4. Velocity in Quantum Mechanics

Is Velocity Meaningless in Quantum Mechanics?

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1. Classical Trajectory vs. Quantum Uncertainty

In classical Newtonian mechanics, a particle has simultaneously well-defined position $x(t)$ and velocity $v(t) = \frac{dx}{dt}$ .

In quantum mechanics, position $\hat{x}$ and momentum $\hat{p}$ are represented by non-commuting Hermitian operators:

[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar

According to the Robertson-Schrödinger uncertainty relation:

\sigma_x \sigma_p \ge \frac{\hbar}{2}

Because an exact position measurement ( $\sigma_x \to 0$ ) forces infinite momentum dispersion ( $\sigma_p \to \infty$ ), the concept of an instantaneous path velocity $\frac{\Delta x}{\Delta t}$ is fundamentally ill-defined.

2. Expectation Values & Ehrenfest’s Theorem

While point-by-point trajectory ceases to exist, the average or expectation value of momentum obeys a quantum correspondence principle:

\langle p \rangle = \int_{-\infty}^{\infty} \psi^*(x) \left( -i\hbar \frac{\partial}{\partial x} \right) \psi(x) \, dx

By Ehrenfest’s Theorem, the time derivative of the average position recovers classical velocity:

\frac{d\langle x \rangle}{dt} = \frac{\langle p \rangle}{m}