Question

Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?₁ and ?₂ are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states.

a) Write down the expression for Ψ(?,?).

b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² .

c) Calculate 〈?〉 at t>0 .

d) Calculate 〈?〉 at t>0 .

e) Check that Ehrenfest’s theorem holds for this wave function. (Note that Ehrenfest theorem states that ? /?? 〈?〉 = 〈− ?? /?? 〉 )

Q4) Use the recursion formula ?_m+2 = (2(?−?)) /((?+2)(?+1))?_m, to work out ?₄ (?) and ?₅ (?) .

Answer 1

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please check the calculations once and upload the last ques separately.