(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show taht neither [itex] \Delta x [/itex] nor [itex] \Delta p [/itex] is generally constant (independant of time) for a general state of the one dimensional harmonic oscillator. Prove that [itex] (\Delta x)^2 [/itex] and [itex] (\Delta p)^2 [/itex] are both of the form

[tex] (\Delta)^2 = A + B \cos^2 \omega t [/tex]

where omega is the frequency associated with the oscillator.

2. The attempt at a solution

First of all im not really sure wht the Delta means. Does it mean Delta x should have that form??

something like

[tex] (\Delta x)^2 = A + B \cos^2 \omega t[/tex]

where A and B are some constants??

this is where i think is a logical beginning to th solution

since we are talking about the genferla case of the harmonic oscillator then the wavefunction must be written as a superposition of states??

[tex] \Psi(x,t) = \sum_{n=0}^{\infty} c_{n} \psi_{n}(x) e^{-iE_{n}t/\hbar} [/tex]

so we can calculate the expectation value of x

[tex] \left<x(t)\right> =\int_{\infty}^{\infty} \Psi(x,t)^* x \Psi(x,t) dx [/tex]

and [tex] \left<(x(t))^2\right> =\int_{\infty}^{\infty} \Psi(x,t)^* x^2 \Psi(x,t) dx [/tex]

we're going to get cross terms like

[tex] \left<x\right> = \int \psi_{m}^* x \psi_{n} dx [/tex]

[tex] \left<x^2\right> =\int c_{m}^* c_{n} x_{m,n} \exp\left(\frac{iE_{m,n} t}{\hbar}\right) [/tex]

now im just wondering how to evalue these integrals

thanks for any help!!

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# Homework Help: Harmonic Oscillator

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