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2 Quantum Mechanics – I
2.3.8 Uncertainty Principle
2.82 ΔxΔ p x ∼/2
P =
2x
E =
p
2
2m
+ 1/2 mω
2 x
2
=
2
8mx 2 + 1/2mω
2 x
2
The ground state energy is obtained by setting
∂ E
∂ x
= 0
∂ E
∂ x
= −
2
4mx 3 + mω
2 x = 0
whence x
2
=
2mω
∴ E = 1/4ω + 1/4ω =
1
2
ω
2.83 If E and p are to be measured simultaneously their operators must commute.
Now
H = −
2
∇
2
/2m + V and p = −i∇
[H, p] = [−
2
∇
2
/2m + V, −i∇]
= i
3
∇
2
∇/2m − iV ∇ − i
3
∇∇
2
/2m + i∇V
The first and the third term on the RHS get cancelled because ∇
2
∇ = ∇∇
2 .
Therefore
[H, P] = −i(V ∇ − ∇V )
If V = constant, the commutator vanishes. To put it differently energy
and momentum can be measured with arbitrary precision only for unbound
particles.
2.84 Consider the motion of a particle along x-direction.
The uncertainty Δx is defined as
(Δx)
2
=< (x− < x >)
2
>=< x
2
> −2 < x >< x > + < x >
2
=< x
2
> − < x >
2
(1)
Similarly
(ΔP x )
2
=< P
2
x > − < P x >
2
(2)