The Wave Function

2023-01-10T00:00:00

physics quantum mathematics

The quantum state of a particle is described by its wave function $\Psi(x,t)$ , which contains all the information about the particle's quantum state.

Key properties of the wave function:

Normalization: $\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$
Probability density: $|\Psi(x,t)|^2$ gives the probability of finding the particle at position $x$ at time $t$
Complex valued: $\Psi(x,t)$ is generally a complex function

The wave function evolves according to the time-dependent Schrödinger equation:

i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \hat{H}\Psi(x,t)

where $\hat{H}$ is the Hamiltonian operator representing the total energy of the system.

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