Quantum mechanics - Heisenberg picture

[alc95]

Homework Statement

We consider the O2- molecule, with the Hamiltonian and position operator having matrix representations in terms of the Pauli matrices:

In the Heisenberg picture, the position operator is:

(1) Find the eigenvalues and eigenstates of x(t) at time t=pi*hbar/(4A)

(2) The state |-a> remains the same in the Heisenberg picture. Find the probabilities of measuring the two eigenvalues of x(t) at time t=pi*hbar/(4*A)

(3) Now switching to the Schrodinger picture, find the state at time t = pi*hbar/(4A) for initial state |−a>

(4) In the Schrodinger picture the operator x is constant. Find the probabilities of measuring eigenvalues +a and −a of x at time t = pi*hbar/(4*A), given initial state |−a>. Compare these to the result from question 2.

Homework Equations

The equations given above, plus the Born rule for calculating probabilities.

The Attempt at a Solution

(1) This part was ok, I got ± a for the eigenvalues, and the corresponding eigenvectors:
(1/sqrt(2))*[1,∓ i]

(2)
the probabilities will be:
[tex]
\begin{equation}
|<psi|\lambda_+>|^2
\end{equation}
[/tex]
and
[tex]
\begin{equation}
|<psi|\lambda_->|^2
\end{equation}
[/tex]
where the lambdas are the eigenstates corresponding to the positive and negative eigenvalues. Psi is the state of the system, although I'm unsure of what it will be.

(3)
I'm not exactly sure how to interpret this question. Should I try to find an expression for psi in terms of |-a>?

(4)
Am I on the right track with the following?
The probabilities will be:
[tex]
\begin{equation}
|<-a|lambda_+>|^2
\end{equation}
[/tex]
and
[tex]
\begin{equation}
|<-a|lambda_->|^2
\end{equation}
[/tex]

Any help is appreciated :)

 

[anathema]

For part (2), you are correct in using the Born rule to calculate the probabilities. However, you also need to consider the state of the system at time t = pi*hbar/(4A). This can be found by applying the time evolution operator, which in this case is just the identity matrix since the operator x is constant in the Heisenberg picture. So the state of the system at time t = pi*hbar/(4A) is just |-a>. Then you can plug this into the Born rule to calculate the probabilities.

For part (3), you are correct that you need to find an expression for the state of the system in terms of |-a>. This can be done by applying the time evolution operator, which in this case is given by exp(-iHt/hbar), where H is the Hamiltonian. You can use the matrix representations of the Hamiltonian and position operator to calculate this.

For part (4), you are on the right track with using the Born rule again. However, note that in the Schrodinger picture, the state of the system changes with time, so you need to use the time evolution operator to find the state at time t = pi*hbar/(4A). Then you can plug this state into the Born rule to calculate the probabilities of measuring the eigenvalues +a and -a.

Hope this helps!