“All of modern physics is governed
by
that magnificent and thoroughly
confusing discipline called quantum
mechanics...It has survived all tests and
there is no reason to believe that there
is any flaw in it….We all know how to
use it and how to apply it to problems;
and so we have learned to live with the
fact that nobody can understand it.”
Murray GellMann
Lecture 11:
Particles in Finite Potential Wells
n=1
n=2
n=3
n=4
0
L
U
0
I
II
III
U(x)
(x)
AlGaAs
GaAs
AlGaAs
U(x)
x
Act 1
Act 1
(a)
(b)
2.
Which of the following wave functions corresponds to a particle
more likely to be found on the left side?
(x)
0
x
(x)
0
x
(c)
(x)
0
x
Solution
Solution
(a)
(b)
2.
Which of the following wave functions corresponds to a particle
more likely to be found on the left side?
(x)
0
x
(x)
0
x
(c)
(x)
0
x
None of them!
(a) is clearly symmetrical.
(b) might seem to be “higher” on the left
than on the right, but only the absolute
square determines the probability.
0
x
Probabilities
n=1
0
x
L
n=2
0
x
L
0
x
L
n=3
U=
U=
0
x
L
0
x
L
0
x
L
Often what we measure in an experiment is the probability density, 
(x)
2
.
1
( )
sin
n
n
x
B
x
L
Wavefunction =
Probability
amplitude
2
2
2
1
( )
sin
n
n
x
B
x
L
Probability
per unit
length
(in
1dimension)
“Again an idea of Einstein’s gave me the lead. He had tried to make the duality of
particles – light quanta or photons  and waves comprehensible by interpreting the
square of the optical wave amplitudes as probability density for the occurrence of
photons. This concept could at once be carried over to the
function: 

2
ought to
represent the probability density for electrons (or other particles). It was easy to assert
this, but how could it be proved?”
M. Born, Nobel Lecture (1954).
Probability and Normalization
1
(
)
sin
n
n
x
B
x
L
We now know that
.
How can we determine B
1
?
We need another constraint.
It is the requirement that
total probability equals 1
.
The probability density at x is 
(
x
)
2
:
Therefore, the total probability is the integral:
In our square well problem, the integral is
simpler, because
= 0 for x < 0 and x > L:
Requiring that P
tot
= 1 gives us:
0
x
L
n=3
B
1

2
Integral under
the curve = 1
1
2
B
L
2
tot
P
x
dx
2
2
1
0
2
1
sin
2
L
tot
n
P
B
x
dx
L
L
B
Probability
Density
2
2
sin
n
P
x
N
x
L
In the infinite well:
.
(Units are m
1
, in 1D)
Notation:
The constant is typically written as “N”, and
is called the “
normalization constant
”.
For the square well:
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 Winter '16
 Ptot