Please answer all questions (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: Y(x, t=0) = A1 ¢1(x) + A2 2(x).Where A1 and A2 are constants and ¢1(x) and 2(x) are energy eigenfunctions associated withenergies E1 and E2. What condition must A1 and A2 satisfy in order for Y(x, t=0) to benormalized?c) If the particle in the state Y(x,t=0), given above, is 4 times as likely to be found having energyEi than energy E2, find acceptable real positive values of A1 and A2 so that Y(x, t=0) satisfies thisrequirement and is normalized.d) Given Y(x,t=0) above, what is Y(x,t)? Make sure to define terms appearing in your result.e) The time dependence of the expectation value of any operator Â in an arbitrary state Y(x,t) isgiven by the equation below. The equation involves the expectation value of the commutator ofthe operator with the Hamiltonian and the expectation value of the partial derivative of theoperator with respect to time.ĐÂa (A) = < [A,Ã] > +<>Using this equation show that any wave function Y(x,t) that is normalized at time t=0 remainsnormalized for all times t. Make sure to clearly explain your reasonings.

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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: Y(x, t=0) = A1 ¢1(x) + A2 ¢2(x). Where A1 and A2 are constants and g1(x) and o2(x) are energy eigenfunctions associated with energies E1 and E2. What condition must A1 and A2 satisfy in order for Y(x, t=0) to be normalized? c) If the particle in the state Y(x,t=0), given above, is 4 times as likely to be found having energy Ei than energy E2, find acceptable real positive values of A1 and A2 so that Y(x, t=0) satisfies this requirement and is normalized.

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: Y(x, t=0) = A1 ¢1(x) + A2 ¢2(x). Where A1 and A2 are constants and g1(x) and o2(x) are energy eigenfunctions associated with energies E1 and E2. What condition must A1 and A2 satisfy in order for Y(x, t=0) to be normalized? c) If the particle in the state Y(x,t=0), given above, is 4 times as likely to be found having energy Ei than energy E2, find acceptable real positive values of A1 and A2 so that Y(x, t=0) satisfies this requirement and is normalized.

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter11: Quantum Mechanics: Model Systems And The Hydrogen Atom

Section: Chapter Questions

Problem 11.14E

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