All Questions
Tagged with hamiltonian operators
352
questions
3
votes
1
answer
287
views
Time-evolution operator in QFT
I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following ...
- 613
-1
votes
0
answers
39
views
How to get $ H=\int\widetilde{dk} \ \omega a^\dagger(\mathbf{k})a(\mathbf{k})+(\mathcal{E}_0-\Omega_0)V $ in Srednicki 3.30 equation?
We have integration is
\begin{align*}
H =-\Omega_0V+\frac12\int\widetilde{dk} \ \omega\Big(a^\dagger(\mathbf{k})a(\mathbf{k})+a(\mathbf{k})a^\dagger(\mathbf{k})\Big)\tag{3.26}
\end{align*}
where
\...
- 37
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
4
votes
1
answer
78
views
Solving for unitary operation using perturbation theory
Let the time-dependent Hamiltonian be
\begin{equation}
H(t) = H_0(t) + \lambda H_1(t),
\end{equation} where $\lambda$ is a small parameter. In the interaction picture (i.e. rotating frame w.r.t ...
- 93
0
votes
0
answers
51
views
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
0
votes
2
answers
90
views
Energy and momentum operators using Hamilton's equations
The energy operator is:
$${\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}\tag1$$
and the momentum operator is
$${\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}}.\...
- 443
6
votes
1
answer
178
views
Are $\mathcal{PT}$-symmetric Hamiltonians dual to Hermitian Hamiltonians?
I was reading this review paper by Bender, in particular section VI where they show that, despite $\mathcal{PT}$-symmetric Hamiltonians not being hermitian, they can have a real spectra. They go on ...
- 1,544
0
votes
0
answers
39
views
Hamiltonian in Non-Linear Optics
I want to know why we add an additional term known as hermitian conjugate in the hamiltonian of many non-linear optical processes like SPDC. For example the in the equation below,
- 29
10
votes
3
answers
1k
views
Quantum harmonic oscillator meaning
Imagine we want to solve the equations
$$
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
0
votes
2
answers
64
views
Constant of Motion in Quantum Mechanics for explicit time-dependent Operators
I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
1
vote
1
answer
80
views
How to deal with explicit time dependence in the Heisenberg picture?
I am studying for my test in Quantum Mechanics, and there is something I don't quite understand about the Heisenberg picture and Heisenberg's equation of motion. In the lecture, we derived Heisenberg'...
- 111
0
votes
2
answers
78
views
Where does the complex conjugate term generally come from in a Hamiltonian?
I find myself stumbling across Hamiltonians which go like
$$
\hat{H}\sim\alpha\hat{a}+\alpha^*\hat{a}^\dagger
$$
How does this form of Hamiltonian actually come about?
To my knowledge, the Hamiltonian ...
- 692
1
vote
1
answer
82
views
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
8
votes
3
answers
919
views
Property of the Hamiltonian's discrete spectrum
I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
- 147