Wavefunction

In quantum mechanics, a wavefunction $Ψ : 𝐷 \to ℂ$ is a description of the quantum mechanical state of an isolated quantum system. Here $𝐷$ is the space of all degrees of freedom. The wavefunction must satisfy Schrödinger equation, which given an initial condition describes the evolution of the wavefunction over time.

A wavefunction is said to be normalized iff $\int 𝐷 Ψ (𝑥, 𝑡) * Ψ (𝑥, 𝑡) 𝑑 𝑥 = 1$ for all $𝑡$ . Since the Quantum time evolution operator is unitary, this automatically holds for all $𝑡$ . According to Born's statistical interpretation, for a normalized wavefunction $Ψ (𝑥, 𝑡) * Ψ (𝑥, 𝑡)$ gives a probability density of observing the system in state $𝐷$ .¹ All scalar multiples of a normalized wavefunction represent the same physical state, and non-normalizable wavefunctions must be rejected as nonphysical.²

Basis

Wavefunction in position-space

#state/develop | #lang/en | #SemBr

2018. Introduction to Quantum Mechanics, §1.2, p. 17 ↩
It begs the question, why consider non-normalized wavefunctions at all? The answer is that this is required to make the space of wavefunctions into a vector (Hilbert) space, which brings the machinery of linear algebra into the game. ↩

Wavefunction

Basis

Footnotes