simple quantum mechanics exam question

Hi,

I just want to ask a simple question about quantum mechanics...
This question appears in it's original form here:
[Only registered users see links. ] (question 2)

I have had a go and put some notes into the text below... I am stuck on
parts c) (ii) and (iii).
Any help much appreciated! Thanks.
_______________________________________________

A quantum mechanical system is described by a two-dimensional state
space spanned by the orthonormal vectors |1> and |2>. The Hamiltonian
of the system can be defined by:
H|1> = a|1> + 2a|2>
H|2> = 2a|1> + a|2>

with "a" a positive constant.

(a) Write down the matrix representation of H in the basis {|1> |2>}.

Easy:
(a 2a)
(2a a )


(b) Find the energy eigenvalues of the systems and associated
orthonormal eigenvectors.

I get {3, 1/root2*(1,-1)} and {-1,1/root2*(1,1)}, where this has been
expressed in the form {eigenvalue,eigenvector}.


(c) If at time t = 0 the system is described by the state vector |2>:

(i) Find the mean value of the energy <H> at time t = 0.

I get <H> = a.

(ii) What is the probability that the system will be found having its
highest possible energy at time t = 0?

(iii) If the highest energy is measured at time t = 0, calculate the
state vector of the system at time t.
_________________________________

Any ideas? Thanks! Chris.

Hi, Chris,




Haven't you forgotten something here?


Do you know what the equation is for the variation of psi with time?
Write it down, solve it, and then find the values of the unknown
coefficients by equating the general solution at t=0 to the known
initial condition.


The correct answer to this question is clearly:

Wonderful, wonderful Copenhagen
Friendly old girl of a town
'Neath her tavern light
On this merry night
Let us clink and drink one down
To wonderful, wonderful Copenhagen
Salty old queen of the sea
Once I sailed away
But I'm home today
Singing Copenhagen, wonderful, wonderful
Copenhagen for me
I sailed up the Skagerrak
And sailed down the Kattegat
Through the harbor and up to the quay
And there she stands waiting for me
With a welcome so warm and so gay
Wonderful, wonderful Copenhagen
Wonderful, wonderful Copenhagen
Friendly old girl of a town
'Neath her tavern light
On this merry night
Let us clink and drink one down
To wonderful, wonderful Copenhagen
Salty old queen of the sea
Once I sailed away
But I'm home today
Singing Copenhagen, wonderful, wonderful
Copenhagen for me

Cite it in full and you will get full marks, but do not so much as
breathe the words "Hund's paradox" or the name "Shahriar S. Afshar".
Cheers,

Zigoteau.