Explaining Time Dependent Potentials

[pmiranda]

Hello,

I am trying to self learn a little bit of quantum mechanics in order to describe the magnetic resonance phenomenon. I am following Griffiths book and i am understanding most of it.

Now, there is a particular thing that is bogging me.

The Schrödinger equation can be easily solved by separation of variables if the potential does not depend on time, which i understand as:
$\frac{\partial V(x,t)}{\partial t}=0$

in other words it can simply be
$V(x,t)\rightarrow V(x)$

Now i can think of a many motions where the potential is constant at any given position in time. For instance the Harmonic oscillator or an electron orbiting.

Now i can't imagine in my naive approach, a possible scenario where the potential changes with time without changing the position of a particle.

Is there any easy explanation for that?

Ty in advance

 

[Amok]

Now i can't imagine in my naive approach, a possible scenario where the potential changes with time without changing the position of a particle.

Ty in advance

Not sure I understand that question.

If you want a simple example of a potential that varies with time, you can think of any kind of applied electric field. For example one that depends linearly on time or more realistically one that oscillates with a given frequency (a model for monochromatic light). Potentials that are time-dependent are usually so-called "external" potentials because they do not have their origin in the particles themselves (such as coulomb attraction between two charged particles). Maybe that's why you're having a hard time imagining such a potential.

 

[pmiranda]

I didn´t though of it as an "external" thing. That way it makes sense now!
Thanks a lot

 

[meldraft]

An example of a potential that varies over time but leaves the particle in its position would be the potential in the center of the axes, while two charges in opposite sides of the diameter of a circle around the center of the axes, are rotating in circular trajectory.

 

[Amok]

I didn´t though of it as an "external" thing. That way it makes sense now!
Thanks a lot

Yes, a potential like that can be a function that depends explicitly on time. That doesn't mean that potential energy will not vary (the expecation value) with time if the operator V isn't a function of t. I guess the confusion comes from the fact in classical physics, in the case of a harmonic oscillator for example, V would depend implicitly on time, since it would be a function of x(t). In QM the operator V(x) is not a function of time (for the harmonic oscillator), but the expectation value <V> could depend on time since V does not commute with H. Check out the Ehrenfest theorem:

http://en.wikipedia.org/wiki/Ehrenfest_theorem

However, if you are in a stationary state <V> will indeed be time independent, which to me was always kinda weird (when compared to classical mechanics). But I guess that comes from the fact there are such things as stationary states in QM and no trajectories.