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uantum - on the amplitude and(7.30 Now the probability to...

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uantum-Classical Correspondence 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, (7.29) If we substitute this in the energy expression, , we find that the energy depends
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Unformatted text preview: on the amplitude and , (7.30) Now the probability to find the particle at position , where is proportional to the time spent in an area around . The time spent in its turn is inversely proportional to the velocity (7.31) Solving in terms of we find (7.32) Doing the integration of over from to we find that the normalised probability is (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, (7.34) which tells us what to use for each , (7.35) So let us look at an example for . Suppose we choose and such that . We then get the results shown in Fig. 7.2 , where we see the correspondence between the two functions....
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