ia
2. ### Calculating degeneracy of the energy levels of a 2D harmonic oscillator

Too dim for this kind of combinatorics. Could anyone refer me to/ explain a general way of approaching these without having to think :D. Thanks.
3. ### I Time averages for a 2-dimensional harmonic oscillator

I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it... Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\dot x^2}{2}+\frac{\dot y^2}{2}$$ and potential energy is $$U=\frac{ x^2}{2}+\frac{y^2}{2}$$...
4. ### Phase space of a harmonic oscillator and a pendulum

Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
5. ### I Why is this SHM the way it is?

I know four different forms in which an SHM can be represented after solving the differential and taking the superposition acos(wt+Ø) asin(wt+Ø) acos(wt-Ø) asin(wt-Ø) where a- amplitude In the above image they took B as negative in order to arrive at acos(wt+e). If i already knew i wanted...

7. ### A Overlap of nth QHO excited state and momentum-shifted QHO ground state

##\newcommand{\ket}{|#1\rangle}## ##\newcommand{\bra}{\langle#1|}## I have a momentum-shifting operator ##e^{i\Delta p x/\hbar}## acting on the ground state ##\ket{0}## of the QHO, and I want to compute the overlap of this state with the n##^{th}## excited QHO state ##\ket{n}##. Given...
8. ### A rather weird form of a coherent state

As far as I know we can express the position and momentum operators in terms of ladder operators in the following way {\begin{aligned}{ {x}}&={\sqrt {{\frac {\hbar }{2}}{\frac {1}{m\omega }}}}(a^{\dagger }+a)\\{{p}}&=i{\sqrt {{\frac {\hbar }{2}}m\omega }}(a^{\dagger }-a)~.\end{aligned}}....
9. ### Effects of KE & PE of a Harmonic Oscillator under Re-scaling of coordinates

The wavefunction is Ψ(x,t) ----> Ψ(λx,t) What are the effects on <T> (av Kinetic energy) and V (potential energy) in terms of λ? From ## \frac {h^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x,t)\psi(x,t)=E\psi(x,t) ## if we replace x by ## \lambda x ## then it becomes ## \frac...
10. ### I Problem with the harmonic oscillator equation for small oscillations

Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations: 1) x''+y''+g/r*x=0 2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi) the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...
11. ### Green's Function for a harmonic oscillator

I know that due to causality g(t-t')=0 for t<t' and I also know that for t>t', we should get g(t-t')=\frac{sin(\omega_0(t-t'))}{\omega_0} But I can't seem to get that to work out. Using the Cauchy integral formula above, I take one pole at -w_0 and get \frac{ie^{i\omega_0(t-t')}}{2\omega_0} and...
12. ### A position of stable equilibrium, and the period of small oscillations

I tried by taking the derivative of the potential to find the critic points and the I took the second derivative to find which of those points are minimum points. I found that the point is ##x=- a##. I don't understand how to calculate the period, since I haven't seen anything about the harmonic...
13. ### Finding the parameters for Harmonic Oscillator solutions

Homework Statement Using the Schrödinger equation find the parameter \alpha of the Harmonic Oscillator solution \Psi(x)=A x e^{-\alpha x^2} Homework Equations -\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x) E=\hbar\omega(n+\frac{1}{2})...
14. ### MATLAB Code: Stationary Schrodinger EQ, E Spec, Eigenvalues

Hello everyone, For weeks I have been struggling with this quantum mechanics homework involving writing a code to determine the energy spectrum and eigenvalues for the stationary Schrodinger equation for the harmonic oscillator. I can't find any resources anywhere. If anyone could help me get...
15. ### Angular momentum operator for 2-D harmonic oscillator

1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
16. ### Quantum Spherical Pendulum

I have trouble with finding the eigenstates of a spherical pendulum (length $l$, mass $m$) under the small angle approximation. My intuition is that the final result should be some sort of combinations of a harmonic oscillator in $\theta$ and a free particle in $\phi$, but it's not obvious to...
17. ### A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x)...

Homework Statement [/B] A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be? (A)...
18. ### I Phase angle of a damped driven harmonic oscillation

Hello, in every book and on every website (e.g. here http://farside.ph.utexas.edu/teaching/315/Waves/node13.html) i found for driven harmonic osciallation the same solution for phase angle:θ=atan(ωb/(k−mω^2)) where ω is driven freq., m is mass, k is spring constant. I agree with it =it follows...
19. ### Harmonic Oscillator violating Heisenberg's Uncertainity

Homework Statement Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle? Homework Equations $$\sigma_x\sigma_p = \frac{\hbar}{2}$$ The Attempt at a Solution [/B] I worked out the solution for the second state of the harmonic oscillator...
20. ### I Entropy Contradiction for a Single Harmonic Oscillator

Making use of the partition function, it is straight forward to show that the entropy of a single quantum harmonic oscillator is: $$\sigma_{1} = \frac{\hbar\omega/\tau}{\exp(\hbar\omega/\tau) - 1} - \log[1 - \exp(-\hbar\omega/\tau)]$$ However, if we look at the partition function for a single...
21. ### I Distribution of Position in classical & quantum case

I am confused about the difference between the two In Griffith's 2.3 The Harmonic Oscillator, he superimposes the quantum distribution and classical distribution and says What I understand for quantum case is that ##|\Psi_{100} (x)|^2## gives the probability we will measure the particle...