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3. Consider the quantum dynamics of a qubit whose Hamiltonian is given in the matrix representation [H1 H2] = ah (2) H21 H2 where Hmn = (m|Ħ|n) under the orthonormal basis vectors |1) and |2). (Note they are NOT related to SHO in this problem.) Here a is a real parameter and h is the Planck constant. At t = 0, the system is prepared in the following quantum state: |V) = [2) (3) (a) Find the energy eigenvalues and their corresponding normalized eigenvectors. Write down the eigenvectors in terms of [1) and |2). (b) Find the state voctor li(t)\ for t 0 hon compu the ovnoctation voluo of

3. Consider the quantum dynamics of a qubit whose Hamiltonian is given in the matrix representation [H1 H2] = ah (2) H21 H2 where Hmn = (m|Ħ|n) under the orthonormal basis vectors |1) and |2). (Note they are NOT related to SHO in this problem.) Here a is a real parameter and h is the Planck constant. At t = 0, the system is prepared in the following quantum state: |V) = [2) (3) (a) Find the energy eigenvalues and their corresponding normalized eigenvectors. Write down the eigenvectors in terms of [1) and |2). (b) Find the state voctor li(t)\ for t 0 hon compu the ovnoctation voluo of

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3. Consider the quantum dynamics of a qubit whose Hamiltonian is given in the matrix representation Han H [Hu H12] = ah (2) where Hmn = (m|H|n) under the orthonormal basis vectors |1) and |2). (Note they are NOT related to SHO in this problem.) Here a is a real parameter and h is the Planck constant. At t = 0, the system is prepared in the following quantum state: |V) = |2) (3) (a) Find the energy eigenvalues and their corresponding normalized eigenvectors. Write down the eigenvectors in terms of |1) and |2). (b) Find the state vector |V(t)) for t > 0, and then compute the expectation value of S = |1)(2| + |2)(1| in this state, that is (S) = (V(t)|S|V(t)), as a function of t.
Transcribed Image Text:3. Consider the quantum dynamics of a qubit whose Hamiltonian is given in the matrix representation Han H [Hu H12] = ah (2) where Hmn = (m|H|n) under the orthonormal basis vectors |1) and |2). (Note they are NOT related to SHO in this problem.) Here a is a real parameter and h is the Planck constant. At t = 0, the system is prepared in the following quantum state: |V) = |2) (3) (a) Find the energy eigenvalues and their corresponding normalized eigenvectors. Write down the eigenvectors in terms of |1) and |2). (b) Find the state vector |V(t)) for t > 0, and then compute the expectation value of S = |1)(2| + |2)(1| in this state, that is (S) = (V(t)|S|V(t)), as a function of t.
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