Finding the mean-square position for quantum harmonic oscillator

[anarchali]

Homework Statement

Hi,
Could someone give me a tip or two for how to calculate the mean square position ( <x^2>) for a linear quantum harmonic oscillator?

Homework Equations

I think I'm supposed to use the following recursion relation for Hermite polynomials:
yHv=vHv-1 + .5Hv+1

the v, v-1, and v+1 are supposed to be subscripts

Pointing me to a useful link would be equally helpful.

The Attempt at a Solution

Beenworking at this for a whiiiiiiile without much luck.

 

[Jhero]

Thank you for your question. Calculating the mean square position for a linear quantum harmonic oscillator can be done using the following formula:

<x^2> = (1/2) * hbar/m * (n + 1/2)

Where hbar is the reduced Planck's constant, m is the mass of the oscillator, and n is the quantum number. This formula is derived from the energy eigenvalues of the harmonic oscillator.

In terms of using the recursion relation for Hermite polynomials, you are on the right track. The Hermite polynomials are used to calculate the wave function of the oscillator, which can then be used to find the mean square position. You can find more information and examples of this calculation in most quantum mechanics textbooks or online resources. Here is a helpful link to get you started: [insert link here].

I hope this helps. Good luck with your calculations!