Angular momentum system is prepared in the state

    ❙Ψ❭ = 1͟  ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭
         √10        √10      √10       √10 


Possible results of 𝐋² measurement?

The 𝐋̂² operator was developed in lecture and in the text. The related eigenvalue equation is

    𝐋̂²❙lm❭ = l(l+1) ħ²❙lm❭

The possible measurements of 𝐋̂² for this system are those associated with the initial state vector, i.e. the values lm = 11, 10, 22, 20.

For lm = 11 & 10, 𝐋̂² = 2ħ²

For lm = 22 & 20, 𝐋̂² = 6ħ².

Since the same 𝐋̂² is measures for two states, the sum of the probabilities of measuring those states is the probability of measuring that squared angular momentum.

𝓟(𝐋̂²=2ħ²) = │❬11❙Ψ❭│² + │❬10❙Ψ❭│².

    ❬11❙Ψ❭ = ❬11❙⎛  1͟  ❙11❭ - 2͟ ❙10❭ + ι͟2͟ ❙22❭ + ι1͟ ❙20❭ ⎞   
                 ⎝ √10        √10      √10       √10     ⎠.

The eigenstates for this system are LaPlace's spherical harmonic functions, which comprise an orthogonal set, I.E.:

    ❬lm❙l′m′❭ = δₗₗ′ δₘₘ′.

    │❬11❙Ψ❭│² = │❬11❙  1͟  ❙11❭│² = ¹/₁₀.
                │     √10     │    

    │❬10❙Ψ❭│² = │❬10❙  2͟  ❙10❭│² = ⁴/₁₀.
                │     √10     │    

(𝐚)
    𝓟(𝐋̂²=2ħ²) = ½.

No need to calculate the other set: since the vector is normalized, the probability of measuring 𝐋̂²=6ħ² is also

    𝓟(𝐋̂²=6ħ²) = ½.


For 𝐋̂𝓏 the eigenvalue equation is

    𝐋̂𝓏❙lm❭ = mħ❙lm❭,

so the expected measurements are,

for lm = 11 & 21, 𝐋̂𝓏 = ħ.

for lm = 10 & 20, 𝐋̂𝓏 = 0.

In this case, the probabilities will be

    𝓟(𝐋̂𝓏=ħ) = │❬11❙Ψ❭│² + │❬21❙Ψ❭│² and

    𝓟(𝐋̂𝓏=0) = │❬10❙Ψ❭│² + │❬20❙Ψ❭│².

The method has already been demonstrated, so taking the probability components and summing,

(𝐛)

    𝓟(𝐋̂𝓏=ħ) = ²/₁₀.

    𝓟(𝐋̂𝓏=0) = ⁸/₁₀.

Histogram of probabilities:


𝓟                                      𝓟
   ╭─────────────────┬────────────────╮
1  │                 │                │ 1 
   │                 │                │   
.8 │                 │           ▧    │ .8
   │                 │           ▧    │   
   │                 │           ▧    │   
.5 │   ▧        ▧    │           ▧    │ .5
   │   ▧        ▧    │           ▧    │    
   │   ▧        ▧    │           ▧    │    
   │   ▧        ▧    │   ▧       ▧    │    
.1 │   ▧        ▧    │   ▧       ▧    │ .1
   ╰─────────────────┼────────────────╯
       2ħ²     6ħ²   │   0       ħ    
            𝐋̂²       │       𝐋̂𝓏

            h