Quantum Mechanics: Understanding Operator Momentum and Position

Operator Momentum and Position in Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of particles at the smallest scales. In quantum mechanics, the mean value of an operator represents the expectation value of that operator over a given state. This mean value can evolve over time, and its temporal evolution can be determined using the Schrödinger equation and the commutation relation between the operator and the Hamiltonian.

The expression for the temporal evolution of the mean value of an operator O^ over a state described by the wavefunction ψ is given by:

d/dt ⟨O^⟩_ψ = ⟨ψ| (∂O^/∂t) + (iℏ/2) [O^, H] |ψ⟩

To demonstrate that the mean value of the operator momentum is given by the mass of the particle times the temporal derivative of the mean value of the operator position, we consider the momentum operator, p^, and the position operator, x^.

The momentum operator is given by p^ = -iℏ (∂/∂x), and the position operator is given by x^.

The commutation relation between the momentum and position operators is [x^, p^] = iℏ.

By substituting the momentum and position operators and their commutation relation into the expression for the temporal evolution of the mean value, we can simplify the expression and obtain the desired result.

This result shows that the mean value of the operator momentum is related to the mass of the particle and the temporal derivative of the mean value of the operator position.