Lecture 5
30
Oct 23, 2002
4.2.2 Examples
1.
One-dimensional infinite square well
; see the figure. In
practice, this situation is hardly realizable: infinitely steep
“walls” simply do not occur in nature. However, it is a nice
approximation for many real-life occurrences. The infinite
square well potential is formally given by:
(I.82)
0 for 0
()
everywhere else
xL
Ux
≤≤
=
∞
L
0
x
U
(
x
)
∞
∞
The Schrödinger equation will now take on a different form
inside and outside the infinite well:
22
0
2/
2
0
2
0:
0
(
)
(
)
s
i
nc
o
s
2
0,
:
( )
( )
0
2()
km
E
x
xLU
x
x A k
xB
k
x
mE
x
x
xx
L
U
x
x
mx
ϕ
ϕϕ
=
∂
=
⇒
=−
⇒
=
+
∂
∂
<>
=
∞
⇒
=
⇒ =
∞∂
=
=
=
(I.83)
Thus, similar to a classical particle, a quantum particle cannot exist outside the well. Inside the
well, the particle wave function solution contains two constants
A
and
B
; the wave number
k
is
given by the a priori unknown energy
E
of the particle. The challenge is to determine the wave
function constants ánd the value
E
in the problem. We have the following constraints: on both
edges of the well, the wave function solutions left and right of the potential step have to
smoothly match:
2
0
(
0)
(
0)
and
(
)
(
)
(
0
and
(
)
0
1
L
x
L
L
xd
x
↑=
↓
↓
⇒=
=
=
=
=
∫
x
L
(I.84)
The first two so-called “boundary conditions” give us two equations. Moreover, we get an addi-
tional normalization constraint (the third equation) from the fact that there is one particle in the
well, i.e. the probability for finding the particle anywhere in the well is 1. These three equations
suffice to give
A
,
B
, and
E
:
2
2
non-relativistic
2
2
00
(0)
0
( )
sin
0
with
1,2,.
.. ;
;,
o
r
i
n
t
e
r
m
s
o
f
:
8
2
Normalization: 1
( )
sin
2
nn
n
n
LL
B
L
A
kL
kL
n
n
Lp
h
h
kn
E
E
n
Ln
m
m
m
L
nx
L
x
A
d
x A
A
L
L
ϕπ
ππ
λ
λλ
π
==
⇒
=
=
=
=
=
⇒
=
∫∫
(I.85)
So, apart from an unknown phase factor, the wave function ánd a
discrete energy spectrum
are found from Schrödinger equation! The allowed energies are a discrete set of
energy levels
,
quadratically increasing (in the non-relativistic limit) with a natural number
n
, the
quantum
number
in this problem. For
ultra-relativistic
particles:
E
n
=
p
n
c
=
hc
/
n
=
nhc
/(2
L
).
The solutions (i.e. the wave functions) are simple standing waves with zero, one, two, etc.
nodes inside the well, and have value zero at the edges. For the
ground state
1
(
x
) =
√
(2/
L
)