Notes on Superposition and Quantum States

Superposition in Quantum Mechanics

Superposition is a fundamental concept in quantum mechanics.
When two states are superimposed, the resulting state behaves differently than classical averaging; results are probabilistic.
Measurement yields results corresponding to one of the states with certain probabilities.

Key Assumption: Superimposing a state with itself does not change the physics.
- Mathematical Representation:
  - State A is physically equivalent to (A + A) = 2A, -A, or iA.
  - Not equivalent to 0A (the zero state).
- Non-zero coefficients maintain physical equivalency.
- Importance of choosing a convenient form of the state (normalized state).

Light states can be represented by polarization states.
Example: Two types of photons -
- Polarized along the x-axis (state X).
- Polarized along the y-axis (state Y).
General photon state:
- Expressed as: |ψ⟩ = α|X⟩ + β|Y⟩
- Parameters: Has 2 complex parameters (α, β), equating to 4 real parameters.
The overall coefficient doesn’t matter, allowing simplification to:
- |ψ⟩ = (β/α)|Y⟩
- Results in 1 complex parameter (γ = β/α), thus the most general state is characterized by just 2 real parameters (shape of ellipse).

The most general state of polarization of a wave is elliptical polarization.
Key characteristics:
- Two Parameters:
  - Ratio of semi-major axes (a/b).
  - Angle θ of the ellipse.
Size is not a parameter; it depends on the magnitude of the electric field.

Spin: Intrinsic angular momentum of elementary particles.
Measured in specific directions, e.g., z-direction yielding:
- Spin Up: |↑⟩
- Spin Down: |↓⟩
Superposition of spin states:
- Example: |↑⟩ + |↓⟩.
Normalization factor may apply when calculating probabilities.

Measuring spin in z-direction on 1000 particles results in:
- Approximately 50% up and 50% down when measured multiple times.
Einstein’s critique of quantum superposition:
- Suggests realism: If spin is measured as up, it was up before measurement.

Quantum state ensemble vs. classical ensemble (50% up, 50% down):
- When measuring along x-direction:
  - Quantum state ensemble: All pointing up along x.
  - Classical ensemble: 50% up, 50% down along x.
The ability to distinguish these states can indicate the existence of quantum states.

The discussion of superpositions and spins emphasizes the non-classical nature of quantum mechanics, the roles of assumptions, and implications for measurement and observation.