In quantum mechanics, the wave function Ψ(x, t) describes the state of a quantum system as a function of position (x) and time (t). The dimension of the wave function depends on the system being considered.
For a single particle in one-dimensional space, the wave function Ψ(x, t) is a complex-valued function, and its dimension is usually represented as [L]^-1/2, where [L] denotes the dimension of length. This means that the wave function has units of inverse square root of length, such as [m]^-1/2 or [cm]^-1/2, depending on the choice of units.
For a system with multiple particles or in higher-dimensional spaces, the dimension of the wave function becomes more complicated. In three-dimensional space, for example, the wave function Ψ(x, y, z, t) has dimensions of [L]^-3/2, with units such as [m]^-3/2 or [cm]^-3/2.
It's important to note that the wave function itself is not directly observable, but its square modulus, |Ψ(x, t)|^2, gives the probability density of finding a particle at a specific position and time. The normalization condition ensures that the integral of |Ψ(x, t)|^2 over all space is equal to 1, ensuring the probabilities sum up to unity.