2Some examples in one dimension
IB Quantum Mechanics
2.3 Parity
Consider the Schr¨odinger equation for a particle of mass m
Hψ = −
~
2
2m
ψ
00
+ V (x)ψ = Eψ.
with potential
V (x) = V (−x).
By changing variables
x → −x
, we see that
ψ
(
x
) is an eigenfunction of
H
with
energy
E
if and only if
ψ
(
−x
) is an eigenfunction of
H
with energy
E
. There
are two possibilities:
(i)
If
ψ
(
x
) and
ψ
(
−x
) represent the same quantum state, this can only happen
if
ψ
(
−x
) =
ηψ
(
x
) for some constant
η
. Since this is true for all
x
, we can
do this twice and get
ψ(x) = ηψ(−x) = η
2
ψ(x).
So we get that
η
=
±
1 and
ψ
(
−x
) =
±ψ
(
x
). We call
η
the parity, and say
ψ has even/odd parity if η is +1/ − 1 respectively.
For example, in our particle in a box, our states ψ
n
have parity (−1)
n+1
.
(ii)
If
ψ
(
x
) and
ψ
(
−x
) represent different quantum states, then we can still
take linear combinations
ψ
±
(x) = α(ψ(x) ± ψ(−x)),
and these are also eigenstates with energy eigenvalue
E
, where
α
is for
normalization. Then by construction,
ψ
±
(
−x
) =
±ψ
±
(
x
) and have parity
η = ±1.
Hence, if we are given a potential with reflective symmetry
V
(
−x
) =
V
(
x
), then
we can restrict our attention and just look for solutions with definite parity.