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Mathematics for Physics

138 Quantum States

A complete description of a quantum system that is used to compute the probabilities of every measurement, where a complete description is a wavefunction or a ket that determines all expectation values.

The Hilbert-space state. The state of a system is a vector in a Hilbert space. A Hilbert space is a complete inner-product space of possible states. This principle is used to add states and to compute inner products.

Ray equivalence. Two kets that differ by a nonzero complex factor represent the same physical state. This principle is used to work with normalized representatives.

The position-space wavefunction. The wavefunction \(\Psi(x,t)\) is the position representation of the state. This principle is used to compute position probabilities from \(|\Psi|^{2}\).

Superposition. A general state is a superposition of basis states. A superposition is a linear combination of allowed states. This principle is used to expand a state in the energy basis or the spin basis.

A general expansion is

\[ |\psi\rangle = \sum_{n}c_{n}|n\rangle \]

where

  • \(|\psi\rangle\) is the state.
  • \(|n\rangle\) are basis states.
  • \(c_{n}\) are complex coefficients.

The measurement update. A measurement of an observable yields one eigenvalue and leaves the system in the corresponding eigenstate. This principle is used to connect the abstract state to a laboratory outcome.

138.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — kets as states.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. — wavefunctions as states.