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.
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Chapter1: Chemical Foundations
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Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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![(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 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.
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  in an arbitrary state Y(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.
ĐÂ
a (A) = < [A,Ã] > +<>
Using this equation show that any wave function Y(x,t) that is normalized at time t=0 remains
normalized for all times t. Make sure to clearly explain your reasonings.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecaf6b4-e5d5-4dc2-aa49-b8dd3912883b%2F8dd38e3e-a6e9-4ae4-abf2-dfd0dfab7745%2F8nvsr6m_processed.png&w=3840&q=75)
Transcribed Image Text:(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 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.
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  in an arbitrary state Y(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.
ĐÂ
a (A) = < [A,Ã] > +<>
Using this equation show that any wave function Y(x,t) that is normalized at time t=0 remains
normalized for all times t. Make sure to clearly explain your reasonings.
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