Wavefunction

A wavefunction is a basis-dependent function that represents a pure quantum state. For a state vector $\lvert\psi\rangle$ and a continuous generalized basis $\{\lvert q\rangle\}$,

$$\psi(q)=\langle q\vert\psi\rangle.$$

The abstract state and its wavefunction are not two physical states. They are the same state described without and with a chosen representation.

In Symbols

If the basis resolves the identity with measure $d\mu(q)$,

$$\int \lvert q\rangle\langle q\rvert\,d\mu(q)=I,$$

then

$$\lvert\psi\rangle =\int \psi(q)\lvert q\rangle\,d\mu(q).$$

For a normalized state,

$$\int \lvert\psi(q)\rvert^2\,d\mu(q)=1.$$

The probability of an outcome in a measurable region $\Delta$ is

$$\Pr(q\in\Delta) =\int_\Delta \lvert\psi(q)\rvert^2\,d\mu(q),$$

when the basis labels the outcomes of the corresponding ideal measurement. The probability density is defined relative to the measure; $\lvert\psi(q)\rvert^2$ alone does not specify probabilities until $d\mu(q)$ is known.

Position Representation

For one spinless particle on a line,

$$\psi(x)=\langle x\vert\psi\rangle, \qquad \int_{-\infty}^{\infty} \lvert\psi(x)\rvert^2\,dx=1.$$

Because the integral is dimensionless,

$$[\psi(x)]=[\text{length}]^{-1/2}.$$

In $d$ spatial dimensions, the position-space wavefunction has units $[\text{length}]^{-d/2}$ under the Cartesian measure $d^dx$.

Momentum Representation

The same state has momentum-space wavefunction

$$\widetilde\psi(p)=\langle p\vert\psi\rangle.$$

With the convention

$$\langle x\vert p\rangle =\frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar},$$

the two representations are related by

$$\widetilde\psi(p) =\frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar}\psi(x)\,dx.$$

Other Fourier conventions move factors of $2\pi$ and $\hbar$ between the kernel, measure, and normalization. They represent the same state when used consistently.

Time Dependence

In the Schrödinger picture,

$$\Psi(q,t)=\langle q\vert\psi(t)\rangle.$$

For an energy eigenstate with time-independent Hamiltonian,

$$\Psi_n(q,t) =e^{-iE_nt/\hbar}\psi_n(q).$$

Its probability density is stationary even though the wavefunction carries a time-dependent phase. A general superposition has relative phases that evolve and can change observable probabilities.

Canonical Home

See Wavefunctions as Representations for the canonical abstract treatment of bases, measures, generalized eigenvectors, and representation changes.

Coordinate Representation develops the position-basis machinery used in wave mechanics. Wavefunctions and Probability Density focuses on normalization and spatial probabilities.

Context

In a discrete orthonormal basis $\{\lvert n\rangle\}$, the analogous representation is the component sequence

$$\psi_n=\langle n\vert\psi\rangle, \qquad \sum_n\lvert\psi_n\rvert^2=1.$$

The word wavefunction is most often reserved for a continuous representation, but the mathematical idea is the same: a vector becomes components after a basis is chosen.

The wavefunction contains amplitude and phase information. Multiplying the entire function by one constant phase,

$$\psi(q)\longmapsto e^{i\alpha}\psi(q),$$

does not change the physical ray. Position-dependent or component-dependent phase changes generally do affect the state unless they are accompanied by the corresponding change of basis or gauge transformation.

Measure and Coordinate Dependence

In spherical coordinates, normalization is

$$\int_0^\infty \int_0^\pi \int_0^{2\pi} \lvert\psi(r,\theta,\phi)\rvert^2 r^2\sin\theta\, d\phi\,d\theta\,dr =1.$$

The Jacobian belongs to the measure. Calling $\lvert\psi(r,\theta,\phi)\rvert^2$ the probability density is incomplete unless one states whether the density is with respect to $dr\,d\theta\,d\phi$ or the physical volume element $d^3x$.

Changing coordinates can therefore change the displayed function, its units, and the integration measure without changing the underlying state or probabilities.

Hilbert-Space Precision

Ordinary square-integrable wavefunctions belong to an $L^2$ space. Two functions that differ only on a set of measure zero represent the same $L^2$ vector. A point value such as $\psi(x_0)$ is not an invariant property of a general equivalence class unless additional regularity or a preferred representative has been specified.

Position eigenstates and plane waves are not normalizable elements of $L^2(\mathbb R)$; they are generalized states used in spectral expansions. Normalizable wave packets are built from them through integrals.

Boundary conditions and operator domains matter. The same differential expression can define different quantum operators on different domains, so an acceptable wavefunction must satisfy the conditions associated with the chosen Hamiltonian.

Configuration Space and Components

For $N$ particles in three dimensions,

$$\psi(\mathbf x_1,\ldots,\mathbf x_N)$$

is a function on the $3N$-dimensional configuration space, not a scalar field on ordinary three-dimensional space. For identical particles it must also obey the appropriate exchange symmetry.

Internal degrees of freedom add components. A spin-$1/2$ position-space wavefunction may be written

$$\psi(\mathbf x) = \begin{pmatrix} \psi_\uparrow(\mathbf x)\\ \psi_\downarrow(\mathbf x) \end{pmatrix},$$

with normalization

$$\int d^3x\, \left( \lvert\psi_\uparrow(\mathbf x)\rvert^2 +\lvert\psi_\downarrow(\mathbf x)\rvert^2 \right) =1.$$

The component ordering is part of the basis convention.

Pure and Mixed States

One normalized wavefunction represents a pure state, up to global phase. A general mixed state requires a density operator $\rho$. Although a mixed state can be written as an ensemble of wavefunctions,

$$\rho=\sum_jp_j \lvert\psi_j\rangle\langle\psi_j\rvert,$$

that ensemble decomposition is not unique. No single ordinary wavefunction generally contains the same information as $\rho$.

Minimal Example

The normalized Gaussian

$$\psi(x) =\frac{1}{\pi^{1/4}\sqrt{s}} \exp\left(-\frac{x^2}{2s^2}\right)$$

has density

$$\lvert\psi(x)\rvert^2 =\frac{1}{\sqrt\pi\,s} \exp\left(-\frac{x^2}{s^2}\right).$$

The density has units of inverse length, and

$$\int_{-\infty}^{\infty} \lvert\psi(x)\rvert^2\,dx=1.$$

Its position variance is $s^2/2$. The parameter $s$ is therefore not the standard deviation; this illustrates why the definition of a width parameter must be checked rather than inferred from the exponent.

Common Aliases

• state function;
• probability amplitude in a specified representation;
• orbital, in restricted one-particle or mean-field contexts;
• coordinate-space wavefunction, when the basis is position.

An orbital is not always the complete many-body wavefunction, and a field configuration in quantum field theory is not automatically a single-particle wavefunction.

Common Confusions

• The wavefunction is not the abstract state itself; it is a representation.
• $\lvert\psi(x)\rvert^2$ is a density, not the probability of exactly one point.
• Momentum-space and position-space wavefunctions represent the same state in different bases.
• The wavefunction itself is generally complex; its absolute square supplies a probability density for the declared measurement.
• A global phase does not matter, but relative and spatially varying phases can matter.
• Normalization depends on the measure and coordinate Jacobian.
• A many-particle wavefunction lives on configuration space.
• Spin and other internal degrees of freedom can make the wavefunction vector-valued.
• A general mixed state is not represented by one wavefunction.
• Plane waves and position eigenstates are generalized, not square-integrable, wavefunctions.
• Not every arbitrary $L^2$ function lies in the domain of every observable or Hamiltonian.

Related Entries

• Born Rule
• Hilbert Space
• Superposition
• Density Matrix
• Momentum-Space Representation
• Rays and Global Phase
• L2 Spaces
• Common Symbols Index

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, chs. 1 and 2.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 1, 4, and 5.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, ch. 1.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, chs. 2, 5, and 7.