Tagged Questions
13
votes
3answers
182 views
Hilbert space of harmonic oscillator: Countable vs uncountable?
Hm, this just occurred to me while answering another question:
If I write the Hamiltonian for a harmonic oscillator as
$$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$
then wouldn't one set of ...
1
vote
3answers
109 views
Complex energy eigenstates of the harmonic oscillator
Given the Hamiltonian for the the harmonic oscillator (HO) as
$$
\hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,,
$$
the Schroedinger equation can be reduced to:
$$
\left[
...
1
vote
1answer
46 views
Energy eigenvalues of a Q.H.Oscillator with $[\hat{H},\hat{a}] = -\hbar \omega \hat{a}$ and $[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger$
I just finished deriving the commutators:
\begin{align}
[\hat{H}, \hat{a}] &= -\hbar \omega \hat{a}\\
[\hat{H}, \hat{a}^\dagger] &= \hbar \omega \hat{a}^\dagger\\
\end{align}
On the ...
1
vote
1answer
51 views
Eigenfunctions in a harmonic oscillator
This assignment is about the one dimensional harmonic oscillator (HO).
The hamiltonian is just as you know from the HO, same goes for the energies, but I get that the wavefunction of the particle, at ...
4
votes
2answers
94 views
Proof for commutator relation $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}$
I know how to derive below equations found on wikipedia and have done it myselt too:
\begin{align}
\hat{H} &= \hbar \omega \left(\hat{a}^\dagger\hat{a} + \frac{1}{2}\right)\\
\hat{H} &= ...
3
votes
1answer
137 views
Schrödinger equation for a harmonic oscillator
I have came across this equation for quantum harmonic oscillator
$$
W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi
$$
which is often remodelled by defining a new ...
0
votes
0answers
34 views
Quantum harmonic oscilator - book that does it all right [duplicate]
I am dealing with quantum harmonic oscillator.
In every single book or video i have checked out i can read how the mathematical technique for solving this Schrödinger equation:
$$
W\psi = - ...
2
votes
0answers
61 views
Relativistic genarization of Quantum Harmonic Oscillator
I am trying to find out relativistic description of a quantum harmonic oscillator.
For a classical relativistic oscillator mass is a function of co-ordinates(http://arxiv.org/abs/1209.2876).
...
3
votes
1answer
105 views
The issue on existence of inverse operations of $a$ and $a^{\dagger}$
I have asked a question at math.stackexchange that have a physical meaning.
My assumption: Suppose $a$ and $a^\dagger$ is Hermitian adjoint operators and $[a,a^\dagger]=1$. I want to prove that ...
1
vote
1answer
60 views
Why uncertainity is minimum for coherent states?
While reading for quantum damped harmonic oscillator, I came across coherent states, and I asked my prof about them and he said me it is the state at which $\Delta x\Delta y$ is minimum. I didn't ...
0
votes
0answers
58 views
Quantum harmonic oscillator. Finding operators
Problem:
I'm trying to verify that $p_H(T)$ and $x_H(T)$ satisfy the following equations, (by solving the Heisenberg equation):
$x_H(t)=x_H(0)cos(\omega t)+(1/m\omega)p_H(0)sin(\omega t)$
...
2
votes
1answer
124 views
Coordinate representation of quantum ladder operator?
I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.
I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where ...
3
votes
2answers
175 views
Constant magnetic field applied to a quantum harmonic oscillator
I have a spinless particle of mass $m$ and charge $q$ which is an isotropic harmonic oscillator of frequency $\omega_0$, then I apply a constant magnetic field in the $z$ direction. We can show the ...
1
vote
1answer
145 views
Shift operator (integral calculus involving Hermite polynomials) [closed]
I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
1
vote
3answers
479 views
Partition function for quantum harmonic oscillator
Hi guys I'm currently trying to solve a mock exam for an exam in a few days and am a bit confused by the solutions they gave us for this exercise:
Exercise:
A solid is composed of N atoms which ...
3
votes
2answers
165 views
Definition of “Quantizing”
Could anyone explain to me what "quantize" means in the following context?
Quantize the 1-D harmonic oscillator for which
$$H~=~{p^2\over 2m}+{1\over 2} m\omega^2 x^2.$$
I understand that the ...
1
vote
1answer
81 views
Noise spectrum of two systems and interacting Hamiltonian
I've been discovering recently the concept of noise spectrum, defined as:
$$S_{xx}[\omega] = \int dt<x(t)x(0)>\text{e}^{-i\omega t}$$
Roughly the Fourrier transform of the two-point function.
...
2
votes
0answers
97 views
Zero point fluctuation of an harmonic oscillator
In a paper, I ran into the following definition of the zero point fluctuation of our favorite toy, the harmonic oscillator:
$$x_{ZPF} = \sqrt{\frac{\hbar}{2m\Omega}} $$
where m is its mass and ...
2
votes
2answers
358 views
Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]
Possible Duplicate:
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
0
votes
2answers
288 views
Showing that the probability density of a linear harmonic oscillator is periodic
The complete question I am trying to answer is the following:
Show that the probability density of a linear harmonic oscillator in an arbitrary superposition state is periodic with period equal to ...
0
votes
2answers
185 views
Harmonic oscillator and Lorentz symmetry
There is a analog between harmonic oscillator $x=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)$ and quantum field $\phi=\int dp^3\frac{1}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}(a_p e^{ipx}+a^\dagger e^{-ipx})$, ...
5
votes
2answers
399 views
Expectation value of time-dependent Hamiltonian
I'm trying to solve a problem in QM with a forced quantum oscillator. In this problem I have a quantum oscillator, which is in the ground state initially. At $t=0$, the force $F(t)=F_0 \sin(\Omega t)$ ...
4
votes
1answer
626 views
Evolution operator for time-dependent Hamiltonian
When i studyed QM I'm only working with non time-dependent Hamiltonians. In this case unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation
$$
...
0
votes
1answer
185 views
Spectrum of quantum fluctuations in a harmonic oscillator
If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = ...
5
votes
1answer
206 views
Basis transformation between eigenstates of harmonic oscillators with different frequency
Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$.
Is there a compact expression for the ...
3
votes
1answer
145 views
How do eigenstates of harmonic oscillators with different frequencies compare?
Suppose I have a harmonic oscillator with frequency $\Omega_1$ and another one with frequency $\Omega_2$. Is there a simple relationship between the eigenstates of the two? Especially, how would the ...
0
votes
1answer
143 views
Spring-mass physics homework question [closed]
I've been having trouble with my physics homework. The problem is:
You may have measured the properties of a simple spring-mass system in the lab. Suppose you found ks = 0.9 N/m and m = 0.01 kg, ...
1
vote
1answer
180 views
Simulating quantum network of harmonic oscillators
Let's say that I have a system of $n$ particles $p_1,\ldots,p_n\in\mathbb{R}^3$ (where $n$ here is on the order of 10,000). Furthermore, suppose we have a graph $G=(V,E)$ describing some network, ...
3
votes
1answer
187 views
Question on Sakurai's treatment of the Harmonic Oscillator:
In Section 2.3 of the second edition of Modern Quantum Mechanics (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\left|n\right> = (n-1)a\left|n\right>,$$ and states ...
1
vote
2answers
126 views
Simple step in time evolution of position operator in simple harmonic motion
When considering the 'Heisenberg' picture of the harmonic oscillator, I've come across the step:
$$\begin{align}
\left\langle n\left|(\hat{q_H}\hat{H}-\hat{H}\hat{q_H})\right|k\right\rangle &= ...
8
votes
2answers
953 views
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = ...
7
votes
3answers
294 views
Is the quantization of the harmonic oscillator unique?
To put it a little better:
Is there more than one quantum system, which ends up in the classical harmonic oscillator in the classial limit?
I'm specifically, but not only, interested in an ...
2
votes
2answers
191 views
Why are overtones forbidden within the harmonic approximation?
In vibrational spectroscopy only transitions between neighboring vibrational states ($\Delta \nu = \pm 1$, $\nu$ being the vibrational quantum number) are allowed within the harmonic approximation. ...
0
votes
1answer
309 views
Vibrational motion of linear diatomic molecule
This question concerns the following exercise from an old exam:
The vibrational motion of a linear diatomic molecule can be approximated as simple harmonic motion.
A CO molecule has a bond ...
0
votes
3answers
327 views
How can I show that an arbitrary wavefunction in a 1D SHO is periodic in time?
I want to show that an arbitrary wavefunction $f$ in a one dimensional harmonic potential reproduces itself after a period T up to a phase factor: $f(x,t+T)=Af(x,t)$, $|A|=1$
I am not sure if this ...
2
votes
1answer
243 views
Expected number of quanta in harmonic oscillator states
I'm working my way through A Squeezed State Primer, filling in details along the way.
Let $a$ and $a^\dagger$ be the usual annihilation and creation operators with $[a,a^\dagger]=1$ and ...
2
votes
1answer
1k views
3D Quantum harmonic oscillator
For an isotropic 3D QHO in a potential $V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2)$. I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger equation ...
3
votes
0answers
1k views
Energy Levels of 3D Isotropic Harmonic Oscillator (Nuclear Shell Model)
One simple way of detailing the very basic structure of the nuclear shell model involves placing the nucleons in a 3D isotropic oscillator. It's easy to show that the energy eigenvalues are $E = ...
2
votes
2answers
746 views
Degeneracy of states in mixed infinite square well, harmonic oscillator
I'm trying to determine the degeneracy of states given by $g(\epsilon)=g_{0} \epsilon$ for a system that is trapped in a quite specific potential.
In two dimensions, the particle has a potential as ...
