# Eigenvalues, eigenvectors, and time evolution

A. Given matrix A = 1 2 0 where i found the eigenvalues to be 3hw, 0, -3hw.

2 0 2

0 2 -2

then if at t=0 state-vector is 1 THEN compute the state vector at time t.

0

0

Please explain me how to get this, Im very confused. I know its some sort of matrix multiplication but why?

b. If A = 1 0 0 is an observable. FIND its expectation at time t and compute the

0 2 0

0 0 0

probability that measuring A at time t gives the result 1.

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#### Solution Preview

In problem a) the matrix A is presumably the Hamiltonian up to some factor. First you must compute the eigenvalues. You either did that incorrectly or you wrote down A incorrectly. You can see that at once without doing any computations as follows. The trace of a matrix (this is the sum of the elements on the diagonal) is invariant under transformations of the basis vectors. This means that the trace must equal the sum of the eigenvalues. The trace of A is -1, but your set of eigenvalues sums to 0.

Suppose you find the correct eigenvalues and ...

#### Solution Summary

A detailed solution is given.

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.

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