4More results in one dimensions
IB Quantum Mechanics
4.5 General features of stationary states
We are going to end the chapter by looking at the difference between bound
states and scattering states in general.
Consider the time-independent Schr¨odinger equation for a particle of mass
m
Hψ = −
~
2
2m
ψ
00
+ V (x)ψ = Eψ,
with the potential
V
(
x
)
→
0 as
x → ±∞
. This is a second order ordinary
differential equation for a complex function
ψ
, and hence there are two complex
constants in general solution. However, since this is a linear equation, this
implies that 1 complex constant corresponds to changing
ψ 7→ λψ
, which gives
no change in physical state. So we just have one constant to mess with.
As |x| → ∞, our equation simply becomes
−
~
2
2m
ψ
00
= Eψ.
So we get
ψ ∼
(
Ae
ikx
+ Be
−ikx
E =
~
2
k
2
2m
> 0
Ae
κx
+ Be
−κx
E = −
~
2
κ
2
2m
< 0.
So we get two kinds of stationary states depending on the sign of
E
. These
correspond to bound states and scattering states.
Bound state solutions, E < 0
If we want ψ to be normalizable, then there are 2 boundary conditions for ψ:
ψ ∼
(
Ae
κx
x → −∞
Be
−κx
x → +∞
This is an overdetermined system, since we have too many boundary conditions.
Solutions exist only when we are lucky, and only for certain values of
E
. So
bound state energy levels are quantized. We may find several bound states or
none.
Definition
(Ground and excited states)
.
The lowest energy eigenstate is called
the ground state. Eigenstates with higher energies are called excited states.
Scattering state solutions, E > 0
Now
ψ
is not normalized but bounded. We can view this as particle beams, and
the boundary conditions determines the direction of the incoming beam. So we
have
ψ ∼
(
Ie
ikx
+ Re
−ikx
x → −∞
T e
ikx
x → +∞
This is no longer overdetermined since we have more free constants. The solution
for any E > 0 (imposing condition on one complex constant) gives
j ∼
(
j
inc
+ j
ref
j
tr
=
(
|I|
2
~k
m
− |R|
2
~k
m
x → −∞
|T |
2
~k
m
x → +∞
We also get the reflection and transmission probabilities
P
ref
= |A
ref
|
2
=
|j
ref
|
|j
inc
|
P
tr
= |A
tr
|
2
=
|j
tr
|
|j
inc
|
,
where
A
ref
(k) =
R
I
A
tr
(k) =
T
I
are the reflection and transmission amplitudes. In quantum mechanics, “ampli-
tude” general refers to things that give probabilities when squared.