Concise Physical Chemistry

c15 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

PROBLEMS AND EXAMPLES 247

Problem 15.5

Good mathematical software contains a number of operations on matrices that will

be useful in coming chapters. Use Mathcad©C or similar software to calculate the

sum, product, square, inversion, and product with inverse (under certain conditions

equivalent to division). Choosing matrix c from between the vertical lines in the

preceding problem enables one to solve most of these problems simply by inspection

and then allows one to verify (test run) using the software.

Problem 15.6

According to an Einstein theory of electromagnetic radiation, the 656.1-nm radiation

from atomic hydrogen in the first excited state is a particle (later named aphoton).

What is the energy, frequency, wave number, and momentum of this putative particle?

Express and explain the units of the final result for each of the four steps.

Problem 15.7

Show that the operator∇^2 =(1/r^2 )(d/dr)r^2 (d/dr) in spherical coordinates can

be written∇^2 =(d^2 /dr^2 )+(2/r)(d/dr). (This is part of the solution of the radial

Schrodinger equation for the ground state of the hydrogen atom.) ̈

Problem 15.8

Using the operator∇^2 , write the radial form of the Schr ̈odinger equation.

Problem 15.9

Allow the operator ∇^2 =(d^2 /dr^2 )+(2/r)(d/dr) to operate on the function

R(r)=e−rin spherical polar coordinates (r,θ,andφ). What is∇^2 R(r)? What are

(dR(r)/dθ) and (dR(r)/dφ)?