Problem 2
Consider a hypothetical quantum system that has three allowed energy states, denoted as |E1⟩, |E2⟩ and |E3⟩, which are assumed be orthonormal. Before we measure the energy of the system, the system is in a state described by an unnormalized state,
|\psi ̃⟩ = 3 |E1⟩ + 4 |E2⟩ − 2i |E3⟩ . (2)
(a) What is the probability to find the particle having an energy E3 when we measure the system’s energy?
(b) Suppose right after the first measurement, we find the system having an energy E1. What is the probability to find the system having an energy E2 if we make a second energy measurement?
(c) Find the normalized state |\psi ⟩ from the unnormalized state |\psi ̃⟩.
(d) The expectation value of the energy, denoted as ⟨E⟩, is defined as
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⟨E⟩ ≡ XP(Ei)Ei, (3) i=1
where P(Ei) is the probability to find the system having and energy Ei. Find the expectation value of the energy when the system is in the state |\psi ⟩.
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(e) The allowed energy states are eigenstates of the Hamiltonian operator Hˆ. That is, Hˆ |Ei⟩ = Ei |Ei⟩. Show that in quantum mechanics, for a generic normalized quantum state
|\psi ⟩ = X ai |Ei⟩ (4) i
where ai are complex coefficients, ⟨\psi | Hˆ |\psi ⟩ is equal to the expectation value of the energy.