Consider a magnetic ion that can have its magnetic moment in one of two orientations: "up" or "down". Let us call | †) and | 4), respectively, the corresponding eigenstates of the orientation of the magnetic moment of the ion. Answer the following questions, displaying each step of your reasoning: (a) The ion is placed in the linear superposition state |V) = A (2| ↑) – i| 4)). (1) where A is a real, positive constant. This is an example of the Superposition Principle. State this principle. (b) With the ion in the state described by Eq. (I, the orientiation of the magnetic moment is measured. What are the probabilities that the measurement will find the magnetic moment pointing up and down, respectively? Express your answers as functions of A. (c) Determine A by requiring normalisation and re-express the proba- bilities obtained in the previous section as percentages.
Consider a magnetic ion that can have its magnetic moment in one of two orientations: "up" or "down". Let us call | †) and | 4), respectively, the corresponding eigenstates of the orientation of the magnetic moment of the ion. Answer the following questions, displaying each step of your reasoning: (a) The ion is placed in the linear superposition state |V) = A (2| ↑) – i| 4)). (1) where A is a real, positive constant. This is an example of the Superposition Principle. State this principle. (b) With the ion in the state described by Eq. (I, the orientiation of the magnetic moment is measured. What are the probabilities that the measurement will find the magnetic moment pointing up and down, respectively? Express your answers as functions of A. (c) Determine A by requiring normalisation and re-express the proba- bilities obtained in the previous section as percentages.
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Transcribed Image Text:Consider a magnetic ion that can have its magnetic moment in one of
two orientations: "up" or "down". Let us call | ↑) and | 4), respectively,
the corresponding eigenstates of the orientation of the magnetic moment
of the ion. Answer the following questions, displaying each step of your
reasoning:
(a) The ion is placed in the linear superposition state
|4) = A (2| ↑) – il 4))
(1)
where A is a real, positive constant. This is an example of the
Superposition Principle. State this principle.
(b) With the ion in the state described by Eq. (1), the orientiation of
the magnetic moment is measured. What are the probabilities that
the measurement will find the magnetic moment pointing up and
down, respectively? Express your answers as functions of A.
(c) Determine A by requiring normalisation and re-express the proba-
bilities obtained in the previous section as percentages.
(d) Let H be the Hamiltonian of our magnetic ion. Write the equation
an arbitrary state |4) has to satisfy for it to be an eigenstate of H.
(e) Let us now assume that H is described by the matrix
J
1
2i
2
-2i -2
in the basis formed by | t) and | 4). Here J > 0 is a known energy
scale. Express the state V) in Eq. (1) as a column vector in the
same basis and use that expression to prove that |V) is an eigenstate
of the energy, giving a formula for the corresponding eigenvalue E,
expressed as a function of J.
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