There are a total of four terms in the equation. Two are multiplying x^2 (the
terms in the left-hand parentheses) and can be rewritten by substituting
the value of αfrom eqn 11.9:
(11.21)
So these two terms cancel and we are left with the terms in the right-hand
parentheses:
(11.22)
The product of the wavefunction and the terms in the parentheses must
always zero for all values of the wavefunction, including all non-zero
values. This can only be true if the term in the parentheses is always zero.
Thus, we can write:
(11.23)
Thus, substitution of the wavefunction ψ 0 (x) yields a specific energy of
. This is the ground-state energy. Substitution of the 9 th wavefunction
will yield the energy:
(11.24)
In summary, for the simple harmonic oscillator, the energies of
the wavefunctions are proportional to the quantum number
and separated by a constant factor of Zω(Figure 11.4).
Forbidden region
Classically, the mass attached to the spring vibrates back and
forth and is restricted to a narrow region. The maximum dis-
placement of the mass from the equilibrium position, xTP, is where
the total energy is all potentialenergy, so:
(11.25)
E
kA
A
E
k
==
2
2
2
so
E 9 =+9Z
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
1
2
ω
Zω
2
E
mm
mk k
(^0) m
2
2
2
2
12
22 2
/
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
⎛
⎝
⎜
ZZ
Z
Z
α ⎜⎜
⎞
⎠
⎟⎟ =
12
2
/
Zω
ψ
(^0) α
2
()x 2 m 2 E 0 0
Z
−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟=
−+=−
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟+=
ZZ
Z
2
4
2
222 m^220
k
m
mk k
α
228 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
0
4
3
2
1
0
Displacement, x
Potential
energy, V
Energy
Figure 11.4
Energy levels of the
harmonic oscillator
are evenly spaced.