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