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    Eigenvalues, eigenvectors, and time evolution

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    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.

    © BrainMass Inc. brainmass.com May 20, 2020, 3:05 pm ad1c9bdddf
    https://brainmass.com/physics/scalar-and-vector-operations/eigenvalues-eigenvectors-and-time-evolution-113673

    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 ...

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