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.

 

[Demystifier]

The phase of the wave function is analogous to the Hamilton-Jacobi function $S(x,t)$ in classical mechanics. If $S(x,t)$ is a solution of the HJ equation (which is the classical analogue of the Schrodinger equation), then $S'(x,t)=S(x,t)+\theta$ is also a solution, with the same physical content.

 

[QuarkyMeson]

There are classical mechanical formalisms based on Hilbert spaces like this: Koopman–von Neumann classical mechanics.

As a result, the phase of a classical wave function does not contribute to observable averages. Contrary to quantum mechanics, the phase of a classical wave function is physically irrelevant.

 

[hokhani]

What causes the phase difference between two non-orthogonal quantum states to increase? Or is any physical interpretation for phase difference between non-orthogonal quantum states?

 

[QuarkyMeson]

What causes the phase difference between two non-orthogonal quantum states to increase? Or is any physical interpretation for phase difference between non-orthogonal quantum states?

You need to be specific on what kind of phase you're talking about.

Like @PeterDonis said: A single pure state has no physical absolute phase, since a pure state is a ray in Hilbert space, $|\gamma\rangle$ and $e^{i\theta}|\gamma\rangle$ are the same physical state.

A physically meaningful phase appears only in a relational context, for example between components of the same coherent superposition, or in an interferometric comparison between two non-orthogonal states. In that context you can discuss the phase associated with their overlap. How such a phase evolves depends on the dynamics and on how the comparison is defined.

An important example is the geometric phase, which plays a role in condensed matter physics, quantum optics, and quantum information. Most of the fault tolerant QI research these days is very concerned with geometric phases.