© 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.