What Are the Dynamical Variables in Quantum Mechanics of Time?

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Discussion Overview

The discussion revolves around the role of time as a dynamical variable in quantum mechanics, comparing its function in non-relativistic quantum mechanics to that in classical mechanics. Participants explore definitions and implications of dynamical variables, parameters, and their relationships to observables.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that time plays the same role in quantum mechanics as in Newtonian mechanics, suggesting it is a dynamical variable.
  • Others argue that time is a parameter rather than a dynamical variable, noting that a dynamical variable satisfies an equation of motion and has a corresponding operator, which time does not.
  • One participant points out that generalized coordinates and momenta are examples of dynamical variables, while questioning the distinction between dynamical variables and generalized coordinates.
  • Another participant clarifies that in both classical mechanics and non-relativistic quantum mechanics, dynamical variables are functions of time, which is treated as an independent variable.
  • There is a discussion about the differences in how dynamical variables are treated in Hamiltonian and Lagrangian mechanics.

Areas of Agreement / Disagreement

Participants express differing views on whether time is a dynamical variable or a parameter, indicating a lack of consensus on this issue. Some agree on the definitions of dynamical variables, while others challenge the initial claims made about time.

Contextual Notes

Participants reference the definitions of dynamical variables and observables, highlighting the complexity of these concepts in both classical and quantum contexts. There are unresolved distinctions regarding the treatment of time and its implications in different mechanical frameworks.

Amir
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does anyone know?
time = ?
 
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Time plays the exact same role in (non-relativistic) quantum mechanics as it does in Newtonian mechanics. It's just a dynamical variable.

- Warren
 
Originally posted by chroot
Time plays the exact same role in (non-relativistic) quantum mechanics as it does in Newtonian mechanics.

That's true.

It's just a dynamical variable.

Nope, it's a parameter. A "dynamical variable" satisfies an equation of motion in classical mechanics, and its expectation value satisfies the same equation in nonrelativistic quantum mechanics (NRQM). Also, an operator can be constructed for any dynamical variable in NRQM, but there is no sensible way to construct an operator whose eigenvalue is time.
 
Originally posted by chroot
It's just a dynamical variable.
no t hat, remember?
 
Eek, you're right.

- Warren
 
I'm aware that time is not an observable, has no corresponding operator, and so on, of course. I just goofed up and forgot the definition of the phrase "dynamical variable."

- Warren
 
Originally posted by chroot
I'm aware that time is not an observable, has no corresponding operator, and so on, of course. I just goofed up and forgot the definition of the phrase "dynamical variable."

- Warren

Right, but I think we're talking over a lot of people's heads here. Rather than type out the math (still haven't mastered LaTeX), I posted a link to a reference.
 
Oh by the way...
A "dynamical variable" satisfies an equation of motion in classical mechanics
How does a dynamical variable then differ from a generalized coordinate (or velocity, or whatever)? Is a generalized coordinate an example of a dynamical variable?

- Warren
 
  • #10
Perimeter for what?
 
  • #11
Originally posted by chroot
Is a generalized coordinate an example of a dynamical variable?

Yes, generalized coordinates and generalized momenta together make up the set of dynamical variables.
 
  • #12
Originally posted by Amir
Perimeter for what?

No, it's not "peri[/color]meter", it's "para[/color]meter". In both classical mechanics and NRQM, the dynamical variables can be considered functions (dependent variables) of the parameter, time (the independent variable).

In relativity, position gets demoted to the status of a parameter as well.
 
  • #13
Originally posted by Tom
Yes, generalized coordinates and generalized momenta together make up the set of dynamical variables.
Great, thanks for clearing that up. I won't louse it up again. :wink:

- Warren
 
  • #14
oppsss brain to hand signaling problem ...
so is t constant, or relatively constant ?
 
Last edited:
  • #15
Originally posted by Tom
Yes, generalized coordinates and generalized momenta together make up the set of dynamical variables.

in Hamiltonian mechanics, yes, positions and conjugate momenta make up the dynamical variable.

in Lagrangian mechanics, its positions and velocities instead.
 

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