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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
Erwin Schrödinger (1887-1961)
A careful analysis of the process of observation in atomic physics has
shown that the subatomic particles have no meaning as isolated
entities, but can only be understood as interconnections between the
preparation of an experiment and the subsequent measurement.
- Erwin Schrödinger
Announcement:
First exam will be on Wednesday Oct. 7th at 2:30pm in room 203
You will be given a formula sheet with Lorentz transformations,
other basic equations
The exam will probably consist of 3 problems which you must solve
(not multiple choice)
Focus will be relativity and elements of quantum mechanics
Opinions on quantum mechanics
I think it is safe to say that no
one understands quantum
mechanics. Do not keep saying
to yourself, if you can possibly
avoid it, “But how can it be like
that?” because you will get
“down the drain” into a blind
alley from which nobody has yet
escaped. Nobody knows how it
can be like that.
- Richard Feynman
Those who are not shocked
when they first come across
quantum mechanics cannot
possibly have understood it.
Richard Feynman (1918-1988)
- Niels Bohr
Particle in a Box
A particle (wave) of mass m is in a one-dimensional
box of width ℓ.
The box puts boundary conditions on the wave. The
wave function must be zero at the walls of the box
and on the outside.
In order for the probability to vanish at the walls, we
must have an integral number of half wavelengths in
the box:
n

2
The energy:
or
n 
2
n
(n  1, 2,3,...)
2
2
p
h
E  K  12 mv2 

2m 2m 2
The possible wavelengths
are quantized and hence
so are the energies:
Probability of the particle vs. position
Note that E0 = 0 is not a
possible energy level.
The concept of energy
levels has surfaced in a
natural way by using
waves.
The probability of
observing the particle
between x and x + dx in
each state is:
P( x)   ( x)
2
Properties of Valid Wave Functions
Conditions on the wave function:
1. In order to avoid infinite probabilities, the wave function must be
finite everywhere.
2. The wave function must be single valued.
3. The wave function must be twice differentiable. This means that it
and its derivative must be continuous. (An exception to this rule
occurs when V is infinite.)
4. In order to normalize a wave function, it must approach zero as x
approaches infinity.
Solutions that do not satisfy these properties do not generally
correspond to physically realizable circumstances.
Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is
given in the equation
P( x)dx   ( x, t )( x, t )dx
The probability of the particle being between x1 and x2 is given by
x2
P    dx
x1
The wave function must also be normalized so that the probability
of the particle being somewhere on the x axis is 1.



 ( x, t )( x, t )dx  1
Expectation Values
In quantum mechanics, we’ll compute expectation values.
The expectation value, x , is the weighted average of a
given quantity. In general, the expected value of x is:
x  P1 x1  P2 x2 
 PN xN 
P x
i
i
i
If there are an infinite number of possibilities, and x is continuous:

x  P( x) x dx
Quantum-mechanically:
x 



 ( x) x dx   ( x)  * ( x) x dx   * ( x) x  ( x) dx
2
And the expectation of some function of x, g(x):
g ( x) 

 * ( x) g ( x)  ( x) dx
Bra-Ket Notation
This expression is so important that physicists have a special
notation for it.

g ( x)   * ( x) g ( x)  ( x) dx 
The entire expression is called a bracket.
And  | is called the bra with |  the ket.
The normalization condition is then:
|   1
|g|
The Schrödinger Wave Equation
The Schrödinger Wave Equation for the wave function (x,t) for a
particle in a potential V(x,t) in one dimension is:
2

2
i

V 
2
t
2m x
where
i  1
The Schrodinger Equation is the fundamental equation of
Quantum Mechanics.
Note that it’s very different from the classical wave equation.
But, except for its inherent complexity (the i), it will have similar
solutions.
General Solution of the Schrödinger Wave
Equation when V = 0
2
2
i
Try the usual solution:



t
2m x 2
( x, t )  Aei ( kx t )  A[cos(kx  t )  i sin(kx  t )]

 i Aei ( kx t )  i 
t

i
 (i )(i )   
t
This works as
long as:
2
k 2 p2


2m 2m
2
2


k

2
x
2 2
 2 2
k


2
2m x
2m
which says that the total
energy is the kinetic energy.
General Solution of the Schrödinger
Wave Equation when V = 0
In free space (with V = 0), the wave function is:
( x, t )  Aei ( kxt )  A[cos(kx  t )  i sin(kx  t )]
which is a sine wave moving in the x direction.
Notice that, unlike classical waves, we are not taking the real part
of this function.  is, in fact, complex.
In general, the wave function is complex.
But the physically measurable quantities must be real.
These include the probability, position, momentum, and energy.
Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time: V = V(x).
The Schrödinger equation’s dependence on time and position can then
be separated. Let:
 ( x, t )   ( x) f (t )
2

2
And substitute into: i

V 
2
t
2m x
which yields:
f (t )
 2 f (t )  2 ( x)
i ( x)

 V ( x) ( x) f (t )
2
t
2m
x
Now divide by (x) f(t):
1 df (t )
 2 1  2 ( x)
i

 V ( x)
2
f (t ) t
2m  ( x) x
The left side depends only on t, and the right side
depends only on x. So each side must be equal to
a constant. The time-dependent side is:
i
1 df
B
f t
1 df
i
B
f t
Time-Independent Schrödinger
Wave Equation
f
Multiply both sides by f /iħ:
t
 B f /i
which is an easy differential
equation to solve:
f (t )  e Bt / i  eiBt /
But recall our solution for the free particle:  ( x, t )  e
i  kx t 
in which f(t) = exp(-it), so:  = B / ħ or B = ħ, which means that: B = E !
f (t )  eiEt /
So multiplying by (x), the spatial Schrödinger equation becomes:
d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx
2
Stationary States
The wave function can now be written as:
( x, t )   ( x)eiEt /   ( x)eit
The probability density becomes:
*   * ( x) eit  ( x) eit
  ( x)
2
The probability distribution is constant in time.
This is a standing-wave phenomenon and is called a stationary state.
Most important quantum-mechanical problems will have stationary-state
solutions. Always look for them first.
Operators
d 2

 V  E
2
2m dx
2
The time-independent Schrödinger wave equation is as
fundamental an equation in quantum mechanics as the timedependent Schrödinger equation.
So physicists often write simply:
Ĥ  E
where:
2

Hˆ  
V
2
2m x
2
Ĥ is an operator yielding
the total energy (kinetic
plus potential energies).
Operators
 2 d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx
Operators are important in quantum mechanics.
All observables (e.g., energy, momentum, etc.) have
corresponding operators.
The kinetic energy operator is:
2
K 
2m x 2
2
Other operators are simpler, and some just involve multiplication.
The potential energy operator is just multiplication by V(x).
Momentum Operator
To find the operator for p, consider the derivative of the wave function
of a free particle with respect to x:

 i ( kx t )

[e
]  ikei ( kx t )  ik
x x

 p
With k = p / ħ we have:
 i 
x
 
This yields:

p   i
x
This suggests we define the momentum operator as: pˆ  i
The expectation value of the momentum is:

p  i   * ( x, t )

 ( x, t )
dx
x

.
x