Projection Operators and Measurement Outcomes


Projection Operators and Measurement Outcomes

1. Single-Particle Spin Measurement

Consider a quantum system where a spin-1/2 particle (e.g., an electron) is measured along the z-axis.

Observable: Spin along z-axis (Sz)

The spin operator Sz is:

Sz=ℏ2[100−1]

The possible measured values (eigenvalues) are:

  • +ℏ/2 (Spin up, |+⟩)
  • −ℏ/2 (Spin down, |−⟩)

Projection Operators:

The corresponding projection operators are:

P+=|+⟩⟨+|=[1000]
P−=|−⟩⟨−|=[0001]

Measurement Probabilities:

If the quantum state is |ψ⟩=α|+⟩+β|−⟩, the probability of measuring +ℏ/2 (spin up) is:

P(+ℏ/2)=⟨ψ|P+|ψ⟩=|α|2

Similarly, the probability of measuring −ℏ/2 (spin down) is:

P(−ℏ/2)=⟨ψ|P−|ψ⟩=|β|2

Measurement Process Visualization:

       Spin Measurement Device (Stern-Gerlach)
                      |
      ↑ ( +ℏ/2 )      |       ↓ ( -ℏ/2 )
  --------------------->--------------------
        |ψ⟩ = α|+⟩ + β|−⟩

2. Two-Particle Entangled State

Now, consider a system of two entangled spin-1/2 particles in the Bell state:

|Φ+⟩=1√2(|+⟩A|+⟩B+|−⟩A|−⟩B)

Observable: Total Spin along z-axis

The total spin operator is:

Stotalz=SAz+SBz

The possible measured values are:

  • +ℏ (Both particles spin up)
  • −ℏ (Both particles spin down)
  • 0 (One particle spin up, one spin down)

Projection Operators:

For these measurement outcomes, the projection operators are:

P+ℏ=|+⟩A|+⟩B⟨+|A⟨+|B
P0=|+⟩A|−⟩B⟨+|A⟨−|B+|−⟩A|+⟩B⟨−|A⟨+|B
P−ℏ=|−⟩A|−⟩B⟨−|A⟨−|B

Measurement Probabilities:

For the Bell state |Φ+⟩, we calculate:

P(+ℏ)=⟨Φ+|P+ℏ|Φ+⟩=12
P(−ℏ)=⟨Φ+|P−ℏ|Φ+⟩=12
P(0)=⟨Φ+|P0|Φ+⟩=0

Entanglement Measurement Visualization:

        Particle A                        Particle B
       -----------                      -----------
       |  +⟩   -⟩ |                      |  +⟩   -⟩ |
       |   |    |  |                      |   |    |  |
       ------------------                 ------------------
                 |  Bell State: |Φ+⟩ = 1/√2 (|+⟩|+⟩ + |−⟩|−⟩)
                 |  
                 |  If A is measured as +ℏ/2, then B must be +ℏ/2.
                 |  If A is measured as -ℏ/2, then B must be -ℏ/2.

3. Conclusion

  • Single-particle case: Projection operators extract probabilities of spin measurements.
  • Two-particle case: Projection operators help analyze entanglement, showing how quantum correlations affect measurement outcomes.