130_notes.dvi

If we also consider the spin of the electron in the Hydrogen atom, wefind that we need to add one
more commuting operator to label the states and to compute the energies accurately. If we also add
the spin of the proton to the problem, the we need still one more quantum number to describe the
state.

If it is possible, identifying the commuting operators to be used before solving the problem will
usually save time.

11.3 Uncertainty Principle for Non-Commuting Operators

Let us nowderive the uncertainty relation for non-commuting operatorsAandB. First,
given a stateψ, theMean Square uncertaintyin the physical quantity represented is defined as

(∆A)^2 = 〈ψ|(A−〈A〉)^2 ψ〉=〈ψ|U^2 ψ〉

(∆B)^2 = 〈ψ|(B−〈B〉)^2 ψ〉=〈ψ|V^2 ψ〉

where we define (just to keep our expressions small)

U = A−〈ψ|Aψ〉

V = B−〈ψ|Bψ〉.

Since〈A〉and〈B〉are just constants, notice that

[U,V] = [A,B]

OK, so much for the definitions.

Now we will dotUψ+iλV ψinto itself to get some information about the uncertainties. The dot
product must be greater than or equal to zero.

〈Uψ+iλV ψ|Uψ+iλV ψ〉≥ 0

〈ψ|U^2 ψ〉+λ^2 〈ψ|V^2 ψ〉+iλ〈Uψ|V ψ〉−iλ〈V ψ|Uψ〉≥ 0

This expression contains the uncertainties, so lets identify them.

(∆A)^2 +λ^2 (∆B)^2 +iλ〈ψ|[U,V]|ψ〉≥ 0

Choose aλto minimize the expression, to get the strongest inequality.

∂

∂λ

= 0

2 λ(∆B)^2 +i〈ψ|[U,V]|ψ〉= 0

λ=

−i〈ψ|[U,V]|ψ〉

2(∆B)^2