PHYS212-071
HW # 9 Solutions (Chapter 7)
(Numbers refer to 2nd Edition of Textbook)
5. Assume that the nucleus of an atom can be regarded as a three-dimensional box of
width 2 x 10-14 m. If a proton moves as a particle in this box, find (a) the ground state
energy of the proton in MeV and (b) the energies of the first and second excited states.
What are the degeneracies of these states?
L  2  10 14 m  2  10 5 nm
 2 2 2 2 2
 2 c 2
E0 
1

1

1

3
 1.53MeV
2mL2
2mc 2 L2
 2 2 2 2
 2 c 2
2
E1 
1 1  2  6
 3.06MeV , threefold degenerate
2mL2
2mc 2 L2
 2 2 2
 2 c 2
2
2
E2 
1

2

2

9
 4.59 MeV , threefold degenerate
2mL2
2mc 2 L2






12. The normalized ground state wavefunction for the
electron in the hydrogen atom is
3/ 2
1  1 
  e  r / a0
 r ,  ,   
  a0 
Where r is the radial coordinate of the electron and a 0 is the Bohr radius. (a) Sketch the
wavefunction versus r. (b) Show that the probability of finding the electron between r and
2
r + dr is given by  r  4r 2 dr . (c) Sketch the probability versus r and from your sketch
find the radius at which the electron is most likely to be found. (d) Show that the
wavefunction as given is normalized. (e) Find the probability of locating the electron
between r1  a0 / 2 and r2  3a0 / 2 .
a)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
1
2

0
0
3
b) d  sin dd   d  d sin   4
 r ,  ,   dV 
2
1  1

  a 0
  2 r / a0 2
4
 e
r drd  Pr   3 r 2 e  2 r / a0
a0

c)
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
3
3.5
The probability is maximum when r  a 0 .

d)  drPr  
0
e) p 



4
4
1
r 2 e 2 r / a0 dr  3 a03  x 2 e 2 x dx  4 x 2 e 2 x dx  4   1
3 
4
a0 0
a0 0
0
3 a0 / 2
3/ 2
 Pr dr  4  x e
2
a0 / 2
2 x
dx  4  0.124  0.5
1/ 2
19. A hydrogen atom is in the 6g state. (a) What is the principal quantum number? (b)
What is the energy of the atom? (c) What are the values for the orbital quantum number
and the magnitude of the electron’s orbital angular momentum? (d) What the possible
values for the magnetic quantum number? For each value, find the corresponding z
component of the electron’s orbital angular momentum and the angle that the orbital
angular momentum vector makes with the z-axis.
a) n  6
13.6eV
 0.38eV
62
c)   4;   1  2 5
d) m  4,3,2,1,0,1,2,3,4
b) E  
cos  
m
  1

m
2 5
  deg rees   26.6,47.9,63.4,77.1,90,116.6,137.9,153.4,167.1
2
29. As shown in Example 7.9, the average distance of the electron from the proton in the
hydrogen ground state is 1.5 bohrs. For this case calculate r , the uncertainty in the
distance about the average value, and compare it with the average itself. Comment on the
significance of your result.
P1s 
4
 r 2 e  2 r / a0
3
a0


0
0
2 1/ 2
r   rPr   4a0  x 3 e 2 x dx  4a0 

r  r 2  r

r
2
  r 2 Pr dr 
0
r 

3 3
 a0
8 2
3 2
a0
4
r
3
a 0  0.866  a 0 
2
2
3