Page 73 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

64 Schro Èdinger wave mechanics

Problems
2.1 Consider a particle in a one-dimensional box of length a and in quantum state n.
What is the probability that the particle is in the left quarter of the box
(0 < x < a=4)? For which state n is the probability a maximum? What is the
probability that the particle is in the left half of the box (0 < x < a=2)?
2.2 Consider a particle of mass m in a one-dimensional potential such that
V(x) ˆ 0, ÿa=2 < x < a=2
ˆ1, x , ÿa=2, x . a=2
Solve the time-independent Schrodinger equation for this particle to obtain the
È
energy levels and the normalized wave functions. (Note that the boundary
conditions are different from those in Section 2.5.)
2.3 Consider a particle of mass m con®ned to move on a circle of radius a. Express
the Hamiltonian operator in plane polar coordinates and then determine the energy
levels and wave functions.
2.4 Consider a particle of mass m and energy E approaching from the left a potential
barrier of height V 0 , as shown in Figure 2.3 and discussed in Section 2.6. However,
suppose now that E is greater than V 0 (E . V 0 ). Obtain expressions for the
re¯ection and transmission coef®cients for this case. Show that T equals unity
2
2 2 2
when E ÿ V 0 ˆ n ð " =2ma for n ˆ 1, 2, .. . Show that between these periodic
maxima T has minima which lie progressively closer to unity as E increases.
2.5 Find the expression for the transmission coef®cient T for Problem 2.4 when the
energy E of the particle is equal to the potential barrier height V 0 .