Problem

Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• The wave functions of the harmonic oscillator, like those of a particle in a finite well, are nonzero in the classically forbidden regions, outside the classical turning points. In this question you will find the probability that a quantum particle which is in the ground state of an SHO will be found outside its classical turning points. The wave function for this state is in Table 7.1, and its normalization constant A0 is given in Problem 1. (a) What are the turning points for a classical particle with the ground-state energy in an SHO with ? Relate your answer to the constant b in Eq. (7.99). (b) For a quantum particle in the ground state, write down the integral that gives the total probability for finding the particle between the two classical turning points. The form of the required integral is given in Problem 2, Eq. (7.122). To evaluate it, change variables until you get an integral of the form this is a standard integral of mathematical physics (called the error function) with the known value 1.49. What is the probability of finding the particle between the classical turnin points? (c) What is the probability of finding it outside the classical turning points?

Problem 1

• The wave function ψ0(x) the ground state of a harmonic oscillator is given in Table 7.1. Show that its normalization constant A0 is

A0 = (πb2)−1/4(7.123)

You will need to know the integral , which can be found in Appendix B.

Problem 2

•• If a particle has wave function ψ(x), the probability of finding the particle between any two points b and c is

For a particle in the ground state of a rigid box, calculate the probability of finding it between x = 0 and x = a/3 (where a is the width of the box). Use the hint in Problem 3.

Problem 3

•• Show that the integral which appears in the normalization condition for a particle in a rigid box has the value