Introduction: The Rules of the Game
Quantum mechanics rests on a small set of postulates—axioms that can't be proven but lead to experimentally verified predictions. Here we state them precisely.
The Postulates
- State Postulate: A quantum system is completely described by its state vector \(|\psi\rangle\) in Hilbert space.
- Observable Postulate: Every measurable quantity corresponds to a Hermitian operator.
- Measurement Postulate: Measurement yields an eigenvalue; the state collapses to the corresponding eigenstate.
- Probability Postulate: The probability of outcome \(a\) is \(|\langle a|\psi\rangle|^2\).
- Evolution Postulate: States evolve via \(i\hbar\partial_t|\psi\rangle = \hat{H}|\psi\rangle\).
The Wavefunction
In position representation, the state becomes the wavefunction:
\[\psi(x, t) = \langle x|\psi(t)\rangle\]It contains all information about the system.
Worked Example
A particle is in state \(|\psi\rangle = \frac{1}{\sqrt{2}}(|E_1\rangle + |E_2\rangle)\).
- Measure energy → get \(E_1\) or \(E_2\), each with probability 1/2
- After measuring \(E_1\), state collapses to \(|E_1\rangle\)
- Subsequent energy measurements give \(E_1\) with certainty
The Quantum Connection
These postulates are the foundation. Everything else—energy levels, tunneling, entanglement—follows from applying these rules to specific Hamiltonians. The postulates are minimal: no interpretation is assumed, just the mathematical machinery needed to make predictions.