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Conserved Quantities Example II: Parity operator and symmetric potentials

ˆ ψ ( x ) = ψ (−x )
Definition of a parity operator: Π
What are the eigenfunctions and eigenvalues of the parity operator:

ˆ ( x ) = λu ( x ) ⇒ Π
ˆ Πu
ˆ ( x) = Π
ˆ λu ( x ) ⇒ Π
ˆ Πu
ˆ ( x ) = λΠu
ˆ ( x ) ⇒ u ( x ) = λ 2 u ( x ) ⇒ λ = ±1
Πu
The eigenfunctions of the parity operator all are either odd or even.
f (−x)= f (x) even
f (−x)−= f (x) odd
Does Hamiltonian for Simple Harmonic Oscillator commute with the parity operator?
2
 2 ∂2 1
∂2
1
 2 ∂2 1
2
2 2
2
ˆ (−x ) = − 
Hˆ ( x ) = −
+
m
ω
x

H
+
m
ω
−x
=

+ mω 2 x 2 = Hˆ ( x )
( )
2
2
2
2m ∂x 2
2m ∂ (−x ) 2
2m ∂x 2

Let’s check the commutator:

( 2 2
+
( 2 2
+
&quot;Π,
ˆ Hˆ \$ψ ( x ) = Π
ˆ Hˆ ψ ( x ) − Hˆ Π
ˆ ψ ( x) = Π
ˆ * −  ∂ + 1 mω 2 x 2 -ψ ( x ) − * −  ∂ + 1 mω 2 x 2 -ψ (−x ) =
#
%
2
2
) 2m ∂x 2
,
) 2m ∂x 2
,
(  2 ∂2
+
(  2 ∂2 1
+
1
2
2
= ** −
+
m
ω
−x
ψ
−x

+ mω 2 x 2 -ψ (−x ) = 0
( ) - ( ) *−
2
2
) 2m ∂x 2
,
) 2m ∂ (−x ) 2
,

ˆ . In fact, last
This means that one can always find a set of eigenfunctions common to Hˆ and Π
lecture we have shown that SHO eigenfunctions are always even or odd.