Questions tagged [hamiltonian]
The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.
190
questions
3
votes
1
answer
283
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 ...
- 521
-1
votes
0
answers
38
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
\...
- 31
3
votes
1
answer
140
views
How to get a lower bound of the ground state energy?
The variational principle gives an upper bound of the ground state energy. Thus it is quite easy to get an upper bound for the ground state energy. Every variational wave function will provide one.
...
- 1,947
0
votes
0
answers
35
views
When is the derivative of Hamilton flow respect to initial conditions independent of time?
Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
- 4,827
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
4
votes
1
answer
77
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
36
views
Mean energy measurement in an arbitrary quantum state
I've gone through many papers looking for a way to measure a mean energy in an arbitrary state $\langle \psi | H | \psi \rangle$. I am interested in a theoretical protocol or an exemplary experimental ...
- 1
5
votes
1
answer
282
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
47
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
2
votes
1
answer
111
views
Derivation of Dirac Hamiltonian
In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads
$$
L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi.
...
- 495
0
votes
2
answers
87
views
Energy and momentum operators using Hamilton's equations
The energy operator is:
$${\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
0
votes
0
answers
51
views
Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
- 193
0
votes
1
answer
32
views
Eigenstates of the Laplacian and boundary conditions
Consider the following setting. I have a box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$, for some $L> 0$. In physics, this is usually the case in statistical mechanics or some problems in quantum ...
- 1,123
0
votes
0
answers
43
views
Math in Hamiltonian of the hyperquantization of EM field
1. Background: I encounter this when looking into the hyperquantization of EM field.
We have the secondly quantized field as below:
$$\hat{E}^{(+)}(t)=\mathscr{E} e^{-iwt+i\vec{k}\cdot\vec{r}}\hat{a}=\...
