Wavefunction

A wavefunction is a complex amplitude whose squared modulus gives a probability density.

Wavefunction normalization makes the total Born rule probability equal to one. In an orthonormal basis of energy eigenstates, $\psi ={\sum }_n{c}_n{\varphi }_n$ is normalized precisely when ${\sum }_n|c_n{|}^2=1$.

A matter wave is the quantum wave associated with a material particle. Its phase can interfere after coherent paths recombine.

Matter-wave interference occurs when coherent spatial branches of a particle's wavefunction recombine and their relative quantum phase changes the detection probabilities.

The de Broglie wavelength of a particle with momentum magnitude $p$ is $\lambda =h/p$.

A neutron interferometer coherently splits and recombines neutron matter waves, allowing path-dependent phases to be measured through the output intensity.

The Colella–Overhauser–Werner experiment observed the relative phase acquired by neutron paths at different heights in the Earth's gravitational field.

A plane wave has constant amplitude and a phase linear in space and time. It is a momentum eigenstate and is not spatially normalizable.

A probability amplitude is a complex coefficient whose squared modulus gives the probability of the associated measurement outcome.

The Born rule assigns probability $|⟨a|\psi ⟩{|}^2$ to outcome $a$ when a normalized state $|\psi ⟩$ is measured in an orthonormal eigenbasis.

Every bijection of pure-state rays that preserves transition probabilities is induced by either a unitary or an antiunitary operator on the Hilbert space.

Because state vectors differing by a quantum phase represent the same ray, a unitary realization of a group action need only satisfy

An ideal energy measurement returns an eigenvalue of the Hamiltonian with probability equal to the squared norm of the state's projection onto its eigenspace. After obtaining a nondegenerate energy, the state is the corresponding energy eigenstate.

A wavefunction is normalizable when the integral of its squared modulus is finite, so multiplication by a constant can make its total probability one.

A probability density is a nonnegative function whose integral over an event gives its probability.

For a nonrelativistic particle of mass $m$, the probability current is

For a wavefunction satisfying the Schrodinger equation with a real potential,This continuity equation expresses local conservation of probability.

If the probability current of a normalizable wavefunction vanishes at spatial infinity, integrating the probability continuity equation over all space shows that the total probability $\int |\psi {|}^2$ is constant in time.

A Gaussian wave packet is a wavefunction whose spatial amplitude is a Gaussian function. Free quantum evolution preserves its Gaussian form while changing its complex width and spreading its probability density.