Solving the time dependant schrodinger eqn in dirac (bra ket) notation

[rwooduk]

given:

at t=0 |PSI(0)> = 1/2 |PSI1> + (SQRT3)/2 |PSI2>

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my attempt so far:

we can write

|PSI1> = 1/2 |UP> + 1/2 |DOWN>

|PSI2> = (SQRT3)/2 |UP> + (SQRT3)/2 |DOWN>

therefore

|PSI(0)> = 1/2 |UP> + 1/2 |DOWN> + (SQRT3)/2 |UP> + (SQRT3)/2 |DOWN>


but then what do i do? the lecturer hasnt shown us how to solve the time DEPENDENT schrodinger eqn in ket notation, infact i don't recall her showing us how to solve the time independant eqn either, only something like this:

|PSI(x)> = I(x)|PSI> = INTEGRAL BETWEEN -INF AND +INF of dx|x><x|PSI>

so do i use the Identity to give some sort of integral for |PSI(O)>??

I'm at a total loss and spent another 2+ hours going around in circles.

If someone could point me in the right direction, or even suggest a web page that details how to solve the time dependant schrodinger eqn in dirac notation, it would REALLY be appreciated.

edit

I found the solution on the web, well kind of, my question needs in terms of base vectors, but;

at t=o |PSI> = SUM Cn|PSIn>

at t>0 |PSI> = SUM Cn|PSIn> EXP (-iEt)/h-bar

where does the EXP term come from?? I know where it comes from in normal notation but how to get an exp term from |UP>'s and |DOWN>'s ?? and they are only vectors anyway, how can an EXP term be directional?? totally confused.

 

[rwooduk]

it's ok found something in the book

t=0 |PSI(0)> = SUM αi|PSIi>

t>0 |PSI(t)> = SUM αi EXP (-iEit/h-bar) |PSIi>

from

ih-bar d/dt |PSI> = H |PSI>therefore my answer (i think) will simply be

|PSI(t)> = (1/2 |PSI1> + (SQRT3)/2 |PSI2>)EXP (-iEit/h-bar)

= (1/2 (|UP> + |DOWN>) + (SQRT3)/2 (|UP> + |DOWN>) ) EXP (-iEit/h-bar)or maybe I am way off? will update when the answers come into help others

 

[vela]

The homework template is there for a reason. Could you please provide the complete problem statement?