Momentum Operator

The symbols $\hat p$ and $\hat{\mathbf p}$ denote momentum observables. In ordinary nonrelativistic quantum mechanics, canonical momentum is the self-adjoint generator of spatial translations. Its familiar derivative form is a representation of that generator, not a complete definition independent of domain and boundary conditions.

Default Meaning

On the real line in position representation,

$$(\hat p\psi)(x) = -i\hbar \frac{d\psi}{dx}(x).$$

In three Cartesian dimensions,

$$\hat{\mathbf p} = -i\hbar\nabla,$$

or componentwise,

$$\hat p_i = -i\hbar \frac{\partial}{\partial x_i}.$$

The bold symbol is a vector of three operators. Each component is an observable, and the Cartesian components commute in the absence of gauge-covariant replacements:

$$[\hat p_i,\hat p_j]=0.$$

Mathematical Type and Domain

On $L^2(\mathbb R)$, the standard momentum operator is self-adjoint on the Sobolev-space domain

$$\mathcal D(\hat p) = H^1(\mathbb R) = \left\lbrace \psi\in L^2(\mathbb R) : \psi'\in L^2(\mathbb R) \right\rbrace,$$

where $\psi'$ may be understood as a weak derivative. The operator is unbounded and has continuous spectrum $\mathbb R$.

The formal eigenvalue equation is

$$\hat p\lvert p\rangle = p\lvert p\rangle.$$

Momentum eigenkets on the line are generalized eigenvectors rather than normalizable Hilbert-space vectors. With delta normalization,

$$\langle p\rvert p'\rangle = \delta(p-p'), \qquad \int_{-\infty}^{\infty} \lvert p\rangle \langle p\rvert\,dp =I.$$

The same letter $p$ labels the generalized ket and denotes its real eigenvalue. The hat and ket delimiters carry the type information.

Units and Wave Number

Momentum has dimensions

$$[\hat p] = MLT^{-1}.$$

For a plane wave with wave number $k$,

$$\hat p\,e^{ikx} = \hbar k\,e^{ikx},$$

so

$$p=\hbar k.$$

Wave number has units of inverse length; momentum does not. In units with $\hbar=1$, the numerical distinction disappears, but it must be restored for dimensional calculations. If the Fourier kernel is written using spatial frequency $\nu_x$ rather than angular wave number $k$, then $p=h\nu_x$.

Position and Momentum Representations

Use the convention

$$\langle x\rvert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} \exp\left( \frac{ipx}{\hbar} \right).$$

For

$$\widetilde\psi(p) = \langle p\rvert\psi\rangle,$$

the momentum operator acts by multiplication:

$$(\hat p\widetilde\psi)(p) = p\widetilde\psi(p).$$

The position operator in the same convention is

$$(\hat x\widetilde\psi)(p) = i\hbar \frac{d\widetilde\psi}{dp}(p).$$

Reversing the signs in the Fourier kernels reverses the associated derivative signs. A derivative formula should never be transferred between sources without checking the transform convention.

The probability of obtaining momentum in a region $\Delta$ is

$$\Pr(p\in\Delta) = \int_\Delta \lvert\widetilde\psi(p)\rvert^2\,dp.$$

Its spectral projector is

$$P_p(\Delta) = \int_\Delta \lvert p\rangle \langle p\rvert\,dp.$$

Generator of Translations

For the active convention,

$$T(a) = \exp\left( -\frac{ia\hat p}{\hbar} \right)$$

acts on wavefunctions as

$$(T(a)\psi)(x) = \psi(x-a).$$

A wavepacket centered at $x_0$ is thereby moved to $x_0+a$. Consistently,

$$T^\dagger(a)\hat xT(a) = \hat x+aI.$$

In three dimensions,

$$T(\mathbf a) = \exp\left( -\frac{i}{\hbar} \mathbf a\cdot\hat{\mathbf p} \right).$$

Some sources define passive translations instead. Their exponent or wavefunction argument may carry the opposite sign. The operator, wavefunction action, and transformed position must be compared as a set.

Canonical Commutators

The canonical Cartesian relations are

$$[\hat x_i,\hat p_j] = i\hbar\delta_{ij}I,$$ $$[\hat x_i,\hat x_j]=0, \qquad [\hat p_i,\hat p_j]=0.$$

In one dimension,

$$[\hat x,\hat p] = i\hbar I.$$

These relations imply the Robertson uncertainty bound

$$\Delta x\,\Delta p \ge \frac{\hbar}{2}$$

for states in the domains needed to define the variances and commutator. Because $\hat x$ and $\hat p$ are unbounded, the commutator is not an unrestricted matrix identity valid on every vector.

Canonical and Kinetic Momentum

For a particle of charge $q$ in a vector potential $\mathbf A$, the canonical momentum is $\hat{\mathbf p}$, while the kinetic or mechanical momentum is

$$\hat{\boldsymbol{\pi}} = \hat{\mathbf p} -q\mathbf A(\hat{\mathbf r},t).$$

The minimally coupled Hamiltonian is

$$H = \frac{ \hat{\boldsymbol{\pi}}^2 }{2m} +q\phi(\hat{\mathbf r},t).$$

For this nonrelativistic Hamiltonian, the velocity operator is

$$\hat{\mathbf v} = \frac{\hat{\boldsymbol{\pi}}}{m}.$$

Canonical momentum generates ordinary coordinate translations. Kinetic momentum is directly related to velocity and is gauge covariant. In a magnetic field its components generally fail to commute:

$$[\hat\pi_i,\hat\pi_j] = iq\hbar \epsilon_{ijk}B_k(\hat{\mathbf r},t).$$

Thus replacing $\hat{\mathbf p}$ by $m\hat{\mathbf v}$ is valid only under the relevant Hamiltonian and coupling assumptions.

Conservation

Momentum conservation is tied to translation symmetry. If

$$[H,\hat{\mathbf p}]=0$$

and $\hat{\mathbf p}$ has no explicit time dependence, then its expectation value is constant.

For

$$H = \frac{\hat p^2}{2m} +V(\hat x),$$

the Heisenberg equation gives

$$\frac{d}{dt} \langle\hat p\rangle = -\left\langle \frac{dV}{dx}(\hat x) \right\rangle.$$

A free particle or a spatially translation-invariant system conserves momentum. A potential that depends on position generally does not. In a background gauge field, the conserved generator may be canonical momentum, pseudomomentum, or a magnetic-translation generator depending on the symmetry.

Finite Intervals and Boundaries

The expression $-i\hbar\,d/dx$ does not define the same operator on every configuration space. On an interval $[0,L]$, a family of self-adjoint momentum operators is obtained from quasiperiodic boundary conditions

$$\psi(L) = e^{i\theta}\psi(0), \qquad 0\le\theta<2\pi.$$

The corresponding eigenvalues are

$$p_n = \frac{\hbar}{L} \left( 2\pi n+\theta \right), \qquad n\in\mathbb Z.$$

Periodic boundary conditions are the case $\theta=0$. Dirichlet conditions at both endpoints make the derivative expression symmetric on a restricted domain but do not make it the same self-adjoint momentum operator used on the line.

On a half-line, the minimal symmetric first-derivative operator has unequal deficiency indices and no self-adjoint extension. This is one reason that “momentum on a half-line” cannot be treated as a routine restriction of full-line momentum.

Crystal and Quasimomentum

In a periodic lattice, Bloch states are labeled by crystal momentum

$$\hbar\mathbf k.$$

The wavevector $\mathbf k$ is defined modulo a reciprocal-lattice vector:

$$\mathbf k \sim \mathbf k+\mathbf G.$$

Crystal momentum is a translation quantum number, not generally the expectation value of mechanical momentum. It can be conserved modulo reciprocal lattice vectors even when continuous spatial translation symmetry is absent.

Relativistic Notation

Relativistic momentum is assembled into a four-vector, commonly

$$p^\mu = \left( \frac{E}{c}, \mathbf p \right),$$

with signs in $p_\mu$ depending on the metric convention. The Dirac contraction is written

$$p\!\!\!/ \equiv \gamma^\mu p_\mu.$$

The four-momentum, the three-momentum operator, and its numerical eigenvalue may all be denoted by variants of $p$. Indices, boldface, hats, and context must be read together.

Typography and Symbol Collisions

Form or context	Usual meaning
$p$	Momentum eigenvalue, classical momentum, or operator with hat suppressed
$\hat p$	One-dimensional momentum operator
$\mathbf p$	Three-momentum vector or vector eigenvalue
$\hat{\mathbf p}$	Vector of momentum operators
$\lvert p\rangle$	Generalized momentum eigenket
$p_i$ in Hamiltonian mechanics	Canonical momentum conjugate to $q^i$
$p(x)$ in probability	Probability density or mass function
$p^\mu$	Relativistic four-momentum
$\hbar\mathbf k$	Crystal momentum or plane-wave momentum, depending on context
$P$	Often total momentum or a translation generator

Lowercase $p(x)$ in a probability chapter usually means a probability distribution and has no operator hat. Capital $P$ may mean momentum, parity, a projector, or probability. The surrounding equation should identify its mathematical type.

Hat Policy

This reference uses hats when an operator could be confused with an eigenvalue or classical variable. Many texts instead write

$$p=-i\hbar\frac{d}{dx}$$

and let the equation reveal that $p$ is an operator. In momentum representation, the same glyph then appears on both sides:

$$(p\widetilde\psi)(p) = p\widetilde\psi(p).$$

Keeping a hat on the operator makes the left and right roles explicit, but either convention is sound when used consistently.

Canonical Home

Canonical Commutation Relations owns the position-momentum algebra. Momentum Operator as Generator derives the derivative representation and fixes the active translation convention.

The longer Momentum operator card collects operational facts. Momentum Eigenstates treats generalized plane-wave states, and Fourier-Transform Conventions fixes normalization and signs. Minimal Coupling develops canonical and kinetic momentum in gauge fields.

Convention Warnings

• $\hat p$ is canonical momentum unless a source states otherwise.
• Canonical momentum and kinetic momentum differ in a vector potential.
• Momentum and wave number differ by $\hbar$.
• Plane waves on the line are generalized eigenstates, not square-normalizable states.
• Fourier-transform signs determine derivative-operator signs.
• The differential expression $-i\hbar\,d/dx$ is incomplete without a domain.
• Boundary conditions change self-adjointness and the momentum spectrum.
• Crystal momentum is defined modulo reciprocal lattice vectors and need not equal mechanical momentum.
• Active and passive translation conventions use different-looking signs.
• $p(x)$ often denotes a probability distribution rather than momentum.
• $p^\mu$ is a four-vector; metric conventions determine the lowered components.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 1, 4, and 12.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chs. 1 and 4.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, vol. I, Wiley, 1977, complements A and B.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975, chs. IX and X.