Bound StatesBound States

1.1. A quick review on the chapters 2 to A quick review on the chapters 2 to 5.5.

2.2. Quiz 10.9.Quiz 10.9.3.3. Topics in Bound States:Topics in Bound States:

The Schrödinger equation. The Schrödinger equation. Stationary States.Stationary States. Physical conditions: well-behaved Physical conditions: well-behaved

functions.functions. A review of classical bound states.A review of classical bound states. The infinite potential well.The infinite potential well. The finite potential well.The finite potential well. The simple harmonic oscillation.The simple harmonic oscillation. Expectation values, uncertainties, and Expectation values, uncertainties, and

operators.operators.

today

Review Review

22

2

Ψ ΨΨ

2

x,t x,tU x x,t i

m x t

LargeF ma

Small

Slow Fast

Classical physics

x' x vt

Special Relativity

Quantum mechanics

vx' x vt 2vt ' v c x t

1

21 xx x

u vu ' u vc

0vt t 0 vL L

1 cossource

obsv

ff

p uum

2uE mc

2 2 2 2 4E p c m c

p k E

2xp x

2E t

22

22

d xU x x E x

m dx

2

all phase space

Ψ 1x,t dx

smooth wave function

EM wave particle

S.R.

particle wave

Schrödinger Equation

What is here?

Relativistic quantum mechanicsQuantum electrodynamics,……

p um

21

2KE mu

Review of Special RelativityReview of Special Relativity SS and and S’S’ system: system:

For a particle with velocity in S:For a particle with velocity in S:

The Doppler effect:The Doppler effect:

x

x'y

z

S

y'

z'

S '

S’ moves with velocity v in S along the x-axis.

vx' x vt

y' y

z' z

2v

vt' x t

c

1

21 xx x

u vu ' u vc

1

21 xy y v

u vu ' uc

1

21 xz z v

u vu ' uc

2

1

1v

v,

c

p uum

12 21u u c

2

uE mc 21uKE mc

propervt t

proper vL L

u

2 2 2 2 4E p c m c

1 cossource

obsv

ff

When When θ =0, the course is =0, the course is moving away from the moving away from the observer.observer.

v

Review of EM wave Review of EM wave particle particle EM wave behave as particle:EM wave behave as particle:

Proof: Proof: Blackbody radiation. Blackbody radiation.

Photoelectric effect. Photoelectric effect.

Compton Scattering.Compton Scattering.

Energy (EM wave) converts into Energy (EM wave) converts into matter (particle)matter (particle) Pair production. Pair production.

1 cose

h'

m c

23

8

1Bhf k T

dU hf Vf

df e c

maxKE hf

Review: matter waves and Review: matter waves and the free particle Schrödinger equationthe free particle Schrödinger equation

The de Broglie wavelength of a particle:The de Broglie wavelength of a particle: The frequency:The frequency: The h-bar constant:The h-bar constant: The connection between particle and The connection between particle and

wave:wave: momentummomentum energyenergy

Wave number and angular frequency:Wave number and angular frequency:

The The free particle Schrödinger equation:free particle Schrödinger equation:

And the plane wave solution:And the plane wave solution:

The Heisenberg Uncertainty PrincipleThe Heisenberg Uncertainty Principle

h p

f E h

2k 2 2T f

2h

p k

E

22

2

Ψ Ψ

2

x,t x,ti

m x t

Ψ i kx tx,t Ae

2xp x

2E t

The Schrödinger equationThe Schrödinger equation

22

2

Ψ Ψ

2

x,t x,ti

m x t

From the free particle Schrödinger Equation

22

2

Ψ ΨΨ

2

x,t x,tU x x,t i

m x t

We understand this equation as energy accounting

Ψ i kx tx,t Ae

Ψ ΨKE x,t E x,t

Ψ ΨKE U x x,t E x,t

So the time dependent Schrödinger Equation

And its plane wave solution

And this leads to the equation that adds an external potential

Solve for with the knowledge of , for problems in QM. Ψ x,t U x

Solve Solve the Schrödinger equationthe Schrödinger equation

When the wave function can be expressed as

22

22

d xU x x E x

m dx

Ψ x,t x t

the time independent Schrödinger Equation

We have

The solution of this equation is the stationary states because

The probability of finding a particle does not depend on time:

22 2

Ψ i E tx,t x e x

i E tt e

and

Normalization:

2

all phase space

Ψ 1x,t dx

The wave function be smooth the continuity of the wave function and its first order derivative.

Two conditions

Solving the Schrödinger equation. Solving the Schrödinger equation. Case 1: The infinite potential wellCase 1: The infinite potential well

0 0 or

2sin 0

x x L

x nx x L

L L

Equation and Solution: Energy and probability

2 22 , 1 2 3

2E n n , , ,...

mL

2 22sin

nx x

L L

0 x L

1. Standing wave. 2. The QM ground-

state. A bound state particle cannot be stationary, although its wave function is stationary.

3. Energy ratio at each level: n2.

4. With very large n, QM CM.

Examples 5.1, 5.2

22

2

0 0 or

from: 02

x x L

x d xE x x L

m dx

Solving the Schrödinger equation. Solving the Schrödinger equation. Case 2: The infinite potential wellCase 2: The infinite potential well

0U U L

x0 L

E KE

00U U

x0 L

E KE

00U U

The change 0 0

0

x LU x

x ,x L

0

0 0

0

x LU x

U x ,x L

22

2

220

02

0 0 20

2

d xE xx L m dxU x

U x ,x L d xU x E x

m dx

The change

Equations:

Solving the Schrödinger equation. Solving the Schrödinger equation. Case 2: The finite potential wellCase 2: The finite potential well

00U U

x0 L

E KE

00U U

22

22

d xE x

m dx

22

022

d xU x E x

m dx

2

2sin cos

mEx A kx B kx , k

0x x

x x

x Ce De , x

x Fe Ge , x L

0

sin cos 0

x

x

Ce x

x A kx B kx x L

Ge x L

2cotk

kLk

0

1

2m U E

What is the idea?How would you verify this?

2

0 22 2

2d x m U Ex x

dx

solutionsEnergy quantization Penetration depth

Penetrating into classically forbidden regions depth

Compare Case 1 and Case 2: Compare Case 1 and Case 2: The infinite and finite potential wellsThe infinite and finite potential wells

0

sin cos 0

x

x

Ce x

x A kx B kx x L

Ge x L

2cotk

kLk

0 0 or

2sin 0

x x L

x nx x L

L L

2 22 , 1 2 3

2E n n , , ,...

mL

0

1

2m U E

Finite well

Solving the Schrödinger equation. Solving the Schrödinger equation. Case 3: The simple harmonic oscillatorCase 3: The simple harmonic oscillator

21

2U x x

22

22

1

2 2

d xx x E x

m dx

This model is a good approximation of particles oscillate about an equilibrium position, like the bond between two atoms in a di-atomic molecule.

Example 5.3

Solve for wave function and energy level

Solving the Schrödinger equation. Solving the Schrödinger equation. Case 3: The simple harmonic oscillatorCase 3: The simple harmonic oscillator

102

00 1 2 3

E n

n , , , ,..., m

Energy are equally spaced, characteristic of an oscillator

Wave function at each energy level

Gaussian

Expectation values, uncertainties, and Expectation values, uncertainties, and operatorsoperators

2probability density = x 2

found within at = p dx x x dx

expectation value of x

2 *

all space all space

x x x dx x x x dx

2 2*

all space

x x x x dx

22 2 2 all space

x x x x dx x x uncertainty

operator x x *

all space

ˆx x x x dx

Probability weighted average

Probability weighted average of x2 is

Expectation values, uncertainties, and Expectation values, uncertainties, and operatorsoperators

expectation value (observables)

2 2 x x x

uncertainty

Basic operator

x x *

all space

ˆx x x x dx

Functions of operator

p ix

*

all space

ˆp x p x dx

22 2 2

2

1

2 2 2

pKE i

m x m x

E it

2 2 p p p

*

all space

ˆE x E x dx 2 2 E E E

Q *

all space

ˆQ x Q x dx 2 2 Q Q Q

So Ψ Ψ Ψˆ ˆKE x,t U x,t E x,t

Review questionsReview questions

Follow the definition of operator and Follow the definition of operator and expectation value, how do you expectation value, how do you understand the Schrödinger Equation understand the Schrödinger Equation now? now?

How do you now understand the How do you now understand the tunneling effect mentioned often in tunneling effect mentioned often in semiconductor fabrication industry?semiconductor fabrication industry?

For operator , how do you calculate For operator , how do you calculate its expectation value and uncertainty?its expectation value and uncertainty?

Please summarize this chapter Please summarize this chapter yourself.yourself.

Q

Preview for the next class Preview for the next class (10/21)(10/21)

Text to be read:Text to be read: In chapter 6:In chapter 6:

Section 6.1Section 6.1 Section 6.2Section 6.2 Section 6.3Section 6.3 Section 6.4Section 6.4

Questions:Questions: Have you heard of the “tunneling effect” in the Have you heard of the “tunneling effect” in the

EE department (only for EE students)? EE department (only for EE students)? What is a wave phase velocity? What is a wave What is a wave phase velocity? What is a wave

group velocity? group velocity?

Homework 7, due by 10/16Homework 7, due by 10/16

1.1. Problem 5 on page 186.Problem 5 on page 186.

2.2. Problem 24 on page 187.Problem 24 on page 187.

3.3. Problem 28 on page 187.Problem 28 on page 187.

4.4. Problem 34 on page 188.Problem 34 on page 188.