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(b) At a particular moment in time the state of rotation of the conden- sate is measured and it is found to be rotating counter-clockwise. After some time interval t has passed, we measure again. Find the probability P, (t) that the second mesaurement again finds the con- densate rotating counter-clockwise. How long should t be to max- imise the chances that the second measurement finds the condensate in a different rotational state?

(b) At a particular moment in time the state of rotation of the conden- sate is measured and it is found to be rotating counter-clockwise. After some time interval t has passed, we measure again. Find the probability P, (t) that the second mesaurement again finds the con- densate rotating counter-clockwise. How long should t be to max- imise the chances that the second measurement finds the condensate in a different rotational state?

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A three-state system. Consider an atomic condensate that can be described as a three-state quantum system in terms of its rotation around a particular axis. The possible states of rotation are the following: |R) |N) |L) | rotating counter-clockwise rotating clockwise not rotating (a) In the basis formed by the above three states, the Hamiltonian is represented by the following matrix: -2 1 ÎĤl = hw -2 -1 1 -1 1 where the angular frequency w > 0 is a known quantity. The three eigenvectors |0), |1), |2) of this matrix can be written as 1 |0) V2 1 ; 1) and |2) V3 in order of increasing eigenvalue. Obtain their respective eigenvalues Eo, E1, E2. (b) At a particular moment in time the state of rotation of the conden- sate is measured and it is found to be rotating counter-clockwise. After some time interval t has passed, we measure again. Find the probability P, (t) that the second mesaurement again finds the con- densate rotating counter-clockwise. How long should t be to max- imise the chances that the second measurement finds the condensate in a different rotational state? Hint: The evolution with time t of an energy eigenstate |n) is given by the formula |n (t)) = exp (-it) |n), where En is the correspond- ing energy eigenvalue.
Transcribed Image Text:A three-state system. Consider an atomic condensate that can be described as a three-state quantum system in terms of its rotation around a particular axis. The possible states of rotation are the following: |R) |N) |L) | rotating counter-clockwise rotating clockwise not rotating (a) In the basis formed by the above three states, the Hamiltonian is represented by the following matrix: -2 1 ÎĤl = hw -2 -1 1 -1 1 where the angular frequency w > 0 is a known quantity. The three eigenvectors |0), |1), |2) of this matrix can be written as 1 |0) V2 1 ; 1) and |2) V3 in order of increasing eigenvalue. Obtain their respective eigenvalues Eo, E1, E2. (b) At a particular moment in time the state of rotation of the conden- sate is measured and it is found to be rotating counter-clockwise. After some time interval t has passed, we measure again. Find the probability P, (t) that the second mesaurement again finds the con- densate rotating counter-clockwise. How long should t be to max- imise the chances that the second measurement finds the condensate in a different rotational state? Hint: The evolution with time t of an energy eigenstate |n) is given by the formula |n (t)) = exp (-it) |n), where En is the correspond- ing energy eigenvalue.
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