All Questions
Tagged with coherent-states homework-and-exercises
24
questions
2
votes
3
answers
245
views
Finding the wavefunction of coherent state in 2D oscillator
Suppose I have a two-dimensional harmonic oscillator, $H= \hbar\omega(a_x^{\dagger}a_x+a_y^{\dagger}a_y)$. We define the operator $b=\frac{1}{\sqrt{2}}(a_x+ia_y)$.
If eigenkets of the hamiltonian are $...
- 81
0
votes
0
answers
141
views
Evolution of Quantum Harmonic Oscillator into coherent state
Why does a quantum harmonic oscillator that is driven by an electromagnetic wave in cosine form with its frequency equal to the resonance frequency of the oscillator evolve from its groundstate into a ...
- 95
1
vote
0
answers
86
views
Trouble proving Wigner function identity [closed]
I am trying to prove $$\int d^2 \alpha W(\alpha)=1$$ where $W(\alpha)$ represents the Wigner funcion. However, I have trouble solving it. I tried solving it as follows but I think I have done some ...
- 157
1
vote
1
answer
106
views
Equation of motion for space and momentum of a $\textbf{coherent state}$ [closed]
Given the coherent states
$$| \alpha \rangle\, e^{\textstyle -|\alpha|^2/2}\,\sum_{n = 0}^\infty \dfrac{\alpha^n}{\sqrt{n!}}\,|n\rangle$$
that satisfy the eigenvalue-equation: $A|\alpha\rangle=\alpha\,...
- 462
0
votes
1
answer
86
views
How Do I Do This Integral? [closed]
I am trying to derive a boson coherent path integral and one part of the derivation is to evaluate/prove
$$
\int d\Psi(\tau) d\Psi^*(\tau) |\Psi(\tau)|^{2n} \exp(-|\Psi(\tau)|^2) = (n!) \pi.
$$
This ...
user261609
0
votes
1
answer
165
views
Angular momentum coherent states
$\renewcommand\bm[1]{\mathbf{#1}}$
$\renewcommand\h{\hbar}$
$\renewcommand\ket[1]{|#1\rangle}$
$\renewcommand\mean[1]{\langle #1 \rangle}$
$\renewcommand\norm[1]{||#1||}$
Let $\bm{J}$ be an angular ...
- 127
0
votes
1
answer
97
views
Coherent states - scalar product [closed]
$\newcommand\norm[1]{\lVert#1\rVert}$
$\newcommand\ket[1]{|#1\rangle}$
$\newcommand\mean[1]{\langle #1\rangle}$
$\newcommand\braket[2]{\langle #1|#2\rangle}$
$\newcommand\ketbra[2]{|#1\rangle\langle #...
- 127
2
votes
1
answer
998
views
Time evolution of a coherent state
$\newcommand\norm[1]{\lVert#1\rVert}$
$\newcommand\ket[1]{|#1\rangle}$
I consider an Hamiltonian of the Harmonic Oscillator $\hat{H} = \frac{P^2}{2m}+\frac{1}{2}m\omega^2 X^2$.
I proved already if the ...
- 127
2
votes
3
answers
949
views
Expectation value on coherent states of $(\hat{a}+\hat{a}^\dagger)^n$
I need to evaluate the following expectation value
$$
\langle \alpha \vert (\hat{a}+\hat{a}^\dagger)^n \vert \alpha\rangle
$$
The formulation is very easy, but I can not tackle the problem. Any hint?
- 565
0
votes
1
answer
52
views
Computing $\langle (\Delta L_z)^2\rangle$ for coherent states of $SU(2)$
This is a follow-up to a previous question. I am trying to compute $\langle (\Delta L_z)^2\rangle$, for a general coherent state in the coherent state system for $SU(2)$, where I get all the coherent ...
user282846
1
vote
1
answer
56
views
Computing dispersion of $L_3$ in spin coherent states
I am trying to compute $\langle \left(\Delta L_3\right)^2\rangle$ for coherent states of $SU(2)$. I understand that a set of coherent states can be be formed from rotations of the the state $|j,j\...
user282846
3
votes
2
answers
170
views
What does this kind of notation mean?
Trying to understand quantum information. Need some help :(
What does this notation $$ \langle\alpha|\hat{n}|\alpha\rangle $$ mean? Here $$|\alpha\rangle$$ is a coherent state and $$\hat{n}$$ is ...
- 55
0
votes
2
answers
1k
views
Proof that coherent states are eigenstates of annihilation operator [closed]
My goal is to prove that, for $|\lambda\rangle=N\exp(\lambda\hat{a}^\dagger)|0\rangle$ is an eigenvector of the operator $\hat a$.
I took 2 approaches, but both make sense to me and I get different ...
- 637
2
votes
1
answer
609
views
Coherent states of the form $|{-\alpha}\rangle$
I've a brief question about coherent states in quantum mechanics.
As everyone knows, a coherent state is just the proper state of the anhilitation operator $a$, thus they're defined with the ...
- 1,172
-1
votes
1
answer
780
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Exponential of ladder operators acting on vacuum state [closed]
How would I solve expressions of the following nature:
$$<0|e^{Vt(a+a^\dagger)}|0>$$ and $$<0|e^{\omega aa^\dagger t}|0>~?$$
My intuition is that I have to expand the exponent as a ...
- 63