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Let us consider a shifted harmonic oscillator, given by the Hamiltonian Hˆ = pˆ ^2/2m + 1/2 mω2xˆ 2 + cxˆ,

where c is a real constant.

(a) Compute the eigenvalues and eigenfunctions of this Hamiltonian as functions of c.

(b) For the ground state, compute the expectation values <xˆ> and <pˆ> as functions of c.

(c) What is the physical meaning of the constant c?

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a ) In general the Hamiltonian for a harmonic oscillator
2's given by H = P2 + 1 muse2
Here for a shifted harmonic oscillator the Hamiltonian i's,
2 m
= P +1 mw2 / 20 2 + 2C
2 m
= $ 2
$ 2
( m w 2 )...

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