1) Wavefunction as an amplitude

• The wavefunction Ψ is not a probability distribution; it is a complex probability amplitude.
• For a single particle, |Ψ(x)|² gives the probability density of finding the particle at position x.
• The phase of Ψ encodes relative information important for interference; it has no direct probability meaning by itself.

So:

• Amplitude = a complex number attached to each possible configuration.
• Probability = squared modulus of that amplitude.

2) Entangled pair case

The joint wavefunction lives in a tensor-product space: Ψ(xA, xB).

• The amplitude Ψ(xA, xB) is the complex probability amplitude for “A at xA and B at xB simultaneously.”
• With entanglement, the amplitude cannot factor into single-particle pieces:
  Ψ(xA, xB) ≠ ψA(xA) ψB(xB)
• Instead, it encodes joint correlations (e.g., large amplitudes only for correlated spin outcomes).

3) What the amplitude represents physically

• An amplitude is the contribution of a specific configuration (e.g., “A here with spin-up, B there with spin-down”) to the overall quantum state.
• The magnitude squared of that amplitude is the probability of observing that configuration (in the chosen measurement basis).
• Relative phases between different configurations govern interference and determine correlation patterns under basis changes.

Example (Bell state):

|Ψ⟩ = (1/√2) ( |↑⟩A|↓⟩B − |↓⟩A|↑⟩B )

• Amplitude for |↑A↓B⟩ is +1/√2.
• Amplitude for |↓A↑B⟩ is −1/√2.
• Each squared magnitude equals 1/2 (equal probabilities).
• The relative minus sign is crucial: it controls the type of correlation seen when measuring along different axes.

State: |Ψ⟩ = (1/√2) ( |0⟩A|1⟩B − |1⟩A|0⟩B )

Key takeaways

• The wavefunction assigns a complex amplitude to each joint configuration.
• Probabilities come from squared magnitudes of those amplitudes.
• Relative phases (like the minus sign above) do not change raw probabilities for a fixed basis, but they do determine interference and correlation behavior when you change the measurement basis.
• Entanglement appears because this table cannot be factored into a product of “A-only” and “B-only” parts.