Harmonic oscillator Definition and 699 Threads

I worked these problems out: [x, H] = xH - Hx = 0 [p, H] = pH - Hp = non-zero H is the harmonic oscillator Hamilitonian, x and p are the position and momentum operators, respectively. My question is, why doesn't p commute with H, but x does?

Ok - here goes: I see the energy levels of the harmonic oscillator as E = hw(n+1/2) = hwn + hw/2 (please ignore the lack of cool symbols) Now the hw/2 is something called the ground state - fine - no problem. Should I interpret homework as the fundamental harmonic? 2hw as the...

Hello... We have 3 fermions (s=1/2) at the ground state of a harmonic oscillator moving over the x-axis with a the classic hamiltonian for a three particle oscillator : H =(1/2m)*(P1)^2 +((1/2)*m(w^2)((x1)^2)) +(1/2m)*(P2)^2 +((1/2)*m(w^2)((x2)^2)) +(1/2m)*(P3)^2 +((1/2)*m(w^2)((x3)^2)) we...

Suppose I have 4 bosons in a one-dimensional harmonic oscillator potential and that the total energy is E_\text{tot} = 8 \hbar \omega. Recall, E_n = (n+1/2)\hbar\omega. Question: How many quantum states exist? (assume no spin degeneracy) After accounting for the ground state, we have 6...

energy values of a "half" harmonic oscillator This is the full question: Find the solution to the "half" harmonic oscillator: http://img241.imageshack.us/img241/9181/02bt7.jpg Compare the energy values and wave functions with those of the full harmonic oscillator. Why are some of the full...

Hello all, if I have an ion trapped in a harmonic oscillator potential with a resonant frequency 0f 11 MHz and the ion cooled to a temperature of T=0.48mK, how do I find the probability that the oscillator is in its ground state? I know that the ground state energy is 1/2 \hbar \omega, but...

Hey! Can someone explain to why the energy of the harmonic oscillator must be at least: \frac{(\Delta p)^2}{2m}+\frac{1}{2}m \omega^2 (\Delta x)^2 I mean, \Delta x and \Delta p represents the uncertainty in the position and momentum, and therefore it does not really have anything to do...

okay this is kind of a quick one when youre solving a forced harmonic oscillator for an initial condition by the method of undetermined coefficients, you do the part for the homogenoues equation first and come up with somethign like ke^at + ce^bt. Now when you goto solve for those constants k...

Can someone check my work please I'm pretty sure I don't have the right answer but I can't figure out what I have wrong. The question is: A simple harmonic oscillator has total energy E=1/2kA^2 where A is the amplitude of oscillation. For what value of the displacement does the kinetic...

Hi, I've been scouring through many textbooks to try find some kind of solution to a question I have been asked for a problem sheet and was wondering if any1 would be able to help. The question is as follows; The simple harmonic oscillator with hamiltonian H = (p^2/2m) + (1/2(mw^2x^2) is...

A simple harmonic oscillator has a total energy of E. (a) Determine the kinetic and potential energies when the displacement is three-fourths the amplitude. (Give your answer in terms of total energy E of the oscillator.) Kinetic energy ______________ x E <----(times E) Potential energy...

I have the U(x) functions for the ground state and first excited state of the simple harmonic oscillator. I also have the psi (x,0) wave function for this situation. How do I find the probability the particle is in a particular state? Is it simply the integral of psi(x,0) * u(x) dx evaluated...

I have a wave function which is the ground state of a harmonic oscillator (potential centered at x=0)... but shifted by a constant along the position axis (ie. (x-b) instead of x in the exponential). How does this decompose into eigenfunctions?? I know it's an infinite sum... but I can't...

I'm stuck on this problem: The initial conditions for a two-dimensional isotropic oscillator are as follows: t=0, x=A, y=4A, v=0i +3wAj (vector) where w is the angular frequency. Find x and y as functions of t. Where do I even begin with this problem. I take it A = constant. Can anyone...

I need your help to solve this problem on coupled harmonic oscillators. Two masses m1 and m2 are attached to two rigid supports by means of springs of force constants k1 and k2 respectively. The masses are connected to a third spring of force constant k3. The masses are free to move along the...

why is the energy levels of the harmonic oscillator En=(N+1/2)hf?

Please I don't understand this problem at all: Consider a driven damped harmonic oscillator.Calculate the power dissipated by the damping force? calculate the average power loss, using the fact that the average of (sin(wt+phi) )^2 over a cycle is one half? Please can I have some help for...

why is the lowest allowed energy not E=0 but some definite minimum E=E0?

why for that equation does alpha= 2E/(hf)?

I've been given a question which asks to calculate the probability of finding an electron is an excited state for a Harmonic oscillator perturbed by an electric field pulse E(t) as t tends to infinity. E= -exA\exp{(\frac{-t}{\tau})} I knew I had to use the Time dependent perturbation...

Ok, fairly basic quantum mechanics assignment. One question deals with (I think) the coefficients of the Hermite polynomial. Unfortunately, the lecturer hasn't told us anything about this method, so I donn't know what it's called or what the point of it is, and it's not in any of the examples...

Driven, damped harmonic oscillator -- need help with particular solution Consider a damped oscillator with Beta = w/4 driven by F=A1cos(wt)+A2cos(3wt). Find x(t). I know that x(t) is the solution to the system with the above drive force. I know that if an external driving force applied...

Hi. I'm given a problem with a harmonic oscillator where the potential is V= (kx^2)/2 with a mass m (KE = 1/2 mv^2). I have to use the Heisenberg Uncertainty principle to show what the minimum energy is, but I'm not sure where to start... I think I have to combine KE + V and minimize that, but...

Please help 1) The mechanism which powers the clock (using a harmonic oscillator to keep time with friction) Since the clock has friction the oscillation amplitude decreases in time. If the oscillation amp is unity at t=0 what will the amp be after 10 sec? I believe we need to use the...

Hi, I am wondering how i would go about calculating the canonical partition function for a system of N quantum harmonic oscillators. The idea of the question is that we are treating photons as oscillators with a discrete energy spectrum. I'm confused as whether to use Maxwell-Boltmann...

why is the expectation value of the position of a harmonic oscillator in its ground state zero? and what does it mean that it is in ground state? is ground state equal to n=0 or n=1?

This might be another problem that our class hasn't covered material to answer yet - but I want to be sure. The question is the following: Find the eigenfunctions and eigenvalues of a two-dimensional isotropic harmonic oscillator. Again, I need help simply starting.

Question: (a) Show that the total mechanical energy of a lightly damped harmonic oscillator is E = E_0 e^{-bt/m} where E_0 is the total mechanical energy at t = 0. (b) Show that the fractional energy lost per period is \frac{\Delta E}{E} = \frac{2 \pi b}{m \omega_0} = \frac{2...

Hi, I have to find the expectation values of xp and px for nth energy eigenstate in the 1-d harmonic oscillator. If I know <xp> I can immediately find <px>since [x,p]=ih. I use the ladder operators a_{\pm}=\tfrac1{\sqrt{2\hslash m\omega}}(\mp ip+m\omega x) to find <xp>, but I get a complex...

The equation of motion of an undamped harmonic oscillator with driving force F=F_ocos(\omega*t) is x(t) = Acos(\omega_0*t) + Bsin(\omega_0*t) + \frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2} I am to determine the initial conditions such that the undamped oscillator begins steady...

For some reason this problem has me stuck. It isn't homework, but it might be on the exam Tommorrow. If anyone is still awake, please steer me in the right direction. Thank you A simple harmonic oscillator has a total energy of E. (a) determine the kinetic and potential energies when...

Hi friends I would be grateful if someone could point me to a mathematical treatise (on the internet) about the two body simple harmonic oscillator (classical mechanics only, but no Lagrangian/Hamiltonian...just energy, momentum, Newton's Laws). I am googling right now but all I find is...

Question: A damped harmonic oscillator loses 5.0 percent of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency \omega_0 = \sqrt{k/m}? (b) After how may periods will the amplitude have decreased to 1/e of its original value? So, for...

Why does the half of a harmonic oscillator potential allow only odd values of n? Thx

Hi, I'm having a lot of trouble with a damped harmonic oscillator problem: A damped harmonic oscillator consists of a block (m=2.00kg), a spring (k=10 N/m), and a damping force (F=-bv). Initially it oscillates with an amplitude of 25.0cm. Because of the damping force, the amplitude falls...

a damped simple harmonic oscillator has mass m = 260 g, k = 95 N/m, and b = 75 g/s. Assume all other components have negligible mass. What is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles (Adamped / Ainitial)? having trouble getting...

hi, I was going through my homework and i came to a problem that i can't seem to get. Consider the mass attached to four identical spring. Each spring has the force constant k and unstreched length L_0, and the length of each spring when the mass is at the origin is a(not necessarily the same...

Ok so i have been instructed to normalize N*x*exp(-ax^2), so i squared the function and trying to take the integral. I am a) assuming that the integration should run from the negative value of the amplitude or -A to the positive value of the Amplitude of A, i have a formula for A. My assumption...

Dealing with the one-dimensional harmonic oscillator I'm trying to find a general formula for \int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi there H_n(\xi) and H_k(\xi) are hermite polynomials and p is an integer ( p\geq 0). I can found the answer for p=0 and p=1 but I can't find...

Hi, From knowing that the 3D harmonic oscillator has 3 degrees of freedom, how do you conclude that the average total energy of the oscillator has energy 3kT? Thanks, Ying

For a simple damped oscillator... \text {Apparently if } \beta \ll \omega_0 } \text { then ...} \omega_d \approx \omega_0[1-\frac {1}{2}(\beta/\omega_0)^2]} Given that: \beta=R_m/2m \text { (where } R_m= \text {mechanical resistance) } \text { and } \omega _d=\sqrt{(\omega...

Question: A damped harmonic oscillar loses 5.0 percent of its mechanical energy per cycle. (a) By what percentage does its frequency differ from the natural frequency \omega_0 = \sqrt{k/m}? (b) After how many periods will the amplitude have decreased to 1/e of its original value? (a) Let E(t)...

In my mechanics textbook is given an exemple of how to find the general solution of the of the equation of motion for a force -kx (the simple harmonic oscillator problem). He begin his analysis and finds that e^(iwt) and e^(-iwt) are both solutions. Hence C1*e^(iwt) and C2*e^(-iwt) are also...

If the universe is a simple harmonic oscillator then it must be symmetrical and divided into two halves , each half with approximately 10^52 kg of mass at its centre of mass. If the universe reaches about 10^26 metres in about 10^18.5 seconds then using E = ( n + 1/2)h w for an oscillator we...

If the universe oscillates between a Big Bang and a Big crunch, can two particles at opposite ends of it, be considered to be undergoing simple harmonic oscillation? If the potential energy of the oscillator is given by G m1 m2 /r and m1 is the mass of the universe,10^52 kg,r = 10^26 metres...

This is (another!) question I cannot solve The ground state wavefunction for the harmonic oscillator can be written as $\chi _0 = \left( {\frac{\alpha } {\pi }} \right)^{\frac{1} {4}} \exp \left( {\frac{{ - \alpha x^2 }} {2}} \right)$ where $\alpha = \sqrt {\frac{{mk}} {{\hbar ^2...

The question is: Solve for the motion of the undamped harmonic oscillator with an applied force F, treated in class, when the force is no longer constant but has the form F=F0+kT, where Fo and k are constants. Use the intial conditions x(0)=d and x'(0) =v0. I'm trying to solve this problem...

Here's the problem: A one dimensional harmonic oscillator has mass m and frequency w. A time dependent state psi(t) is given at t=0 by: psi(0)=1/sqrt(2s)*sum(n=N-s,n=N+s) In> where In> are the number eigenstates and N>>s>>1. Calculate <x>. Show it varies sinusoidally; find the...

can anyboy show to me why the formula for the harmonic oscillator is f"(x)+W^2 * X(t)=0. Please I spent a whole afternoon trying to figure it out and I just wasted my time. Thanks