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12 Quantum theory: techniques and
applications
Solutions to exercises
Discussion questions
E12.1(b) The correspondence principle states that in the limit of very large quantum numbers quantum
mechanics merges with classical mechanics. An example is a molecule of a gas in a box. At room
temperature, the particle-in-a-box quantum numbers corresponding to the average energy of the gas
molecules ( 1 kT per degree of freedom) are extremely large; consequently the separation between the
2
2
levels is relatively so small (n is always small compared to n , compare eqn 12.10 to eqn 12.4) that
the energy of the particle is effectively continuous, just as in classical mechanics. We may also look at
these equations from the point of view of the mass of the particle. As the mass of the particle increases
to macroscopic values, the separation between the energy levels approaches zero. The quantization
disappears as we know it must. Tennis balls do not show quantum mechanical effects. (Except those
served by Pete Sampras.) We can also see the correspondence principle operating when we examine
the wavefunctions for large values of the quantum numbers. The probability density becomes uniform
over the path of motion, which is again the classical result. This aspect is discussed in more detail in
Section 12.1(c).
The harmonic oscillator provides another example of the correspondence principle. The same
effects mentioned above are observed. We see from Fig. 12.22 of the text that probability distribution
for large values on n approaches the classic
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