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.