Physics 215– Elementary Modern Physics
Equations for Test 3
The following equations you should have memorized, and understand how to use
them:
For three dimensions
For one dimension
Expectation Value of Operator

  
*

Uncertainty
2
     2  
 x    x  dx
Schrödinger Equations

2  2

i   
  V  or i   H 
2
t
2m x
t
2
2
 d
E  
  V or E  H
2m dx 2
Momentum
Operators (3D):
 
px 
i x
 
py 
i y
 
pz 
i z
Angular
Momentum
Values:
2
L  2  l 2  l 
Lz  m
Filling up Atoms:
Time-dependence
  x, t   e iEt   x 
2
Momentum
Operator
 
pop 
i x
Hamiltonian
1 2
H
pop  V
2m
E H
Schrödinger Equations in 3D:
  2

2
2 



  V  or i   H 
i   
 2
2
2 
t
t
2m  x y z 
2
E  
2   2
2
2 



  V
2m  x 2 y 2 z 2 
The Hamiltonian
1 2
H
pop  V
2m
E H
or
Restrictions on
Quantum Numbers:
n  1, 2,3,
l  0,1, 2,3, , n  1
m  l , l  1, , 0, , l
ms   12
Hydrogen Wave Functions
 n ,l ,m  r ,  ,    Rn,l  r  Yl ,m  ,  
1s2 2s2 2p6 3s2 3p6 4s2 …
1s2
2s2
3s2
4s2
5s2
6s2
7s2
2p6
3p6
4p6
5p6
6p6
7p6
E  H
Spin Values
s  12
S 2  2  s 2  s 
 34  2
S z  ms
3d10
4d10
5d10
6d10
7d10
4f14
5f14
6f14
7f14
The following equations you need not memorize, but you should know how to use
them if given to you:
h  6.626  1034 J  s  4.136 1015 eV  s
  1.055 1034 J  s  6.582 1016 eV  s
1D square well:
 2 2 n2
En 
2mL2
2
  nx 
sin 
 n  x 

L
 L 
n  1, 2,3,
Harmonic Osc.
En    n  12 
n  0,1, 2,
Barrier penetration:
E
E
T  16 1   e 2 L ,
V0  V0 

2m V0  E 

Hydrogen
13.6 eV  Z 2

En  
n2
Reflection off a step:
 E  E  V
0

R   E  E  V0


1

2

 if E  V0


if E  V0
Layout of the exam: Below is an outline of the exam
This test consists of three parts. Please note that in parts II and III, you can skip one
question of those offered.
Part I: Multiple Choice [20 points]
For each question, choose the best answer (2 points each)
[questions 1-10]
Part II: Short answer [20 points]
Choose two of the following three questions and give a short answer (1-3
sentences) (10 points each).
[questions 11-13]
Part III: Calculation: [60 points]
Choose three of the following four questions and perform the indicated
calculations (20 points each)
[questions 14-17]