Chap12 soln

Physical Chemistry

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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 2 kT per degree of freedom) are extremely large; consequently the separation between the levels is relatively so small ( n is always small compared to n 2 , 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 classical picture of the motion. (Look at the graph for n = 20.) E12.2(b) The physical origin of tunnelling is related to the probability density of the particle which according to theBorninterpretationisthesquareofthewavefunctionthatrepresentstheparticle. Thisinterpretation requires that the wavefunction of the system be everywhere continuous, event at barriers. Therefore, if the wavefunction is non-zero on one side of a barrier it must be non-zero on the other side of the barrier and this implies that the particle has tunnelled through the barrier. The transmission probability depends upon the mass of the particle (specifically m 1 / 2 , through eqns 12.24 and 12.28): the greater the mass the smaller the probability of tunnelling. Electrons and protons have small masses, molecular groups large masses; therefore, tunnelling effects are more observable in process involving electrons and protons. E12.3(b) The essential features of the derivation are: (1) The separation of the hamiltonian into large (unperturbed) and small (perturbed) parts which are independent of each other. (2) The expansion of the wavefunctions and energies as a power series in an unspecified parameters, λ , which in the end effectively cancels or is set equal to 1.
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