Unbound StatesUnbound States

1.1. A review on calculations for the A review on calculations for the potential step.potential step.

2.2. Quiz 10.23Quiz 10.233.3. Topics in Unbound States:Topics in Unbound States:

The potential step. The potential step. Two steps: The potential barrier Two steps: The potential barrier

and tunneling. and tunneling. Real-life examples: Alpha decay Real-life examples: Alpha decay

and other applications.and other applications. A summary: Particle-wave A summary: Particle-wave

propagation.propagation.

today

The potential step: solve the The potential step: solve the equationequationKE E

00U x U

0x

0KE E U E

0 0U x

Initial condition: free particles moving from left to right.

x

0U UWhen

22

22

d xU x x E x

m dx

The Schrödinger Equation:

When 0U

2

22 2

2d x mEx k x

dx

20 2

2 2

2d x m E Ux k' x

dx

ikx ikxx Ae Be Solution:

Inc.

Refl.

Trans.

Apply normalization and wave function smoothness

0E UWhen

ik' xx Ce

The potential step: solve the The potential step: solve the equationequationKE E

00U x U

0x

0 0KE E U

E

0 0U x

Initial condition: free particles moving from left to right.

x

22

22

d xU x x E x

m dx

The Schrödinger Equation:

0U UWhen

2

0 22 2

2d x m U Ex x

dx

ikx ikxx Ae Be Solution:

Inc.

Refl.

When 0U

2

22 2

2d x mEx k x

dx

0E UWhen

xx Ce

0E U

The potential step: transmission The potential step: transmission and reflectionand reflection

0x x

Transmission probability:

Reflection probability:

0

2

0

4E E U

TE E U

2

0

2

0

E E UR

E E U

0E UWhen0E UWhen

Transmission probability:

1 0T R

Reflection probability:

1*

*

B BR

A A

Penetration depth: 0

1

2m U E

E

0x x

E0U 0U

Two steps: The potential barrier Two steps: The potential barrier and tunneling.and tunneling.

00U x L U

0x

E

0 0U x

Initial condition: free particles moving from left to right.

xx L

0U x L

0E UWhen 0E UWhen Tunneling

0: ikx ikxx x Ae Be Solution:

Inc.

Refl. 0 : ik' x ik' xx L x Ce De

: ikxx L x Fe Trans.

0: ikx ikxx x Ae Be Solution:

Inc.

Refl. 0 : x xx L x Ce De

: ikxx L x Fe Trans.Apply normalization and

wave function smoothness

Two steps: The potential barrier Two steps: The potential barrier and tunneling.and tunneling.

0E UWhen 0E UWhen Tunneling

Results:

Results:

20

20 0 0

sin 2

sin 2 4 1

m E U LR

m E U L E U E U

0 0

20 0 0

4 1

sin 2 4 1

E U E UT

m E U L E U E U

2 2 20

0 2

2when: or

2

m E U nL n E U

mL

0R

Resonant transmission.Thin film optics analogy.

20

20 0 0

sinh 2

sinh 2 4 1

m E U LR

m E U L E U E U

0 0

20 0 0

4 1

sinh 2 4 1

E U E UT

m E U L E U E U

Tunneling through a wide Tunneling through a wide barrierbarrier

Wide barrier: 02

1m E UL

L L

02 2

0 0

16 1L m E UE E

T eU U

Tunneling:

Transmission probability is very sensitive to barrier width L and the energy E. This leads to some wonderful applications of QM.

Real-life examples: Alpha decay Real-life examples: Alpha decay and other applications.and other applications.

Who took my cheese? Who took the energy from my alphas?

Scanning Tunneling Scanning Tunneling MicroscopeMicroscope

Please read about the tunneling diode, field emission and the SQUIDS yourself.We will discuss about the STM here.

A summary: Particle-wave A summary: Particle-wave propagation.propagation.

Review questionsReview questions

Please review the solutions to the Please review the solutions to the Schrödinger equation with the step Schrödinger equation with the step and two steps condition and make and two steps condition and make sure that you feel comfortable with sure that you feel comfortable with the results.the results.

Preview for the next class Preview for the next class (10/28)(10/28)

Text to be read:Text to be read: Please skim from 7.1 to 7.8. If you have Please skim from 7.1 to 7.8. If you have

difficulty in understanding the materials, see difficulty in understanding the materials, see the slides by next Monday.the slides by next Monday.

Questions:Questions: What is the fundamental change to move the What is the fundamental change to move the

Schrödinger equation from 1-D to 3-D? Schrödinger equation from 1-D to 3-D? What is the quantization condition for the What is the quantization condition for the zz

component of angular momentum? component of angular momentum? According to QM, can you have a visual According to QM, can you have a visual

presentation for the electron’s whereabouts in presentation for the electron’s whereabouts in a hydrogen atom? a hydrogen atom?

Homework 9, due by 10/30Homework 9, due by 10/30

1.1. Problem 21 on page 224.Problem 21 on page 224.

2.2. Problem 24 on page 225.Problem 24 on page 225.

3.3. Problem 32 on page 225.Problem 32 on page 225.