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Consider a particle of mass m in a one-dimensional harmonic potential. Hamiltonian H is written as H = Answer the following questions. Here, x, p, and w are the position operator, the momentum operator, and the angular frequency, respectively. For the normalized wave function (x) and the eigenenergy En, the following Schrödinger equation holds. Hon(x) = En On (x) Here, n is a non-negative integer (n = 0, 1, 2, ...), and n = 0 represents the ground state. Define the operators A and A+ as follows. A = p² 2 1 2m 2 At = =+= mw²x². mw 2ħ mw 2ħ = x + x- Here, ħ is the Planck constant divided by 27. The following formulae hold. 1 Pn+1(x) Pn (x) ip mω, ip mw/ √n + 1 1 √n + 1 =A+Qn(x) =A@n+1(x) (1) Calculate A¹A, and represent H by using A† and A. Note that the commutation relation [x, p] = iħ holds. (2) Obtain Eo. Note that Apo = 0. (3) Calculate the commutation relation [A, A+], and obtain E₁. (4) Obtain the expectation value of x for the state On (n|x|4n).
Let o be the state for time t < 0. Consider the situation that a perturbed potential V(x) = Fx is added for t≥0. Here, F is a positive constant representing force. Assuming that F is very small, use the first order perturbation. Then, the wave function at time t is written as ) = e¯¹ t¶₁(x) + c₁(t)e-/ªq₁(x). Y(x, t) = Here, the coefficient c₁(t) is written as C₁ (t) = 1 * (P₁V \o de tout dt'. iħ (5) Calculate c₁(t), and represent the probability of finding the particle in the state ₁ at time t(≥ 0) using m, w, ħ, and F. (6) Represent the expectation value of x at time t(≥ 0), (y|x|y), using m, w, and F. Draw a graph of (4|x|4) as a function of t in the range of 0 ≤ t ≤ ²7 W