Harmonic oscillator special state (QM)

[faen]

Homework Statement

\psi(x,0) = N exp[-\alpha(x-a)^2]

(1):This wavefunction is a solution to the time dependent schrödinger equation for a harmonic oscillator, but not to the time independent one. How is that possible?

(2):Explain without calculating how would you find the time dependent wave function \psi(x,t)?

Homework Equations

The Attempt at a Solution

(1)According to my book it seems to explain that any solution to the schrödinger equation can be explained by a linear combination of the basis states.

I would assume that if \psi(x,t) is a linear combination of time dependent basis states, that \psi(x,0) would also be a linear combination of time independent basis states, and that \psi(x,t) consist of the same basis states as \psi(x,0) except that each basis state is multiplied with the time dependent factor exp[-iEn*t/h]. In that case, \psi(x,0) would be a solution, but it is not. So I am stuck in this contradiction.

(2)To find the time dependent state, i would try to identify the basis time independent states describing the wave function. Then multiply each basis state with the corresponding exp[-iEn*t/h] factor. However i can't identify any basis time independent states if any in \psi(x,0).

 

[Mindscrape]

I think that really what 1) wants you to say is that the solution to the time independent schrödinger equation is just one part of the separable solution, and you need to express the other separable part for the whole thing.

Question two is probably just asking about how you solve for the coefficients and wants you to talk about linear combinations and completeness.

 

[faen]

Thats what i thought too, but i can't find any other separable states in the wave function in the form of a linear combination. Also if the linear combination of the states were a solution to the time dependent SE, wouldn't the same linear combination (without time factors) have to be a solution to the time independent SE? Feels like I am stuck in some kind of paradox :p

The whole state seem to be in the state of the harmonic oscillator ground state at specific time tho, but otherwise its traveling. I suspect that has something to do with it perhaps.

 

[Mohandas]

Yo dude. Did you find N? I just copied some crap from the book and got (pi*h_bar/mk)^1/4, assuming i could normalize the first eq. on this page for a=0. Kinda hard to think straight right now

 

[Mohandas]

Actually in 1a) i assume that E = hw/2 in one of the steps to show that 'a' must be zero. But then they say 'find E for a=0'.. .. ?

The way i see it, it's not a solution of the time ind. SE because it can't be solved for any other case than a=0, but again, I am assuming the E=hw/2

Still trying to figure out what to say on d)... but hey its only 4am, plenty of time still! :)

 

[Mohandas]

in b) Psi(x,t) for a=0 is just (eq. 4) * exp(-iEt/h_bar), where eq. 4 is the first eq on this page.

?

 

[faen]

in b) Psi(x,t) for a=0 is just (eq. 4) * exp(-iEt/h_bar), where eq. 4 is the first eq on this page.

?

...

 

[Mohandas]

But how can there be a general case for a? The time independent SE is only valid for a=0, so what are the other Psi_n(x) that you can tack exp(-i*E_n*t/h_bar) to? I'm not sure this makes any sense, but I am too tired to care now..

 

[faen]

Yeah exactly same thing which doesn't make sense to me :/

 

[Mohandas]

In which case the only state is the ground state, with 100% probability to find E=hw/2 at t=0 ?

 

[faen]

Hm not sure about that one either :/

Ill take the night and go to bed, 4:40 am. Gl and good nite dude :)