Download 6. Quantum Mechanics II

CHAPTER 5
The Schrodinger Eqn.
5.1
5.2
5.3
5.4
5.5
The Schrödinger Wave Equation
Expectation Values
Infinite Square-Well Potential
Finite Square-Well Potential
Three-Dimensional InfinitePotential Well
5.6 Simple Harmonic Oscillator
5.7 Barriers and Tunneling
Homework due Wednesday Oct.
Read Chapters 4 and 5 of Kane
Chapter 4: 2, 3, 5, 13, 20
Chapter 5: 3, 4, 5, 7, 8
First exam this Friday in class
1st
Erwin Schrödinger (1887-1961)
The Scanning-Tunneling Microscope
Electrons must tunnel through the
vacuum (barrier) from the surface
to the tip. The probability is
exponentially related to the
distance, hence the ultrahigh
resolution.
Image of a molecule
Alpha-Particle Decay
Tunneling explains alpha-particle decay of heavy, radioactive nuclei.
Outside the nucleus, the Coulomb force dominates.
Inside the nucleus, the strong,
short-range attractive nuclear
force dominates the repulsive
Coulomb force. The potential
is ~ a square well.
The potential barrier at the nuclear
radius is several times greater than
the energy of an alpha particle.
In quantum mechanics, however,
the alpha particle can tunnel
through the barrier.
This is radioactive decay!
Radioactive
Decay
The number of
radioactive nuclei as
a function of time.
The time for the
number of nuclei to
drop to one half its
original value is the
well-known half life.
Three-Dimensional InfinitePotential Well
y
The wave function must be a function of all
three spatial coordinates.
So consider momentum as an
operator in three
dimensions:
p2  p2 
x

pˆ x  i
x
x
z
py2  pz2

pˆ y  i
y

pˆ z  i
z
The three-dimensional Schrödinger wave equation is:
 2   2  2  2 
  2  2  2   V  E
2m  x
y
z 
or:

2
2m
2  V  E
The 3D infinite potential well
It’s easy to show that:
 ( x, y, z )  A sin(k x x) sin(k y y ) sin(k z z )
where:
k x   nx / Lx
 nx2 n y2 nz2 
E
 2  2  2 
2m  Lx Ly Lz 
2
and:
2
When the box is a cube:
E
2
2
2
2mL
kz   nz / Lz
k y   n y / Ly
n
2
x
Lz
Ly
n n
2
y
y
2
z

z
x
Lx
Degeneracy
E
2
2
2mL2
n
2
x
 ny2  nz2 
Try 10, 4, 3 and 8, 6, 5
Note that more than one wave function can have the same energy.
When more than one wave function has the same energy,
those quantum states are said to be degenerate.
Degeneracy results from symmetries of the potential energy
function that describes the system.
A perturbation of the potential energy can remove the degeneracy.
Examples of perturbations include external electric or magnetic
fields or various internal effects, like the magnetic fields due to the
spins of the various particles.
Simple Harmonic Oscillator
Simple harmonic
oscillators describe
many physical
situations: springs,
diatomic molecules
and atomic lattices.
Consider the Taylor expansion of an
arbitrary potential function:
1
V ( x)  V0  V1  [ x  x0 ]  V2  [ x  x0 ]2  ...
2
Near a minimum, V1[xx0] ≈ 0.
Simple Harmonic
Oscillator
Consider the second-order term
of the Taylor expansion of a
potential function:
Letting x0 = 0.
V ( x)  12  ( x  x0 ) 2  12  x 2
Substituting this into
Schrödinger’s equation:
 2 d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx

 m x 2 2mE 
d 2
2m   x 2
  2 
 E    2  2 
We have:
2
dx
 2



2mE
m
Let   2 and   2 , which yields:


2
d 2
2 2


x  
2
dx


The Parabolic
Potential Well
The wave function solutions
are:  n ( x)  H n ( x) exp( x 2 / 2)
where Hn(x) are Hermite
polynomials of order n.
n
|n |2
The Parabolic
Potential Well

Classically, the
probability of finding the
mass is greatest at the
ends of motion (because
its speed there is the
slowest) and smallest at
the center.


Classical
result

Contrary to the classical
one, the largest
probability for the lowest
energy states is for the
particle to be at (or near)
the center.
Correspondence Principle for the
Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase,
however, the solution approaches the classical result. This confirms the
Correspondence Principle for the quantum-mechanical simple
harmonic oscillator.
Classical
result
The Parabolic Potential Well
The energy levels are given by:
1
1
En  (n  )  / m  (n  )
2
2
The zero point
energy is
called the
Heisenberg
limit:
1
E 0  
2