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Physics LibreTexts

10.4: Simple Example

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The best way to clarify this abstract discussion is to consider the quantum mechanics of the Harmonic oscillator of mass m and frequency ω,

ˆH=22md2dx2+12mω2x2

If we assume that the wave function at time t=0 is a linear superposition of the first two eigenfunctions,

ψ(x,t=0)=12ϕ0(x)12ϕ1(x)ϕ0(x)=(mωπ)1/4exp(mω2x2)ϕ1(x)=(mωπ)1/4exp(mω2x2)mωx

(The functions ϕ0 and ϕ1 are the normalised first and second states of the harmonic oscillator, with energies E0=12ω and E1=32ω.) Thus we now kow the wave function for all time:

ψ(x,t)=12ϕ0(x)e12iωt12ϕ1(x)e32iωt.

In figure 10.4.1 we plot this quantity for a few times.

The best way to visualize what is happening is to look at the probability density,

P(x,t)=ψ(x,t)ψ(x,t)=12(ϕ0(x)2+ϕ1(x)22ϕ0(x)ϕ1(x)cosωt)

This clearly oscillates with frequency ω.

Question: Show that P(x,t)dx=1.

clipboard_e429ef8f0ed28ca0d787504a71cee45cf.png
Figure 10.4.1: The wave function (10.10) for a few values of the time t. The solid line is the real part, and the dashed line the imaginary part.

Another way to look at that is to calculate the expectation value of ˆx :

ˆx=P(x,t)dx=12ϕ0(x)2xdx=0+12ϕ1(x)2xdx=0cosωtϕ0(x)ϕ1(x)xdx=cosωtmω1πy2ey2=12mωcosωt.


This once again exhibits oscillatory behaviour!


This page titled 10.4: Simple Example is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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