A coordinate transformation is performed to express the harmonic oscillator problem
involving two masses coupled by a spring in terms of the motion of the system’s center of
mass and the motion of the system’s reduced mass.
a. Calculate the reduced mass of the following pairs of atoms:
i. C and O
ii. C and H
b. What general conclusions can you reach about the relationship between the relative sizes
of the two masses and the corresponding reduced mass?
c. Locate the position of the center of mass for the pairs of atoms given below. Place the
carbon atom at the origin of the coordinate system.
i. C and O. A typical C–O bond length is 1.43Å
ii. C and H. A typical C–H bond length is 1.10Å.
d. What can you conclude about the position of the center of mass when the two masses are
nearly equal in contrast with the situation where there is a significant difference between
the masses?
Document Preview: CEM
 483/CEM
 881
 –
 Homework
 #5
 (due
 Friday,
 October
 11,
 2013
 at
 the
 beginning
 of
 class)
Â
Please write legibly. Show all of your work to receive full credit. We are just as interested in the
process and reasoning that you followed to reach your answer as the final answer.
1. Show that
! !
?? ?? ?? ?? ?? ??
! !
- =- ?? -?? - ?? - ?? -?? - ??
! ! ! ! ! !
! !
???? ???? ?? ??
! !
is equal to
!
?? ??
?? +????= 0
!
????
using the definitions for the relative coordinate and reduced mass.
2. A coordinate transformation is performed to express the harmonic oscillator problem
involving two masses coupled by a spring in terms of the motion of the system’s center of
mass and the motion of the system’s reduced mass.
a. Calculate the reduced mass of the following pairs of atoms:
i. C and O
ii. C and H
b. What general conclusions can you reach about the relationship between the relative sizes
of the two masses and the corresponding reduced mass?
c. Locate the position of the center of mass for the pairs of atoms given below. Place the
carbon atom at the origin of the coordinate system.
i. C and O. A typical C–O bond length is 1.43Å
ii. C and H. A typical C–H bond length is 1.10Å.
d. What can you conclude about the position of the center of mass when the two masses are
nearly equal in contrast with the situation where there is a significant difference between
the masses?
3. Show whether the following functions are even, odd, or neither:
a. cos??
b. ??sin??
!
c. ?? +4
! !
d. ?? +7?? +2??
! !!
e. ?? ??
!
! !!
f. ?? ??4. This problem will consider the harmonic oscillator wave function corresponding to ??= 4.
a. The solutions to the time-independent Schrödinger equation for the harmonic oscillator
have the general form
!
!/! !!! /!
?? ?? =?? ?? ?? ?? ??
! ! !
where
!/!
????
??=
!
?
The normalization constant ?? is given by the expression
!
!/!
1 ??
?? =
!
! !/!
2 ??! ??
and the fourth Hermite...