What Are the Dynamical Variables in Quantum Mechanics of Time?

[Amir]

does anyone know?
time = ?

 

[chroot]

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

- Warren

 

[quantumdude]

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.

 

[lethe]

Originally posted by chroot
It's just a dynamical variable.

no t hat, remember?

 

[chroot]

Eek, you're right.

- Warren

 

[quantumdude]

Originally posted by lethe
no t hat, remember?

For a quick, yet clear, discussion of why that is, check out page 2 of http://xxx.lanl.gov/PS_cache/hep-th/pdf/9505/9505152.pdf.

 

[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

 

[quantumdude]

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.

 

[chroot]

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

 

[Amir]

Perimeter for what?

 

[quantumdude]

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.

 

[quantumdude]

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.

 

[chroot]

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.

- Warren

 

[Amir]

oppsss brain to hand signaling problem ...
so is t constant, or relatively constant ?

 

[lethe]

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.