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```Recap I
Lecture 40
Recap II
Today:
• More quantum mechanics
• Heisenberg’s
uncertainty principle
• 1-D infinite square well
• Finite square well,
tunneling…
Werner Heisenberg
Infinite 1-D Square Well: Wave functions and Quantized Energy
 n x 
 n ( x)  A sin 
, for 0  x  L.
L


 h2 
  n 2 E1 , n  1, 2, 3, .
En  n 
2 
 8mL 
2
|
1(x
|2)
11.1
0.9
11(x
( x))
0.7
0.5
0.8
Energy
0.3
0.3
-0.2
0.1
L1
0
-0.1
L1
0
-0.7
2
|
1(x
21.1|)
-1.2
2 (x
( x))
0.9
0.8
0.7
16E1
0.5
0.3
-0.2
0.3
L1
0
0.1
-0.1
-0.7
-1.2
L1
0
2
|
1(x
31.1|)
33(x
( x))
0.8
E1
0.7
0.5
L1
0
0.3
0.1
-0.7
-0.1
-1.2
Position x
9E1
4E1
0.9
0.3
-0.2
25E1
L1
0
Position x
Electron (de Broglie Waves) in
an Infinite 1-D Square Well
Which set of energy levels
corresponds to the larger
value of well size L?
A.
B.
C.
Set 1
Set 2
The spacing of energy levels does not depend on L
Finite 1-D square well:
For an electron in a potential well of finite depth we must solve
the time-independent Schrödinger equation with appropriate
boundary conditions to get the wave functions.
10
10
8
8
6
6
4
4
2
2
0
-2
-1
0
En
 (x)
V(x)
-0.5
0
x/L
0.5
1
-2
-1
En
 (x)
V(x)
-0.5
0
x/L
0.5
1
Quantum Tunneling
Particle can “tunnel”
through a barrier
that it classically
could not surmount
Example: Scanning Tunneling Microscope (STM)
Example: Scanning Tunneling Microscope (STM)
Spin
• Electrons (and many other particles) have an intrinsic property

(like mass & charge) called spin angular momentum, S, (or
just called spin).

• The component of S measured along any axis is quantized.
• For electrons:
S  ss  1
h
2 ,
• Have spin
where s = 1/2 is the spin
quantum number of an electron
• Spin component along any axis can only have one the
h
following two values:
S z  ms
2
with spin magnetic quantum number ms = ½ or ½.
The Pauli Exclusion Principle
(Wolfgang Pauli, 1925):
No two electrons confined to the same trap can have the
same set of values for their quantum numbers (n, ms …).
 In a given trap, only two electrons, at most, can occupy a
state with the same (single-particle) wave function
(solution of the Schrödinger equation). One must have
ms  ½ and the other must have ms  ½.
```
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