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Mathematics for Physics

155 Wavefunctions

A complex-valued function that is used to describe the physical state of a quantum system, where the absolute square of the function is the probability density of finding the particle.

The wavefunction as the state. The state of a particle in one dimension is a wavefunction \(\Psi(x,t)\). This principle is used to replace a classical trajectory by a function of position and time.

The Born rule. The probability of finding the particle in an interval is the integral of \(|\Psi|^{2}\) over that interval. This principle is used to compute all position probabilities from one function.

The Born rule is

\[ P(a\leq x\leq b) = \displaystyle\int_{a}^{b}\lvert\Psi(x,t)\rvert^{2}\,dx \]

where

  • \(P\) is the finding probability.
  • \(\Psi\) is the wavefunction.

Normalization. The wavefunction must be normalizable so that the total probability is one. This principle is used to discard solutions that grow at infinity.

The normalization condition is

\[ \displaystyle\int_{-\infty}^{\infty}\lvert\Psi(x,t)\rvert^{2}\,dx = 1 \]

where

  • \(\Psi\) is the wavefunction.

Schrödinger evolution. The wavefunction evolves according to the Schrödinger equation. This principle is used to compute \(\Psi\) at a later time.

Irrelevance of a global phase. A global phase factor \(e^{i\alpha}\) does not change any probability. This principle is used to treat wavefunctions that differ by a constant phase as the same physical state.

155.1 References

  1. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §1.2–1.4 — wavefunctions and the statistical interpretation.
  2. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — position-space wavefunction.