The equivalent concept of phase change in classical mechanics

[hokhani]

TL;DR

How to make correspondence for phase change in classical mechanics?

In quantum mechanics phase change, as a coefficient $e^{i\theta}$, would not change the quantum state. I would like to know whether we have such a concept for classical systems.

 

[Nugatory]

Something similar happens with waves in classical physics: the phase is determined by our choice of where we choose $t=0$. I'm not sure how illuminating this correspondence is.

 

[hokhani]

Something similar happens with waves in classical physics: the phase is determined by our choice of where we choose $t=0$. I'm not sure how illuminating this correspondence is.

Also in quantum, the time evolution of an eigenstate appears as phase coefficient, and it seems that the phase change to be related to the origin of time.

 

[Haborix]

The closest parallel I can think to draw is to canonical transformations in classical mechanics. Strictly speaking, a phase change does change the quantum state, but it leaves other things invariant. Similarly, canonical transformations may change the phase space coordinates, but they leave the form of Hamilton's equations the same.

 

[PeterDonis]

Strictly speaking, a phase change does change the quantum state

Not in the usual formulation of non-relativistic QM, where a "quantum state" is a ray in Hilbert space, not a vector. Multiplying a vector in Hilbert space by $e^{i \theta}$ gives another vector in the same ray, so it doesn't change the quantum state.

 

[Haborix]

Not in the usual formulation of non-relativistic QM, where a "quantum state" is a ray in Hilbert space, not a vector. Multiplying a vector in Hilbert space by $e^{i \theta}$ gives another vector in the same ray, so it doesn't change the quantum state.

You're correct. I got muddled up thinking about a ray in terms of real numbers multiplying a vector.