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

exp
- x2 , |
|
(x) |
= |

exp
- x2 x. |
(10.9) |
(The functions
and
are the normalised first and
second states of the harmonic oscillator, with energies
E0 = 

and
E1 = 

.)
Thus we now kow the wave function for all time:
(x, t) |
= |
 (x)e- i t -  (x)e- i t. |
(10.10) |
In figure 10.1 we plot this quantity for a few times.
Figure 10.1:
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.
|
|
The best way to visualize what is happening is to look at the
probability density,
(x, t) |
= |
(x, t)* (x, t) |
|
| |
= |
  (x)2 + (x)2 - 2 (x) (x)cos t . |
(10.11) |
This clearly oscillates with frequency
.
Question: Show that

(x, t)dx = 1.
Another way to look at that is to calculate the expectation value of
:
   |
= |
 (x, t)dx |
|
| |
= |

+ 
- cos t (x) (x)xdx |
|
| |
= |
-cos t  y2e-y2 |
|
| |
= |
-  cos t. |
(10.12) |
This once again exhibits oscillatory behaviour!
Next: 10.5 Wave packets (states
Up: 10. Time dependent wave
Previous: 10.3 Completeness and time-dependence