All Questions
15 questions
3
votes
0
answers
169
views
Quantum Harmonic Oscillator: find a constant $\beta$ such that $U=\exp(\beta a^{\dagger} - \beta^*a)$ diagonalize $H$ [closed]
Given Hamiltonian of Quantum Harmonic Oscillator,
$$H = \frac{p^2}{2m}+\frac{1}{2}m\omega^2 x^2-\gamma x$$
I have to find a constant $\beta$ such that the unitary operator $U=\exp(\beta a^{\dagger} - \...
- 91
4
votes
1
answer
215
views
What is the probability to find the system in the ground state? [closed]
I previously posted a question related to this Hamiltonian, but the original concern was different:
We examine the following Hamiltonian:
\begin{equation}
H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2 -...
user372470
3
votes
2
answers
373
views
Does the state change, when the Hamiltonian changes?
Consider the Hamiltonian
\begin{equation}
H = \frac{p^{2}}{2m} + \frac{1}{2} m\omega^{2}x^{2} - \theta(t) qEx
\end{equation}
where $\theta(t)$ is $0$ for $t = 0$ and $1$ for $t > 0$. If at $t = ...
user372470
0
votes
1
answer
255
views
How to diagonalize a single particle hamiltonian? [closed]
$$H=\hbar\omega \left(a^\dagger a+\frac{1}{2}\right)+\hbar \omega_0\left(a^\dagger+a\right)$$ How to diagonalize $H$ and find its eigenenergies?
- 11
2
votes
0
answers
40
views
Propagator for radial force field?
The propagator $K(x,y;t)$ is well known for the (1D) harmonic oscillator:
$$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + \frac{m}{2}\omega^2 x^2$$
is there a simple closed form solution ...
user84158
1
vote
0
answers
110
views
Canonical transformation of the harmonic oscillator‘s Hamiltonian [closed]
I could deduce the Hamiltonian of the damped harmonic oscillator:
$$
H=\frac{p^2}{2m}e^{-2 \gamma t}+\frac{m \omega_0^2 q^2}{2}e^{2 \gamma t}
$$
Using the canonical transformation $Q=e^{\gamma t}q, P=...
- 522
3
votes
1
answer
324
views
Modified quantum harmonic oscillator spectrum and eigenstates
I am trying to find the eigenstates/eigenvalues of the following Hamiltonian
$$
\hat{H} = \hbar \omega \Big(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\Big)+A\big(\hat{a}^{\dagger}\hat{a}^{\dagger}+\hat{a}\...
- 745
1
vote
3
answers
96
views
Why is $\langle n| (\hat{a}+\hat{a}^{\dagger})^2|n\rangle=2n+1$ for the QM harmonic oscillator? [closed]
Consider a one-dimensional quantum-mechanical simple harmonic oscillator of mass $m$ and potential energy $\frac{kx^2}{2}$. The energy levels of this system are $E_n=(n+\frac{1}{2})\hbar\omega $ for $...
- 239
1
vote
1
answer
440
views
Statistical weight for $N$ harmonic oscillators in microcanonical ensemble
I would like to compute the statistical weight for the microcanonical ensemble for $N$ harmonic oscillators.
To do that i use the hamiltonian of the harmonic oszillator:
$$H(q,p)=\sum\limits_{i=1}^N \...
0
votes
1
answer
408
views
Calculating exact energy levels of perturbed Hamiltonian
I wish to find the exact energy levels of the following perturbed hamiltonian.
$$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$
I believe that it can be solved by using the ...
- 558
2
votes
1
answer
2k
views
Transforming simple Hamiltonian to interaction picture
I am trying to follow the math in a paper, and in it they do a lot of transforming into the interaction frame. It has been awhile since I have done these kind of calculations explicitly by hand and I ...
- 1,001
-1
votes
2
answers
1k
views
Hamiltonian approximation of the Coulomb interaction energy of two charged oscillators
I'm adding an excerpt from the book Introduction to Solid State Physics 7th edition by Charles Kittel.
I don't see how they arrived at the approximation of the Hamiltonian (2) by expanding it. If $...
- 443
-2
votes
1
answer
241
views
Canonical Quantization of harmonic oscillator
I have a system of two particles with the usual Lagrangian,
$$L=\frac12M_1{\dot{x_1}}^2+\frac12M_2{\dot{x_2}}^2-\frac12k({x_1}^2+{x_2}^2)$$
I want to find the quantum Hamiltonian of the system. I ...
0
votes
1
answer
421
views
Trick for reformulating in terms of centre of mass and relative variables
I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian
$$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$
I want to express the ...
- 895
1
vote
1
answer
300
views
Change of operator in the Hamiltonian [closed]
We are told that the particle has mass m and charge e and is moving in 2 dimensions.
The position operator $\mathbf{X}=(X_{1},X_{2})$ and momentum operator $\mathbf{P}=(P_{1},P_{2})$
We are given ...
- 99