Simple Nature - Light and Matter

45 Suppose that an electron, in one dimension, is confined to a

certain region of space so that its wavefunction is given by

Ψ =







0 ifx < 0

Asin(2πx/L) if 0≤x≤L

0 ifx > L

Determine the constantAfrom normalization.

√

46 In the following,xandyare variables, whileuandvare

constants. Compute (a)∂(uxln(vy))/∂x, (b)∂(uxln(vy))/∂y√.

47 (a) A radio transmitter radiates powerPin all directions, so

that the energy spreads out spherically. Find the energy density at

a distancer.

√

(b) Let the wavelength beλ. As described in example 8 on p. 876,

find the number of photons in a volumeλ^3 at this distancer.

√

(c) For a 1000 kHz AM radio transmitting station, assuming rea-

sonable values ofP andr, verify, as claimed in the example, that

the result from part b is very large.

48 The wavefunction Ψ of an electron is a complex number.

Make up an example of a value for the wavefunction that is not a

real number, and consider the following expressions: Ψ^2 ,|Ψ|^2 ,|Ψ^2 |.

Which of these would it make sense to interpret as a probability

density? All of them? Some? Only one? .Solution, p. 1052

49 (a) Consider the function defined byf(x,y) = (x−y)^2.

Visualize the graph of this function as a surface. (This is a simple

enough example that you should not have to resort to computer

software.) Use this visualization to determine the behavior of the

sign of the Laplacian, as in example 20 on p. 910.

(b) Consider the following incorrect calculation of this Laplacian.

We take the first derivatives and find

∂f

∂x

+

∂f

∂y

= 0.

Next we take the second derivatives, but those are zero as well, so

the Laplacian is zero. Critique this calculation in two ways: (1) by

comparing with part a; (2) by comparing with a correct calculation.

(c) In general, if we have a functionfof two variables, the quantity

Q=∂f/∂x+∂f/∂ycan never be of physical interest, because it is

not rotationally invariant (sec. 3.4.2, p. 195). Prove this by showing

that by rotating your coordinate system, you can get a completely

different answer than the one calculated in part b.

.Solution, p. 1052

950 Chapter 13 Quantum Physics