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One of the interesting questions raised by the fact that we can solve
both the quantum and the classical problem exactly for
the harmonic oscillator, is
``Can we compare the Classical and Quantum Solutions?''
Figure 7.2:
The correspondence between quantum and classical probabilities
|
|
In order to do that we have to construct a probability for the
classical solution. The variable over which we must average to get
such a distribution must be time, the only one that remains in the solution.
For simplicity look at a cosine solution, a sum of sine and cosines
behaves exactly the same (check!). We thus have, classically,
x = Acos( t), v = - A sin( t). |
(7.29) |
If we substitute this in the energy expression,
E =
mv2 +
m
x2, we find that the energy depends
on the amplitude A and
,
E(A) = mA2 sin2( t) + m A2cos2( t) = m A2 |
(7.30) |
Now the probability to find the particle at position x, where - A < x < A
is proportional to the time spent in an area dx around x. The time
spent in its turn is inversely proportional to the velocity v
Solving v in terms of x we find
| v(x)| =   |
(7.32) |
Doing the integration of 1/v(x) over x from - A to A we find
that the normalised probability is
class(x)dx = dx. |
(7.33) |
We now would like to compare this to the quantum solution. In order
to do that we should consider the probabilities at the same energy,
which tells us what A to use for each n,
A(n) = . |
(7.35) |
So let us look at an example for n = 10. Suppose we choose
m and
such that
= 10-10 m.
We then get the results shown in Fig. 7.2,
where we see the correspondence between the two functions.
Next: 8. The formalism underlying
Up: 7. The Harmonic oscillator
Previous: 7.4 A few solutions