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I just want to ask a simple question about quantum mechanics...

This question appears in it's original form here:

http://www.chriscentral.com/physics/PC3101.PDF (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.

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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.

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Any ideas? Thanks! Chris.

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# Simple quantum exam question

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