A wave function is a mathematical representation of a pure state in quantum mechanics. Usually denoted by (\psi) or (\Psi), it assigns complex-valued probability amplitudes to possible configurations or measurement outcomes. Its squared magnitude determines measurement probabilities, while its phase affects interference. For a single spinless particle in the position representation, (\psi(\mathbf r,t)) depends on position and time; for several particles, it depends on their combined configuration rather than simply on ordinary three-dimensional space. (feynmanlectures.caltech.edu)
Mathematical representation
A wave function is not identical to the abstract quantum state: it expresses that state in a particular representation. In Hilbert space, a pure state is represented by a normalized vector (|\psi\rangle). Its position-space wave function is
[ \psi(\mathbf r)=\langle\mathbf r|\psi\rangle. ]
Choosing a different basis gives different amplitudes for the same state. A momentum-space wave function, for example, describes the amplitudes associated with possible momentum measurements. Systems with discrete outcomes can instead be represented by a sequence of complex coefficients. (ocw.mit.edu)
Multiplying an entire normalized state by a constant phase factor (e^{i\theta}) does not change its physical predictions. Relative phases between components, however, can change measurement probabilities. A general statistical mixture cannot be represented by one pure-state wave function; it requires a density matrix, which also accommodates pure states. (feynmanlectures.caltech.edu)
Probability and normalization
The Born rule connects wave functions with probability. For a spinless particle in three dimensions,
[ \rho(\mathbf r,t)=|\psi(\mathbf r,t)|^2 =\psi^*(\mathbf r,t)\psi(\mathbf r,t) ]
is the position probability density, where the asterisk denotes complex conjugation. The probability of finding the particle in a region (R) is the integral
[ P(R,t)=\int_R|\psi(\mathbf r,t)|^2,d^3r. ]
Thus, the value of (|\psi|^2) at a point is a density, not itself the probability of an exact position. (ocw.mit.edu)
An ordinary normalized wave function satisfies
[ \int_{\mathbb R^3}|\psi(\mathbf r,t)|^2,d^3r=1. ]
For a position-space wave function in three dimensions, normalization gives (\psi) dimensions of length to the power (-3/2). Ideal plane waves extending throughout infinite space are not square-integrable; they are treated using generalized normalization or combined into normalizable wave packets. Appropriate Schrödinger evolution preserves normalization. (ocw.mit.edu)
Time evolution and stationary states
In nonrelativistic quantum mechanics, the Schrödinger equation governs wave-function evolution:
[ i\hbar\frac{\partial\psi}{\partial t} =\hat H\psi. ]
Here (\hat H) is the Hamiltonian operator, representing the system’s total energy, and (\hbar=h/(2\pi)) is the reduced Planck constant. For a spinless particle of mass (m), without electromagnetic vector-potential coupling,
[ \hat H=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r,t), ]
where (V) is potential energy and (\nabla^2) is the Laplacian. Initial conditions, the Hamiltonian, and suitable boundary conditions determine the evolution. (ocw.mit.edu)
For a time-independent Hamiltonian, an energy eigenfunction satisfies
[ \hat H\phi_n=E_n\phi_n. ]
The associated time-dependent solution is (\psi_n(\mathbf r,t)=\phi_n(\mathbf r)e^{-iE_nt/\hbar}). Although its phase evolves, its position probability density remains unchanged. Such a solution is called a stationary state. The energies are eigenvalues of the Hamiltonian; general states need not possess a single definite energy. (feynmanlectures.caltech.edu)
Superposition, interference, and momentum
The linearity of quantum dynamics permits superposition. A state can be expressed as a linear combination
[ \psi=\sum_n c_n\phi_n. ]
In an orthonormal measurement basis, (|c_n|^2) gives the probability of the corresponding outcome. Amplitudes, rather than probabilities, are added when alternatives remain coherent and indistinguishable. Consequently,
[ |\psi_1+\psi_2|^2 =|\psi_1|^2+|\psi_2|^2 +2\operatorname{Re}(\psi_1^*\psi_2). ]
The final term produces interference, making the relative phase experimentally consequential. This distinguishes coherent superposition from an ordinary statistical mixture. (feynmanlectures.caltech.edu)
Position and momentum wave functions are related by a Fourier transform. A narrowly localized wave packet generally requires a broad range of momentum components. This relationship underlies the position–momentum uncertainty principle, commonly written (\Delta x,\Delta p_x\geq\hbar/2), with each uncertainty defined as a standard deviation. It is a constraint on quantum states, not merely a limitation of measuring instruments. (ocw.mit.edu)
Many-particle states and internal degrees of freedom
An (N)-particle position wave function has the form
[ \Psi(\mathbf r_1,\ldots,\mathbf r_N,t). ]
It therefore depends on (3N) spatial coordinates. Additional variables or components describe internal properties such as spin. For an electron, for example, a complete description may require two spin components, not just one scalar spatial function. (feynmanlectures.caltech.edu)
A joint pure-state wave function that cannot be factored into separate subsystem wave functions represents entanglement. Predictions must then be obtained from the joint state; assigning an independent pure wave function to each subsystem is generally inadequate. (feynmanlectures.caltech.edu)
For identical particles in three-dimensional space, exchanging two particles’ complete sets of coordinates leaves bosonic wave functions unchanged and reverses the sign of fermionic wave functions. Fermionic antisymmetry implies the Pauli exclusion principle: two identical fermions cannot occupy the same one-particle state. These symmetry requirements constrain many-particle states even before their detailed dynamics are solved. (feynmanlectures.caltech.edu)
Historical development
Erwin Schrödinger developed wave mechanics in 1926, building on ideas about the wave nature of matter. His equation converted problems such as atomic energy levels into eigenvalue problems. In the same year, Max Born introduced the statistical interpretation of wave-function amplitudes in scattering. This established the distinction between the complex amplitude and the probability obtained from its squared magnitude. Born’s statistical interpretation was explicitly recognized in his share of the 1954 Nobel Prize in Physics. (nobelprize.org)