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In quantum mechanics, momentum is an observable represented by a vector operator. In the position basis, the momentum operator is given by [latex]\hat{\vec{p}} = -i\hbar\nabla[/latex], where [latex]\hbar[/latex] is the reduced Planck constant and [latex]\nabla[/latex] is the gradient operator. The 運動量保存の法則 corresponds to the fact that the ハミルトニアン operator commutes with the momentum operator, [latex][hat{H}, hat{vec{p}}] = 0[/latex], for a system with translational symmetry.
Quantum mechanics replaces classical observables with Hermitian operators acting on a ヒルベルト space of states. The momentum of a particle is no longer a simple number but an operator, [latex]\hat{\vec{p}}[/latex]. Its eigenvalues represent the possible outcomes of a momentum measurement. The famous expression [latex]\hat{\vec{p}} = -i\hbar\nabla[/latex] arises from the canonical commutation relation between the position operator [latex]\hat{\vec{x}}[/latex] and the momentum operator, [latex][\hat{x}_j, \hat{p}_k] = i\hbar\delta_{jk}[/latex], which is a fundamental postulate of quantum theory and the mathematical basis for the Heisenberg uncertainty principle.
The time evolution of the expectation value of an operator is governed by the Ehrenfest theorem. For momentum, it shows that the expectation value of momentum changes according to the expectation value of the force, mirroring Newton’s second law. A quantity is conserved if its operator commutes with the Hamiltonian [latex]hat{H}[/latex], the operator for total energy. If a system’s potential energy is independent of position (i.e., the system has translational symmetry), then [latex][hat{H}, hat{vec{p}}] = 0[/latex], and the expectation value of momentum is conserved. This re-establishes the connection between translational symmetry and momentum conservation, as seen via Noether’s theorem, within the quantum framework.
量子運動量演算子
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