Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

PROBLEMS 265

and

ψ 2 =

√

2sin( 2 πx)

Problem 16.3

If the operatorAˆ=d^2 /dx^2 , find the eigenfunctionφ(x) and the eigenvalueafor the

eigenvalue equation

Aˆφ(x)=aφ(x)

Problem 16.4

According to a theory of Niels Bohr (1913) for an electron to move in a stable

classical orbit, the centrifugal forcemv^2 /rpulling away from the nucleus must be

exactly balanced by the electrostatic force of attraction between the negative electron

and the positive protone^2 /r^2.

(a) Write an expression for the velocity of the electron.

(b) Calculate the velocity of the electron. To get your answer in SI units,

use e^2 / 4 πε 0 for the charge on the electron, where 4πε 0 = 1. 113 ×

10 −^10 C^2 s^2 kg−^1 m−^3 is called thepermittivityin a vacuum.

(c) Give units.

(d) Suppose the reader forgets to take a square root in the last step and arrives at

the resultv= 4. 784 × 1012. How can he immediately know that something

has gone wrong?

Problem 16.5

Normalize the eigenfunction(x)=Asin

2 πx

λ

for the particle in a one-dimensional

box of dimensiona.

Problem 16.6

Suggest a simple classical (macroscopic) mechanical system for which the probability

function varies in a regular way with position. What does the probability function

look like?

Problem 16.7

Carry out the “little algebra” to go from

−

4 π^2

( 2

nl

) 2 =−

2 mE

̄h^2