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 (\( S_z \))

The spin operator \( S_z \) is:

\[
S_z = \frac{\hbar}{2} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
\]

The possible measured values (eigenvalues) are:

  • \( +\hbar/2 \) (Spin up, \( |+\rangle \))
  • \( -\hbar/2 \) (Spin down, \( |-\rangle \))

Projection Operators:

The corresponding projection operators are:

\[
P_+ = |+\rangle \langle +| = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}
\]
\[
P_- = |-\rangle \langle -| = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}
\]

Measurement Probabilities:

If the quantum state is \( |\psi\rangle = \alpha |+\rangle + \beta |-\rangle \), the probability of measuring \( +\hbar/2 \) (spin up) is:

\[
P(+\hbar/2) = \langle \psi | P_+ | \psi \rangle = |\alpha|^2
\]

Similarly, the probability of measuring \( -\hbar/2 \) (spin down) is:

\[
P(-\hbar/2) = \langle \psi | P_- | \psi \rangle = |\beta|^2
\]

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:

\[
|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|+\rangle_A |+\rangle_B + |-\rangle_A |-\rangle_B)
\]

Observable: Total Spin along \( z \)-axis

The total spin operator is:

\[
S_z^{\text{total}} = S_z^A + S_z^B
\]

The possible measured values are:

  • \( +\hbar \) (Both particles spin up)
  • \( -\hbar \) (Both particles spin down)
  • \( 0 \) (One particle spin up, one spin down)

Projection Operators:

For these measurement outcomes, the projection operators are:

\[
P_{+\hbar} = |+\rangle_A |+\rangle_B \langle +|_A \langle +|_B
\]
\[
P_0 = |+\rangle_A |-\rangle_B \langle +|_A \langle -|_B + |-\rangle_A |+\rangle_B \langle -|_A \langle +|_B
\]
\[
P_{-\hbar} = |-\rangle_A |-\rangle_B \langle -|_A \langle -|_B
\]

Measurement Probabilities:

For the Bell state \( |\Phi^+\rangle \), we calculate:

\[
P(+\hbar) = \langle \Phi^+ | P_{+\hbar} | \Phi^+ \rangle = \frac{1}{2}
\]
\[
P(-\hbar) = \langle \Phi^+ | P_{-\hbar} | \Phi^+ \rangle = \frac{1}{2}
\]
\[
P(0) = \langle \Phi^+ | P_0 | \Phi^+ \rangle = 0
\]

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.