r,t)=\psi(r)f(t) $, where $f(t) = e^{\frac{-iEt}{\hbar}}$,
$$ H \psi(r) = E \psi(r) ,$$ where $\psi(r)$ is the eigenstate of a ...
- 45
1
vote
0
answers
72
views
If $H$ anniliates a state, must $Q$ and $Q^\dagger$ also annihilate the state?
Suppose we have a a Hamiltonian, $H$. And suppose also we have some operator $Q$ such that $\{Q, Q^{\dagger}\} = H$, and $Q^2 = 0$.
If we find a state $|\psi \rangle$ such that $Q|\psi \rangle = Q^{\...
- 213
3
votes
2
answers
204
views
Why are these unbounded operators (essentially) self-adjoint?
Can anyone provide exact mathematical reasoning as to why the following fundamental unbounded symmetric operators are essentially self-adjoint? I.e. on, their natural domains, they admit a unique ...
- 30
-3
votes
2
answers
107
views
Multi-particle Hamiltonian for the free Klein-Gordon field
The text I am reading (Peskin and Schroeder) gives the Hamiltonian for the free Klein-Gordon field as:
$$H=\int {d^3 p\over (2\pi)^3}\; E_p\; a^{\dagger}_{\vec p}a_{\vec p}$$
This does not seem to be ...
- 4,399
2
votes
0
answers
81
views
Why Fock representation holds only in a free quantum field theory?
With a quantum system with $N$ degrees of freedom, all the representations are unitarily equivalent to Fock representation. However, if the number of degrees of freedom goes to infinity, there are ...
- 159
1
vote
0
answers
40
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Questions of lower boundness of Hamiltonians in quantum theories
In general spectral analysis, we have examples of unbounded from below hamiltonians with discrete spectrum. Is it okay to say that they have no sense in physical context, because for me it looks like ...
