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1. Consider the problem of a particle of mass m moving in the double oscillator potential V(x) = ½ k ( |x| - a )2 which has two wells centered at x = ±a separated by a barrier whose height at the origin is given by V0 = ½ k a2 . The particle can tunnel from one well to the other.

a) Explain why the eigenstates of this potential must have a definite parity.

b) For large a (or V0 >> ħω) the two minima are well-separated and a good approximation to the wavefunction of the lowest energy states of this particle is a linear combination of the ground states of two separate harmonic oscillator wells centered at x = ±a. Write down the possible wavefunctions and explain which one is the ground state and which one is the first excited state. Hint: plot the wavefunctions.

c) Normalize these wavefunctions.

d) The ground state has an energy E0 = ħω(1/2 – ε) while the first excited state has an energy E1 = ħω(1/2 + ε) where ε2 = (β/π) exp[-2β] with β = 2V0/ħω . Assume that at t = 0 the particle is in the ground state of the oscillator well centered at x = - a. Find ψ(x,t) and find the time that the particle takes to tunnel completely to the well centered at x = +a.

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# Homework Help: Double oscillator potential

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