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256 7 QUANTUM THEORY
Apart from the constant, the leading term is quadratic in the displacement from
the minimum at ϕ = π/3. By analogy with the harmonic potential V = 1
2 kfx
2,
the force constant is 9V0.
�e energy levels are therefore
Eυ = (υ + 1
2 )ħω ω = (9V0/meff)1/2
where meff is the appropriate e�ective mass for the motion.
As the energy increases the amplitude of the motion increases and it begins to
sample parts of the potential which are no longerwell-represented by a quadratic
function. Additional terms are needed to describe the potential, and from the
form of the cosine function it is evident that these terms will �atten out the po-
tential meaning that it rises less steeply than the quadratic function developed
above. As a result the energy levels will get closer together.
P7E.18 �e general form of the harmonic wavefunctions are ψυ = NυHυ(y)e−y
2/2,
where Nυ is the normalization constant and Hυ(y) is a Hermite polynomial of
order υ, expressed in terms of the reduced position variable y. Nodes in the
wavefunction occur when the wavefunction passes through zero. �e wave-
functions go asymptotically to zero at y = ±∞ on account of the term e−y
2/2,
but these do not count as nodes as the wavefunction does not pass through
zero.�erefore, nodes in the wavefunction correspond to those values at which
Hv(y) = 0.
�e �rst six Hermite polynomials are plotted in Figs 7.9 and 7.10; note that the
normalizing factors have been included.
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
−2
−1
0
1
2
y
H
υ(
y)
υ = 0
υ = 1
υ = 2
Figure 7.9