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Quantum Mechanics, time independant solution in Dirac notati

  1. Mar 15, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider the Hamiltonian:
    $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$
    where $C$ is a constant and the magnetic field is given by

    $$\vec{B} = (0,B,0) $$

    and the spin is

    $$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$

    with


    $$\hat{S}_{x} =\frac{\hbar}{2}∣↑⟩⟨↓∣+ \frac{\hbar}{2}∣↓⟩⟨↑∣ $$


    $$\hat{S}_{y} =−\frac{i\hbar}{2}∣↑⟩⟨↓∣+ \frac{i\hbar}{2}∣↓⟩⟨↑∣ $$


    $$\hat{S}_{z} =\frac{\hbar}{2}∣↑⟩⟨↑∣− \frac{\hbar}{2}∣↓⟩⟨↓∣,$$

    where the basis vectors are assumed orthogonal and normalised. Knowing that at t=0

    $$ ∣ψ(0)⟩= \frac{1}{2}∣ψ_1⟩+ \frac{\sqrt{3}}{2}∣ψ_2⟩$$

    where the ${∣ψ_i⟩}$ are the eigenvectors of the Hamiltonian, solve the time-dependent Schrodinger equation. Write the solution in terms of the basis vectors $∣↑⟩$ and $∣↓⟩$.


    2. Relevant equations


    3. The attempt at a solution
    From the question I understand that the Hamiltonian will cancel out the x and z spin operators through the dot product. I have tried following this forward and using a presumed general solution to the time dependent Schrodinger equation but I got no where. Could someone just explain to me how to look at attempting this question?
     
    Last edited: Mar 15, 2015
  2. jcsd
  3. Mar 15, 2015 #2

    DrClaude

    User Avatar

    Staff: Mentor

    I would start by calculating the eigenstates of ##\hat{H}##.
     
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