Wavefunction

A wavefunction is a representation of a quantum state in position space. For a particle on a line, write it as

$$\psi (x,t).$$

The value of $\psi$ is a complex amplitude, not a directly observed quantity. The probability density for finding the particle near $x$ is

$$|\psi (x,t){|}^2.$$

Normalisation requires

$${\int }_{-\infty }^{\infty }|\psi (x,t){|}^{2}dx=1.$$

How to read it

The wavefunction stores both magnitude and phase. Magnitude controls probabilities; relative phase controls interference. The same abstract state in Hilbert space can be represented in other bases, such as momentum space.

Dynamics

The Schrödinger equation tells how the wavefunction changes in time. For bound systems, allowed standing-wave patterns often produce discrete energy values, as in the Quantum harmonic oscillator.

Common pitfalls

• Do not confuse amplitude $\psi$ with probability $|\psi {|}^2$.
• A global phase does not change probabilities, but relative phase can change interference.
• Sharp position means spread-out momentum content, connecting to the Uncertainty principle.

Source trail: Susskind The Theoretical Minimum index; reference book: Quantum Mechanics: The Theoretical Minimum by Leonard Susskind and Art Friedman.