Unbound StatesUnbound States1.1. A review about the discussions A review about the discussions
we have had so far on the we have had so far on the Schrödinger equation.Schrödinger equation.
2.2. Quiz 10.21Quiz 10.213.3. 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
Thurs.
Review: The Schrödinger equationReview: The Schrödinger equation
2 2
2 Ψ Ψ2
x,t i x,tm x t
The free particle Schrödinger Equation:
22
2
Ψ ΨΨ
2x,t x,t
U x x,t im x t
From
Ψ i kx tx,t Ae
Ψ ΨKE x,t E x,t
Ψ ΨKE U x x,t E x,t
The Schrödinger Equation
The 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
The momentum of this particle: p kcompletely defined. The location of this particle: 2 2Ψ i kx t i kx tx,t Ae Ae A
2
2pKEm
and p k E
We understand this equation as energy accounting
undefined.
KE U x x E x
Review: the time independent Review: the time independent Schrödinger equation and the two Schrödinger equation and the two conditions for the wave functionconditions for the wave function
When the wave function can be expressed as
2 2
22d U x x E x
m dx
Ψ x,t x t
and the time independent Schrödinger Equation:
We have found
The solution of this equation is the stationary states becauseThe probability of finding a particle does not depend on time:
22 2
Ψ i E tx,t x e x
i E tt e
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
22
22d x
U x x E xm dx
Review: Solving the Schrödinger Review: Solving the Schrödinger equation. equation.
Case 1: The infinite potential wellCase 1: The infinite potential well
0 0 or
2 sin 0
x x Lx n x x L
L L
Equation and Solution: Energy and probability 2 2
2 , 1 2 32
E n n , , ,...mL
2 22 sin nx xL 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.
22
2
0 0 or
from: 02
x x Lx d x
E x x Lm dx
Solving the Schrödinger equation. Solving the Schrödinger equation. Case 2: The finite potential wellCase 2: The finite potential well
0U U L
x0 L
E KE
x0 L
E KE
00U U 00U U
0 00x L
U xx ,x L
0
0 00x L
U xU 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
The change
Equations:
Review: Solving the Schrödinger Review: Solving the Schrödinger equation. equation.
Case 2: The finite potential wellCase 2: The finite potential well 00U U
x0 L
E KE
00U U
2
2sin cos mEx A kx B kx , k
2
0 22 2
2d x m U Ex x
dx
0x x
x x
x Ce De , xx Fe Ge , x L
0
sin cos 0
x
x
Ce xx A kx B kx x L
Ge x L
Solutions: 2cot kkLk
Energy quantization:
0
12m U E
Penetration depth:
22
22d x
E xm dx
22
022d x
U x E xm dx
Review: Solving the Schrödinger Review: Solving the Schrödinger equation. equation.
Case 3: The simple harmonic oscillatorCase 3: The simple harmonic oscillator
212
U x x
This model is a good approximation of particles oscillate about an equilibrium position, like the bond between two atoms in a di-atomic molecule.
22
22
12 2d x
x x E xm dx
Solve for wave function and energy level
Review Solving the Schrödinger Review Solving the Schrödinger equation. 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
Compare Case 1, Case 2 and 3: Compare Case 1, Case 2 and 3:
0 0 or
2 sin 0
x x Lx n x x L
L L
2 22
2 , 1 2 32
E n n , , ,...mL
Energy levels:
Wave function:
0
sin cos 0
x
x
Ce xx A kx B kx x L
Ge x L
2cot kkLk
0
12m U E
Penetration depth:
Wave function:
Energy levels: 1
02
00 1 2 3
E n
n , , , ,..., m
Energy levels:
New cases, unbound states: the New cases, unbound states: the potential steppotential step
From potential well to a one-side well, or a step:
0x
KE E0KE E U
E
00U x U
0 0U x
Free particle with energy E. Standing waves do not form and energy is no quantized.
Ψ i kx tx,t Ae We discussed about free particle wave function before. Which is: Ψ ikx i t ikx i tx,t Ae Ae e Re-write it in the form
Right moving. Why?Right moving (along
positive x-axis) wave of free particle:
ikxx Ae
Left moving wave of free particle:
ikxx Ae
x
2 0k p k
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
22d x
U x x E xm dx
The Schrödinger Equation:
When 0U
2
22 2
2d x mE x k xdx
2
0 22 2
2d x m E Ux k' x
dx
ikx ikxx Ae Be Solution:
Inc.
Refl.
Trans.
Why no reflection here? Ans: no 2nd potential step on the right.
Are we done? What do we learn here? Any other conditions to apply to the solutions?
Normalization and wave function smoothness
0E UWhen
ik' xx Ce
The potential step: apply The potential step: apply conditionsconditions
0x x
Smoothness requires:
2
Inc.*A A 2
Refl*
.B B 2
Trans*
.C C
2 2# particles # particles distancetime distance time
v k
0 0 00 00 0 : ik ik ik'
x x Ae Be Ce
A B C
0 0 00 0
0 0
: ik ik ik'x x
x x
d d ikAe ikBe ik ' Cedx dx
k A B k' C
Transmission probability:
20Trans.
2 20Inc.
# trans. time 4 incident time
*x
*x
k C Ck' kk 'T# A Akk k k '
Reflection probability:
2 20Refl.
2 20Inc.
# Refl. time 1 incident time
*x
*x
k k k 'B BR T# A Ak k k'
Express B and C in terms of A:
B k k' k k ' A 2C k k k' A
The potential step: transmission The potential step: transmission and reflectionand reflection
0x x
Now using the definitions of k and k’:
02
2m E Uk'
2
2mEk
nair
Reference:for normal incidence, light transmission probability:
2
41nT
n
2
2
1
1
nR
n
Reflection probability:
Transmission probability:
2
4kk 'Tk k '
Reflection probability:
2
2
k k'R
k k '
0
2
0
4E E U
TE E U
2
0
2
0
E E UR
E E U
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
22d x
U x x E xm dx
The Schrödinger Equation:
0U UWhen
20 2
2 2
2d x m U Ex x
dx
ikx ikxx Ae Be Solution:
Inc.
Refl.
When 0U
2
22 2
2d x mE x k xdx
0E UWhen
xx Ce
0E U
The potential step: apply The potential step: apply conditionsconditions
Smoothness requires: 0 0 0
0 00 0 : ik ikx x Ae Be Ce
A B C
0 0 00 0
0 0
: ik ikx x
x x
d d ikAe ikBe Cedx dx
k A B C
Transmission probability:
1 0T R
Reflection probability:
1*
*
B BRA A
Express B in terms of A: B ik ik A
Penetration depth: 0
12m U E
* *B B A AOne can prove:
Review questionsReview questions The plane wave solution of a free The plane wave solution of a free
particle Schrödinger Equation isparticle Schrödinger Equation isCan you normalize this wave function? Can you normalize this wave function?
Try to solve for the wave function and Try to solve for the wave function and discuss about transmission and discuss about transmission and reflection for this situation: reflection for this situation:
Ψ i kx tx,t Ae
KE E 00U x U
0x
0KE E U E
x 0 0U x
Preview for the next class Preview for the next class (10/23)(10/23)
Text to be read:Text to be read: In chapter 6:In chapter 6:
Section 6.2Section 6.2 Section 6.3Section 6.3 Section 6.4Section 6.4
Questions:Questions: How much do you know about radioactive How much do you know about radioactive
decays of isotopes? Have you heard of alpha decays of isotopes? Have you heard of alpha decay, beta decay and gamma sources? decay, beta decay and gamma sources?
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 8, due by 10/28Homework 8, due by 10/281.1. Problem 25 on page 187.Problem 25 on page 187.2.2. Problem 34 on page 188.Problem 34 on page 188.3.3. Problem 15 on page 224.Problem 15 on page 224.4.4. Problem 18 on page 224.Problem 18 on page 224.