Consider that the wave function of a dimensionless harmonic oscillator, whose Hamiltonian is = P+, is...

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Consider that the wave function of a dimensionless harmonic oscillator, whose Hamiltonian is Ĥ= P²+², is given at time t = 0 by y (x, 0) = 1 /8π 1 =po(x) + =p2(x) = 187 1 8π 1 2/2 + √ISπ (1 – 2x²) e-x²/² (a) Find the expression of the oscillator's wave function at any later time t. (b) Calculate the probability Po to find the system in an eigenstate of energy 1/2 and the probability P₂ of finding the system in an eigenstate of energy 5/2. (c) Calculate the probability density, p(x, t), and the current density, J(x, t). (d) Verify that the probability is conserved; that is, show that op/ot + V. J(x, t) = 0. Consider that the wave function of a dimensionless harmonic oscillator, whose Hamiltonian is Ĥ= P²+², is given at time t = 0 by y (x, 0) = 1 /8π 1 =po(x) + =p2(x) = 187 1 8π 1 2/2 + √ISπ (1 – 2x²) e-x²/² (a) Find the expression of the oscillator's wave function at any later time t. (b) Calculate the probability Po to find the system in an eigenstate of energy 1/2 and the probability P₂ of finding the system in an eigenstate of energy 5/2. (c) Calculate the probability density, p(x, t), and the current density, J(x, t). (d) Verify that the probability is conserved; that is, show that op/ot + V. J(x, t) = 0.