Question

(30) 1. a) Briefly explain the physical reasoning for requiring a wavefunction to be normalized, b) The state of a harmonic oscillator is given by the wavefunction: P(x, t0) = A1 01(x) + A2 02(x). Where Al and A2 are constants and 1(x) and 02(x) are energy eigenfunctions associated with energies E, and E. What condition must A1 and A2 satisfy in order for 'Plx, t0) to be normalized? c) If the particle in the state P(x,t=0), given above, is 4 times as likely to be found having energy E, than energy E, find acceptable real positive values of A1 and A2 so that Px,t=0) satisfies this requirement and is normalized. d) Given P(x,t=0) above, what is P(x,t)? Make sure to define terms appearing in your result. e) The time dependence of the expectation value of any operator  in an arbitrary state P(x,t) is given by the equation below. The equation involves the expectation value of the commutator of the operator with the Hamiltonian and the expectation value of the partial derivative of the operator with respect to time. au () = < [1,A] > < > Using this equation show that any wave function P(x,t) that is normalized at time t=0 remains normalized for all timest. Make sure to clearly explain your reasonings.

Answer 1

Question is solved using principles of quantum mechanics.

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