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Quantum Mechanics

Momentum Operator

Operator

Section titled “Operator”

On the line, canonical momentum acts in coordinate representation as

p^=−iℏddx.\hat p = -i\hbar\frac{d}{dx}.

In three dimensions,

p^=−iℏ∇.\hat{\mathbf p} = -i\hbar\nabla.

Momentum generates translations:

U(a)=exp⁡(−iℏa⋅p^).U(\mathbf a) = \exp \left( -\frac{i}{\hbar}\mathbf a\cdot\hat{\mathbf p} \right).

For a plane wave,

p^ eikx=ℏk eikx.\hat p\,e^{ikx} = \hbar k\,e^{ikx}.

Assumptions

Section titled “Assumptions”
  • The derivative expression is paired with a domain and boundary condition.
  • The displayed operator is canonical momentum.
  • Momentum-space normalization follows the stated Fourier convention.
  • In electromagnetic fields, kinetic momentum differs from canonical momentum.

Common Mistakes

Section titled “Common Mistakes”
  • Treating −iℏd/dx-i\hbar d/dx as self-adjoint on any interval without boundary conditions.
  • Confusing canonical momentum p^\hat{\mathbf p} with kinetic momentum p^−qA\hat{\mathbf p}-q\mathbf A.
  • Dropping Fourier-transform convention factors when moving between xx and pp.
  • Treating momentum eigenstates on the line as square-normalizable.
Section titled “Canonical Links”

References

Section titled “References”
  • R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
  • M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.