3 Quantum Harmonic Oscillator Another form of a potential energy function that can be used to describe a molecular bond is: V(y) = where and are constants. (A) Using a Taylor Series expansion, derive an effective harmonic force constant that could be used for a harmonic oscillator approximation to the potential V(y) around the point y=0. (B) What are the eigenvalues of the QHO Hamiltonian utilizing the harmonic force constant you derived in (A)? (C) Calculate for the n-3 QHO wavefunction. You must work through the math to show your solution (either Mathematica or even/odd function relationships are suitable means of evaluating integrals). (D) Calculate for the n=3 QHO wavefunction. You must work through the math to show your solution (either Mathematica or even/odd function relationships are suitable means of evaluating integrals). (E) Calculate the commutator of [H,py] (be sure you include all terms in the QHO Hamiltonian). What does your result mean about the ability to measure H and Py simultaneously? Evaluate the following expressions numerically: <1|PrV3 >, < 2P1516 >, (F) <448|23|452>
3 Quantum Harmonic Oscillator Another form of a potential energy function that can be used to describe a molecular bond is: V(y) = where and are constants. (A) Using a Taylor Series expansion, derive an effective harmonic force constant that could be used for a harmonic oscillator approximation to the potential V(y) around the point y=0. (B) What are the eigenvalues of the QHO Hamiltonian utilizing the harmonic force constant you derived in (A)? (C) Calculate for the n-3 QHO wavefunction. You must work through the math to show your solution (either Mathematica or even/odd function relationships are suitable means of evaluating integrals). (D) Calculate for the n=3 QHO wavefunction. You must work through the math to show your solution (either Mathematica or even/odd function relationships are suitable means of evaluating integrals). (E) Calculate the commutator of [H,py] (be sure you include all terms in the QHO Hamiltonian). What does your result mean about the ability to measure H and Py simultaneously? Evaluate the following expressions numerically: <1|PrV3 >, < 2P1516 >, (F) <448|23|452>
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![3 Quantum Harmonic Oscillator
Another form of a potential energy function that can be used to describe a molecular bond is:
V(y)
=
where and are constants.
(A)
Using a Taylor Series expansion, derive an effective harmonic force constant that
could be used for a harmonic oscillator approximation to the potential V(y) around the point y=0.
(B)
What are the eigenvalues of the QHO Hamiltonian utilizing the harmonic force
constant you derived in (A)?
(C)
Calculate <y> for the n-3 QHO wavefunction. You must work through the
math to show your solution (either Mathematica or even/odd function relationships are suitable
means of evaluating integrals).
(D)
Calculate <py > for the n=3 QHO wavefunction. You must work through the
math to show your solution (either Mathematica or even/odd function relationships are suitable
means of evaluating integrals).
(E)
Calculate the commutator of [H,py] (be sure you include all terms in the QHO
Hamiltonian). What does your result mean about the ability to measure H and Py simultaneously?
Evaluate the following expressions numerically: <1|PrV3 >, < 2P1516 >,
(F)
<448|23|452>](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1822b493-2c18-4a65-99e5-18fca75fe15d%2Fb302e799-2ee5-4841-9eca-b05b1f30ee7d%2Fakpka8_processed.png&w=3840&q=75)
Transcribed Image Text:3 Quantum Harmonic Oscillator
Another form of a potential energy function that can be used to describe a molecular bond is:
V(y)
=
where and are constants.
(A)
Using a Taylor Series expansion, derive an effective harmonic force constant that
could be used for a harmonic oscillator approximation to the potential V(y) around the point y=0.
(B)
What are the eigenvalues of the QHO Hamiltonian utilizing the harmonic force
constant you derived in (A)?
(C)
Calculate <y> for the n-3 QHO wavefunction. You must work through the
math to show your solution (either Mathematica or even/odd function relationships are suitable
means of evaluating integrals).
(D)
Calculate <py > for the n=3 QHO wavefunction. You must work through the
math to show your solution (either Mathematica or even/odd function relationships are suitable
means of evaluating integrals).
(E)
Calculate the commutator of [H,py] (be sure you include all terms in the QHO
Hamiltonian). What does your result mean about the ability to measure H and Py simultaneously?
Evaluate the following expressions numerically: <1|PrV3 >, < 2P1516 >,
(F)
<448|23|452>
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