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© Matthias Liepe, 2012
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
1E
14E
19E
125E
116E
1(x )
-1.2
-0.7
-0.2
0.3
0.8
0 1L
2(x )
-1.2
-0.7
-0.2
0.3
0.8
0 1L
3(x )
-1.2
-0.7
-0.2
0.3
0.8
0 1L
)(2 x
)(3 x
)(1 x
1(x )
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 1L
1(x )
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 1L
1(x )
-0.1
0.1
0.3
0.5
0.7
0.9
1.1
0 1L
21 ||
22 ||
23 ||
Position x Position x
Energy
Infinite 1-D Square Well: Wave functions and Quantized Energy
.,3,2,1 ,8
1
2
2
22
nEn
mL
hnEn
.0for ,sin)( LxL
xnAxn
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. Set 1 B. Set 2 C. 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.
-1 -0.5 0 0.5 1 -2
0
2
4
6
8
10
x/L
En (x) V(x)
-1 -0.5 0 0.5 1 -2
0
2
4
6
8
10
x/L
En (x) V(x)
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, (or
just called spin). ,S
• The component of measured along any axis is quantized.
• For electrons:
• Have spin , where s = 1/2 is the spin
quantum number of an electron
• Spin component along any axis can only have one the
following two values:
with spin magnetic quantum number ms = ½ or ½.
S
2
h1ss S
2
hsz mS
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.