Topic 7

Question 1

How do you simultaneously measure any pair of observables with total precision?

Answer 1

Only in some cases.
If [A, ˆ Bˆ] = 0 then the associated observables A and B can simultaneously
measured with absolute precision.

Question 2

How is ∆B (the uncertainty in an associated obserble) yielded for two observables that commute?

Answer 2

If they commute they have a common set of eigenfunction

Take the system
initially is some state (the sum of the linear combination)

A makes the system collapse into one of the ψ_n’s, yielding
A_n

ψ_n’s are also the eigenfunctions of
Bˆ

B will then yield B_n and the system will remain in the state ψ_ n,

Thus ∆B = 0

Question 3

What is the minimum precision with which one can measure both associated observables?

Answer 3

∆A · ∆B ≥1/2|[A, ˆ Bˆ]|