Time Dependent Schrodinger Equation -> T(t) solution

[karkas]

Could someone guide me step by step from the free SE to $T(t)=e^(iE_n t)/\hbar$ ?

I am not really familiar with PDEs of any kind and I would like slow step by step analysis! I am just confused by the great many ways of getting from there to there I find in books and Internet, so I would like someone to enlighten me!

Thanks in advance!

 

[Matterwave]

LaTeX isn't working, so I can't see what you're asking =/

 

[karkas]

Here http://img404.imageshack.us/img404/4383/testlike.png

This will explain it all to you.

 

[Matterwave]

Oh, simple. Seperate the wave-function into:
Psi(x)Phi(t) which can be done as long as the Hamiltonian is not (explicitly) time dependent.

From that, divide both sides by Psi(x)Phi(t) and you get a function on the left which is only of x, while a function on the left which is only of t. The only way that these two functions can be equal for all x and t is if they are both equal to the same constant.

We call this "seperation constant" E because it turns out this constant is the Energy. We then have a pretty simple equation:

d(phi(t))/dt=-iE/hbar

The solution to which is simple:
phi(t)=e^(-iEt/hbar)
It turns out then that all time-independent Hamiltonians have the same time-dependent part of the solution.
This is not the full solution because we don't have Psi(x) yet, but we can't get Psi(x) until we get the Hamiltonian.

 

[karkas]

Ok I get it. Now let me ask another thing.

Say I am stupid and I want to add another d/dt meaning I add another dimension in time. Does that mean that I finally get ih(d/dt1 + d/dt2)Phi = Phi * E ? (h-> hbar)

But then the solution I get from Mathematica is Phi(t1,t2) = C exp(-iEt1/h) (t2-t1). Is that mathematically correct? Can you even do t2-t1 in such a case?

 

[Matterwave]

Hmmm, I'm not sure you can obtain anything meaningful by doing that. Schroedinger's equation is non-relativistic so t1=t2 (there is absolute time). If you want to work with relativity where there are different times, you may have to go look into the Dirac-equation or the Klein-Gordon equation. I haven't studied relativistic quantum mechanics though so I can't be sure.

 

[karkas]

Oh..you got me out of a dead-end there, it seems :P
Gonna work with Klein-Gordon and see what stuff comes out. Thanks really really really much!

 

[Fredrik]

Actually, the only thing that's non-relativistic about the Schrödinger equation is that it assumes a non-relativistic form of the Hamiltionian (H=p²/2m+V).

I prefer to define H as the generator of translations in time, i.e. you write the time translation operator as U(t)=exp(-iHt), and take this to be the definition of H. This works in relativistic QM too.

If you write f(t)=exp(-iHt)f(0), you can easily see that this function satisfies idf/dt=Hf, which is the Schrödinger equation without the assumption H=p²/2m+V.

Two time dimensions would mean two time translation operators, and therefore two Hamiltonians and two Schrödinger equations.

 

[karkas]

What would be the way to create another Hamiltonian? Do we need to move in another field of Physics to do this or is it just imagination ? :P

 

[Fredrik]

Regarding the definition of a second Hamiltionian...I can't think of anything to add to what I already said in #8.

We can definitely talk about what QM would be like if we had another time dimension. It would be a theory that doesn't agree with experiments. Specifically, it would predict two different energy concepts, let's call them energy and shmenergy. The fact that shmenergy hasn't been detected in experiments falsifies the theory.

 

[karkas]

Your words are rough :) I guess you cancel out String Theory because of this huh?
Well it's natural to criticize theories via their behaviour in experiments, I agree with you.

 

[physlad]

Your words are rough :) I guess you cancel out String Theory because of this huh?
Well it's natural to criticize theories via their behaviour in experiments, I agree with you.

As far as I know, string theory does not introduce a new "time" dimension, only extra spatial dimensions. Also, the predictions of string theory cannot be tested by nowadays experiments. Its predictions take place at the Planck scale (~ 10^19 GeV). So, the theory cannot be proved nor falsified directly.

 

[karkas]

Yes what you say may indeed be true, but what would you say about a theory like Izthak Bars's 2T Physics, where he takes on a n=6 dimensional system, where there are 4 spatial and 2 time dimensions inside String Theory?

 

[Fredrik]

Yes what you say may indeed be true, but what would you say about a theory like Izthak Bars's 2T Physics, where he takes on a n=6 dimensional system, where there are 4 spatial and 2 time dimensions inside String Theory?

I'm not familiar with that theory, but that second time dimension must have some really weird properties.