Motion/momentum of a wave packet

Question

I'm reading Dirac's "Principles of quantum mechanics" right now, being a little confused about the following part: (Chapter V$\S$31: "The Motion of Wave Packets").

He's making the following ansatz:

"Suppose that the time-dependent wave function is of the form $$\psi(\vec{q},t) = Ae^{iS/\hbar}\tag{35},$$ where $A$ and $S$ are real functions of $\vec{q}$ and $t$."

Schrödingers Wave equation gives us $$i\hbar\frac{\partial}{\partial t}Ae^{iS/\hbar}=H(\vec{q},\vec{p})Ae^{iS/\hbar}.$$

Then letting $\hbar \rightarrow 0$, and setting $\vec{p} = \nabla_{\vec{q}} S$, and assuming $\hbar\nabla_{\vec{q}}A \ll A \nabla_{\vec{q}} S$, he derives that

$\frac{d\vec{p}}{dt} = - \nabla_{\vec{q}}H(\vec{q},\vec{p})$.

My question is: How can i make sense of defining $\vec{p} = \nabla_{\vec{q}} S$, dealing with wave packets. I see that in operator terms $<\vec{p}> = \int (A^2\nabla_{\vec{q}} S)dq$, but im wondering if there's some way to interpret $\vec{p}$ for example as a pointwise momentum of the wave packet?

Generally speaking: Is there some other way to talk about momentum or velocity of a wave packet besides the mean value described above?

Answer 1

If you are familiar with the Hamilton Jacobi formulation of classical mechanics, this relationship is natural. $S$ is the action variable. The wavepacket assumption is to obtain Hamilton’s equations of motion for a well defined position and momentum. In general, you would get the classical Hamilton-Jacobi equation. You can find a short introduction of this semiclassical approach in V. Arnold’s Mathematical Methods of Classical Mechanics, Appendix 11 “Short Wave Asymptotics.”

This is analogous to to the eikonal approximation in optics. In the limit of short wavelengths, wave optics including diffraction becomes ray optics. The action variable becomes the optical path and momentum becomes light momentum (usually index of refraction times velocity).

Hope this helps.

Answer 2

im wondering if there's some way to interpret p⃗ for example as a pointwise momentum of the wave packet?

I believe interpreting p⃗ as a pointwise momentum of, say, the center of a symmetric wavepacket then, you are approximating classical mechanics, no?

This invokes the idea that the center of the wavepacket moves with the group velocity, and I believe your specific question about momentum is addressed by Ehrenfrost theorem.

In the second link above momentum is derived in part from the Schrodinger equation in certain conditions to equal the classical mechanics definition of momentum.