Consider a simple harmonic oscillator in one dimension. Introduce the raising and lowering operators, a, and a, respectively. The hamiltonian ÂĤ and wave function Y at t = 0 are: Ĥ = hw a,a_ + ) Y(x,0) = 11) + =12) V5 V5 where |n) denotes the state y „(x) of energy En = hw(n+1/2). a. Time dependent solution to the Schrodinger equation Write the wave function Y(x, t) in terms of |n) (more specifically, in terms of |1) and |2) ). b. Expectation value of the energy Find (E) by using the Dirac notation. c. Expectation value of the time-dependent position The position x can be represented in operators by x = Xo(â_ + â„) where Xp = vħ/2mw is a constant. Derive an expression for the expectation of the time-dependent position (write your answer as a sinusoidal). (x(t)) = (Y(x, t)\x|¥(x, t)) You may need operator expressions such as â-\n) = /ñ\n– 1) and â,\n) = v/n+I]n+1)

Consider a simple harmonic oscillator in one dimension. Introduce the raising and lowering operators, a, and a, respectively. The hamiltonian ÂĤ and wave function Y at t = 0 are: Ĥ = hw a,a_ + ) Y(x,0) = 11) + =12) V5 V5 where |n) denotes the state y „(x) of energy En = hw(n+1/2). a. Time dependent solution to the Schrodinger equation Write the wave function Y(x, t) in terms of |n) (more specifically, in terms of |1) and |2) ). b. Expectation value of the energy Find (E) by using the Dirac notation. c. Expectation value of the time-dependent position The position x can be represented in operators by x = Xo(â_ + â„) where Xp = vħ/2mw is a constant. Derive an expression for the expectation of the time-dependent position (write your answer as a sinusoidal). (x(t)) = (Y(x, t)\x|¥(x, t)) You may need operator expressions such as â-\n) = /ñ\n– 1) and â,\n) = v/n+I]n+1)