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Eigenvalues, eigenvectors, and time evolution

Dear Mitra,

I in fact wrote the wrong matrix but I am still confused after PART A.

(PART A)

The actual matrix is:

H = 1 2 0
2 0 2
0 2 -1

Where the eigenvalues are E1 =0, E2=3hw, E3=-3hw and as you said the trace(H) =0 = sum of eigenvalues.

I also found the eigenvectors using Hx = Ex and they were:

for E1 =0 : |v1> = <-1, 1/2, 1>

for E2 =3: |v2> = <2, 2, 1>

for E3 =-3: |v3> = <1/2, -1, 1>

After this, I am confused. Do I need to normalize those eigenvectors?

if so then I get the normalized vectors |v1>, |v2>,|v3> to be:

|v1> = <-2/3 , 1/3, 2/3>
|v2> = <2/3 , 2/3, 1/3>
|v3> = <1/3 , -2/3, 2/3>

(PART B)

Now I write the initial state as a linear combination of the normalized eigenvectors:

|x> = |v1> + |v2> +|v3>

We are given that at t=0, our initial state is 1
0
0

So as a linear combination:

<1,0,0> = (-2/3)<-2/3 , 1/3, 2/3> + (2/3)<2/3 , 2/3, 1/3> + (1/3)<1/3 , -2/3, 2/3>

=(-2/3)|v1> + (2/3)|v2> +(1/3)|v3>

After this I am totally lost. The question asks me to find the state vector at time t. Can you explicitly do it so I can see. Thanks.

(PART C)

A = 1 0 0
0 2 0
0 0 0

We need to find its expectation at time t and also compute the probability that measuring A at time t gives 1.

My main problem is understanding how the eigenvectors and eigenvalues of the Hamiltonian matrix relate to the different probabilities and states.

Solution Preview

To be compatible with the Dirac notation you should write vectors like e.g.

|-2/3 , 1/3, 2/3>

The dual vector is written the other way around:

<-2/3 , 1/3, 2/3|

You can think of vectors as column vectors and dual vectors as row vectors. An inner product is then just a matrix product of the two. The notation <A> means the average of the operator A. The average of A in some state |x> is given by:

<x|A|x>

Now back to your problems:

You found that:

|x> = (-2/3)|v1> + (2/3)|v2> +(1/3)|v3>

You must indeed normalize the ...

Solution Summary

A detailed solution is given.

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