Solution
The difference in energy of the adjacent states is the same and equals
h ,or
E(6.626
10 ^34 J s)(1.800
1013 s^1 ) 1.193
10 ^20 J
Since the energy of a photon is given by the equation Eh , the calculation
can be reversed to obtain the frequency of the photon necessary. It should be
obvious that the frequency of the photon is thus 1.800
1013 s^1. Using the
equation c
:
2.9979
108 m/s(1.800
1013 s)
0.00001666 m 1.666
10 ^5 m
This corresponds to 16.66 m or 166,600 Å. Calculations using the equations
Eh and c
are common in physical chemistry. Students should al-
ways remember that these equations can be used to convert quantities like E,
, and to corresponding values with other units.11.4 The Harmonic Oscillator Wavefunctions
We return to the wavefunction itself. It has already been established that the
wavefunction is an exponential ex(^2) /2
times a power series that has been ar-
gued to have a limited, not an infinite, number of terms. The final term in the
sum is determined by the value of the quantum number n, which also speci-
fies the total energy of the oscillator. Further, each wavefunction is composed
of either all odd powers ofxin the power series, or all even powers ofx.The
wavefunctions can be represented as
0 ex
(^2) /2
(c 0 )
1 ex
(^2) /2
(c 1 x)
2 ex
(^2) /2
(c 0 c 2 x^2 )
3 ex
(^2) /2
(c 1 xc 3 x^3 ) (11.17)
4 ex
(^2) /2
(c 0 c 2 x^2 c 4 x^4 )
5 ex
(^2) /2
(c 1 xc 3 x^3 c 5 x^5 )
..
.
It should be pointed out that the c 0 constant in 0 does not have the same
value as the c 0 in 2 or 4 , or other ’s. This is also true for the values ofc 1 ,
c 2 , and so on, in the expansions of the summations. The first wavefunction, 0 ,
consists only of the exponential term multiplied by the constant c 0. This
nonzero wavefunction is what allows a quantum number of 0 to be allowed for
this system, unlike the situation for the particle-in-a-box. All the other wave-
functions consist of the exponential term multiplied by a power series in xthat
is composed of one or more terms. Instead of an infinite power series, this set
of terms is simply a polynomial.
Like any proper wavefunction, these wavefunctions must be normalized.
The wavefunction 0 is easiest to normalize since it has only a single term
in its polynomial. The range of the one-dimensional harmonic oscillator is
to , since there is no restriction on the possible change in position.
324 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom