Webpage title: How to Calculate <x> for a One-Dimensional Harmonic Oscillator

In summary, the problem involves a one-dimensional harmonic oscillator with a time-dependent state given at t=0. The calculation of <x> shows it varies sinusoidally with frequency w and amplitude 1/2s. This is compared to the values of a classical harmonic oscillator, where the amplitude would be 1 and the frequency would be w. The approximation N>>s>>1 is used to simplify the calculation, resulting in a sinusoidal function with a coefficient of 2sqrt(N).
  • #1
spdf13
10
0
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 frequency and amplitude. Compare the amlitude and frequency to the corresponding values of a classical harmonic oscillator.


Here's how I proceeded:

<x>=(1/2s) (some constants)*sum(n=N-s,n=N+s)*sum(m=N-s,m=N+s) <n I (a+a') I m> Exp[i(Em-En)t/h]

(note a' is "a dagger")

=(1/2s) (some constants)*sum(n=N-s,n=N+s)*sum(m=N-s,m=N+s) {sqrt(m) <n I m-1> + sqrt(m+1) <n I m+1>} Exp[i(En-Em)t/h]

(note <n I m-1>=delta(n,m-1) and <n I m+1>=delta(n,m+1).

=(1/2s) (some constants)*sum(n=N-s,n=N+s) {sqrt(m+1) Exp[-iwt] + sqrt(m) Exp[iwt]}

This is where I get stuck. I don't know if I'm supposed to make some approximation since N>>s>>1, and approximate the term in the {} as sqrt(m) cos (wt), or if I'm just completely wrong from the start. If someone can help, I'd really appreciate it.
 
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  • #2
Originally posted by spdf13
This is where I get stuck. I don't know if I'm supposed to make some approximation since N>>s>>1, and approximate the term in the {} as sqrt(m) cos (wt), or if I'm just completely wrong from the start. If someone can help, I'd really appreciate it.

If N>>s, then all the numbers in the range of the index of summation [N-s,N+s] are approximately equal to N. That is, it is (approximately) as though you only have a single term.

That single term is going to be of the form:

sqrt(N+1)exp(-i&omega;t)+sqrt(N)exp(i&omega;t)

The thing that is screwing this up from being a sinusoid is the fact that the two terms have different coefficients. Now is the time to invoke N>>1. Do that to approximate as follows:

sqrt(N)[exp(-i&omega;t)+exp(i&omega;t)]=2sqrt(N)cos(&omega;t)
 
  • #3
I think you're right. Thanks for the help.
 

FAQ: Webpage title: How to Calculate <x> for a One-Dimensional Harmonic Oscillator

What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a mathematical model used in quantum mechanics to describe the behavior of a particle or system that experiences a restoring force proportional to its displacement from equilibrium. It is often used to study the behavior of atoms, molecules, and other microscopic systems.

What is the difference between a classical and a quantum harmonic oscillator?

In a classical harmonic oscillator, the energy levels are continuous and the particle can have any energy. In a quantum harmonic oscillator, the energy levels are discrete and the particle can only have certain energies. Additionally, in a classical harmonic oscillator, the particle can have any position and momentum, while in a quantum harmonic oscillator, the position and momentum are subject to the uncertainty principle.

What is the significance of the zero-point energy in a quantum harmonic oscillator?

The zero-point energy is the minimum amount of energy that a quantum harmonic oscillator can have. This is due to the Heisenberg uncertainty principle, which states that the position and momentum of a particle cannot be known simultaneously with certainty. Therefore, even at its lowest energy state, the quantum harmonic oscillator still has some energy due to the uncertainty in its position and momentum.

How is the frequency of a quantum harmonic oscillator related to its energy levels?

The frequency of a quantum harmonic oscillator is directly proportional to the energy levels. This means that as the energy level increases, the frequency of the oscillator also increases. This relationship is described by the Planck-Einstein relation, which states that the energy of a quantum harmonic oscillator is equal to h times its frequency, where h is the Planck constant.

How does the potential energy function of a quantum harmonic oscillator differ from that of a classical harmonic oscillator?

In a classical harmonic oscillator, the potential energy function is a parabola, while in a quantum harmonic oscillator, the potential energy function is a series of discrete energy levels. This is due to the quantization of energy in a quantum system. Additionally, the potential energy function for a quantum harmonic oscillator will never reach zero, even at its lowest energy state, due to the presence of the zero-point energy.

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