bei48482_FM

226 Chapter Six

EXERCISES

To strive, to seek, to find, and not to yield. —Alfred, Lord Tennyson

6.3 Quantum Numbers

1. Why is it natural that three quantum numbers are needed to
   describe an atomic electron (apart from electron spin)?


2. Show that

20 () (3 cos^2 1)

is a solution of Eq. (6.13) and that it is normalized.

3. Show that

R 10 (r) er^ a^0

is a solution of Eq. (6.14) and that it is normalized.

4. Show that

R 21 (r) er^2 a^0

is a solution of Eq. (6.14) and that it is normalized.

5. In Exercise 12 of Chap. 5 it was stated that an important
   property of the eigenfunctions of a system is that they are
   orthogonal to one another, which means that
   
   
   *nmdV 0 nm

r

a 0

^1

2  6 a 03 2

^2

a 03    2

 10 



4

Verify that this is true for the azimuthal wave functions (^) mlof
the hydrogen atom by calculating
2 
0
*ml (^) mld
for mlml.

6. The azimuthal wave function for the hydrogen atom is
   ()Aeiml
   Show that the value of the normalization constant Ais 1  2 
   by integrating | |^2 over all angles from 0 to 2.

6.4 Principal Quantum Number

6.5 Orbital Quantum Number

7. Compare the angular momentum of a ground-state electron in
   the Bohr model of the hydrogen atom with its value in the
   quantum theory.


8. (a) What is Schrödinger’s equation for a particle of mass m
   that is constrained to move in a circle of radius R, so that 
   depends only on ? (b) Solve this equation for and evaluate
   the normalization constant. (Hint: Review the solution of
   Schrödinger’s equation for the hydrogen atom.) (c) Find the
   possible energies of the particle. (d) Find the possible angular
   momenta of the particle.

Example 6.4

A sample of a certain element is placed in a 0.300-T magnetic field and suitably excited. How

far apart are the Zeeman components of the 450-nm spectral line of this element?

Solution

The separation of the Zeeman components is



Since c , dc d ^2 , and so, disregarding the minus sign,

 



2.83 10 ^12 m0.00283 nm

(1.60^10 ^19 C)(0.300 T)(4.50^10 ^7 m)^2

(4)(9.11 10 ^31 kg)(3.00 108 m/s)

eB 2



4 mc

2 



c

eB



4 m

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