Momentum operator

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The momentum operator is the quantum version of momentum. It tells us how quantum states change when momentum is involved.

- In one dimension (1D) and position space: p̂ = -iħ ∂/∂x. If ψ(x,t) is the wave function, then p̂ ψ = -iħ ∂ψ/∂x. In the momentum representation, the operator acts by multiplication: p̂ ψ(p) = p ψ(p).

- In three dimensions (3D): p̂ = -iħ ∇, where ∇ is the gradient. For a plane-wave state ψ(x,t) ∝ e^{i(p·r - Et)/ħ}, the momentum operator gives p̂ ψ = p ψ.

- Canonical vs kinetic momentum: For a charged particle in an electromagnetic field, the canonical momentum p̂ is not gauge invariant. The physically observable quantity is the kinetic momentum P̂ = -iħ ∇ - qA, where A is the vector potential. For electrically neutral particles, canonical and kinetic momentum coincide.

- Hermiticity and spectrum: The momentum operator is Hermitian (symmetric) and, on a suitable domain, self-adjoint. Its eigenvalues are real, and it is unbounded.

- Commutation with position and uncertainty: In 1D, [x̂, p̂] = iħ. This nonzero commutator leads to the Heisenberg uncertainty principle: you cannot precisely know position and momentum at the same time.

- Relation between position and momentum representations: The wave function in position space is ψ(x) = ⟨x|ψ⟩, which can be written as a Fourier transform of the momentum-space wavefunction. In position space, ⟨x|p̂|ψ⟩ = -iħ ∂ψ/∂x. In the momentum basis, ⟨p|x̂|ψ⟩ = iħ ∂ψ/∂p. These reflect how x and p swap roles as derivative and multiplication operators under the Fourier transform.

- Translation as a generator: The translation operator T(ε) shifts states by ε: T(ε)|ψ⟩ = ∫ dx |x+ε⟩⟨x|ψ⟩. For small ε, T(ε) ≈ 1 - (i/ħ) ε p̂, showing that momentum generates spatial translations.

- Relativistic extension: In relativistic quantum theory, energy and momentum are combined in the 4-momentum operator P̂μ. One common form is P̂μ = (E/c, -p) and, in natural units with time and space treated covariantly, P̂μ = iħ ∂μ. The Dirac operator uses γμ P̂μ = iħ γμ ∂μ, linking momentum and energy to relativistic wave equations.

In short, the momentum operator in quantum mechanics acts as a derivative in position space, as multiplication by p in momentum space, and it serves as the generator of translations, with deep connections to the structure of quantum states and their representations.