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Homework Statement
Consider a linear harmonic oscillator with the solution defined by the ladder operators a and a†. Use the number basis |n⟩ to do the following.
a) Construct a linear combination of |0⟩ and |1⟩ to form a state |ψ⟩ such that ⟨ψ|X|ψ⟩ is as large as
possible.
b) Suppose that the oscillator is in the state constructed in a) at time t = 0. Write an expression to describe the time dependence of this state state for t > 0. Evaluate the expectation value ⟨ψ(t)|X|ψ(t)⟩ as a function of time for t > 0.
c) Defining (∆x)2 = ⟨ψ(t)|X2|ψ(t)⟩ − ⟨ψ(t)|X|ψ(t)⟩2. Calculate (∆x)2.
Homework Equations
The Attempt at a Solution
I have a hunch that for part a) I need to consider some |ψ⟩ = A|0⟩ + B|1⟩, then differentiate the expression ⟨ψ|X|ψ⟩. However I'm not sure how to implement that, any hints?
Many thanks