Lecture 23:
Heisenberg Uncertainty Principle
Phy851 Fall 2009
Heisenberg Uncertainty Relation
• Most of us are familiar with the Heisenberg
Uncertainty relation between position and
momentum:
h
Δx Δp ≥
2
• How do we know this is true?
• Are the similar relations between other
operators?
Variance
• The uncertainties are also called ‘variances’
defined as
A2 − A
Δa =
2
• Note that the variance is state-dependent
• What does it tell us about our state?
– Consider a distribution P(a),
– The average of the distribution is:
a = ∑ P(a) a
a
– To estimate the width of the distribution we
might consider the square of the distance from
the mean:
2
2
d (a ) = (a − a )
– The average of this quantity is
____
2
(
d (a ) = ∑ P(a ) a 2 − 2aa + a 2
a
___
2
= a − 2a 2 + a 2
___
2
We then define the
Root Mean Square
distance as:
Or in quantum
language, we say:
= a −a2
d rms :=
Δa =
____
2
___
2
d = a −a2
2
A − A
2
)
Incompatible Observables
• For an observable A, the only way you can
have Δa=0 is if you are in an eigenstate of A
• Consider two incompatible observables, A and
B:
[A, B]= M ≠ 0
• We cannot have ΔA=0 and ΔB=0 at the same
time
– Then we would have a simultaneous eigenstate
of A and B
• So what is the best we can do?
• To derive the Heisenberg Uncertainty for X
and P relation, let us first introduce
X′ = X − X I
P′ = P − P I
[X ′, P′]= [X , P]
X ′2 = Äx 2
P′2 = Äp 2
€
Geometric Proof of Uncertainty
Relation
• Let:
φ := (X ′ + iλP′)ψ
λ is an arbitrary real number
|ψ〉 is an arbitrary state
• For any λ and |ψ〉 we must have:
φ φ ≥0
ψ X ′2 ψ − iλ ψ P′X ′ ψ + iλ ψ X ′P′ ψ + λ2 ψ P′2 ψ ≥ 0
P ′2 λ2 + i[ X ′, P ′] λ + X ′2 ≥ 0
[X ′, P′]= [X , P]
€
X ′2 = Δx 2
P ′2 = Δp 2
2 2
2
Δp λ + i[ X,P ] λ + Δx ≥ 0
Always >0
Parabolas
2 2
2
Δp λ + i[ X,P ] λ + Δx ≥ 0
This is always real
• Consider a general quadratic polynomial with
real coefficients:
€
f (λ ) = aλ2 + bλ + c
• Its graph is a parabola
– If a>0 it opens up:
• The minimum is at:
λmin
• The minimum value is:
• So f(λ) ≥0 requires:
2
2
Δx Δp ≥
€
i[ X,P ]
4
2
b
=−
2a
b2
f (λmin ) = c −
4a
b2
f (λmin )≥ 0 ⇒ ac ≥
4
h
ΔxΔp ≥
2
Generalized Uncertainty Relations
• Note that only at the very end did we make
use of the specific form of the commutator:
[X , P]= ih
• This means that our result is valid in general
for any two observables:
2
2
Δa Δb ≥
i[ A,B]
4
2
⇒ ΔaΔb ≥
[ A,B]
4
• Consider angular momentum operators:
[ Lx , Ly ] = ihLz
€
h
Δl x Δl y ≥
Lz
2
•
In General, the Heisenberg Lower limit
depends on the state.
•
X and P are special in that all states
have the same limit.
•
The Uncertainty relation is not as useful
in the more general cases