2. Consider a one-dimensional quantum harmonic oscillator with the Hamiltonian p- mw2.12 Η + 2m 2 where m and w are the

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2. Consider a one-dimensional quantum harmonic oscillator with the Hamiltonian p- mw2.12 Η + 2m 2 where m and w are the mass and frequency of the oscillator, and [î, ] = iħ. The eigenvalues En and normalised eigenstates (n) of û are found using two operators, mw â= w + V 2h ai mw 1 p. 2mħw V 2h P. V2mħw (3) as + 3) [4] =n ja) = Aecât - (at)n En = hw into In) F10) (n = 0, 1, 2, ...). (4) Vn! (a) Using the basic relation [â, â ] = 1 and mathematical induction, show that â, (at)"] = n(at)n-1 (5) (b) In this part you explore the properties of coherent states |a) defined by "10), (6) where a e C, and A is the normalisation constant (to be determined later). i. By expanding the operator eaat in the Maclaurin series and using Eq. (5), show that [à, ecât] = aegat ii. Using the result of part i., show that (a) is an eigenstate of the operator â, and find the corresponding eigenvalue. iii. Using the Maclaurin expansion of the operator ecât and Eq. (4), show that |a) = A (7) [4] [2] [3] an In In). n=0 [4] iv. Using Eq. (7), find the condition satisfied by A for la) to be normalised ((ala) = 1), and show that the normalisation constant can be chosen as A= e-lal?/2. v. Show that the expectation value of the energy for the state (a) is given by () [3] {E} = (al\a) = "w (lapp +) (c) The dipole polarisability for an electron in the oscillator potential is given by the 2nd-order perturbation-theory expression Kn\@[0)2 ad = 2e2 En – Eo n#0 Use Eqs. (3) and (4) to calculate this sum and find ad. [5]