Question

Problem 3: Two-state system Imagine a system where there are just two linearly independent states: |1) = (1 0) and |2) = (0 1). The most general state is a normalised linear cominbation given by |?) = a|1) + b|2), where |a|^2 + |b|^2 = 1. Suppose the Hamiltonian operator is given by the matrix: (h g g h) (a) Find the normalised states of definite energy. (b) What are the energies of these states? (c) Suppose the system starts out in |1) at t = 0. Derive two expressions of the state vector at time t using the energy eigenstates as the basis and using the |1) and |2) as the basis. (d) Find and plot the probability of finding the system in the states |1) and |2) as a function of time.

          Problem 3: Two-state system

Imagine a system where there are just two linearly independent states:
|1) = (1 0) and |2) = (0 1).

The most general state is a normalised linear cominbation given by
|?) = a|1) + b|2), where |a|^2 + |b|^2 = 1.

Suppose the Hamiltonian operator is given by the matrix:
(h g
g h)

(a) Find the normalised states of definite energy.
(b) What are the energies of these states?
(c) Suppose the system starts out in |1) at t = 0. Derive two expressions of the state vector at time t using the energy eigenstates as the basis and using the |1) and |2) as the basis.
(d) Find and plot the probability of finding the system in the states |1) and |2) as a function of time.

Problem 3: Two-state system

Imagine a system where there are just two linearly independent states:
|1) = (1 0) and |2) = (0 1).

The most general state is a normalised linear cominbation given by
|?) = a|1) + b|2), where |a|^2 + |b|^2 = 1.

Suppose the Hamiltonian operator is given by the matrix:
(h g
g h)

(a) Find the normalised states of definite energy.
(b) What are the energies of these states?
(c) Suppose the system starts out in |1) at t = 0. Derive two expressions of the state vector at time t using the energy eigenstates as the basis and using the |1) and |2) as the basis.
(d) Find and plot the probability of finding the system in the states |1) and |2) as a function of time.

Added by Amanda F.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Problem 3: Two-state system Imagine a system where there are just two linearly independent states: |1) and |2). The most general state is a normalized linear combination given by |Ψ) = a|1) + b|2), where |a|^2 + |b|^2 = 1. Suppose the Hamiltonian operator is given by the matrix: (a) Find the normalized states of definite energy. (b) What are the energies of these states? (c) Suppose the system starts out in |1) at t = 0. Derive two expressions of the state vector at time t using the energy eigenstates as the basis and using the |1) and |2) as the basis. (d) Find and plot the probability of finding the system in the states |1) and |2) as a function of time.