Chapter 16 Lecture 15 - Eigenstates of the Quantum Harmonic Oscillator - Continued

16.1 Hermite Polynominals

Y_n(b ) is generated by applying A n times to Y₀(b ) .

Y_n(b )µ (A

)ⁿY₀(b ) (16.1)

The normalisation of (A)ⁿY₀(b ) results in the normalisation eigenfunctions.

Y_n(b )=

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2ⁿn!

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ø

C₀H_n(b )e

b²

We say that the H_n(b ) 's are the polynominals in b (or x ) of order n and are called Hermite Polynominals.

Some important features of the wave functions

Y_n(x) remains finite in classically forbidden regions ( V(x)>E_n )
Y_n(x)® 0 as x®±¥ because of it begin roughly e^-(^mw^²^/2^x^²⁾
as n (the energy) increases the number of nodes (oscillations) in the wavefunction increases

16.2 Comparision To The Classical Harmonic Oscillator System

We can consider the probability distribution for position in quantum and classical oscillators

Figure 15.1 - For a Low n State

Figure 15.2 - For a High n State

The classical turning points by definition are

E_n=V(x)

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w =

mw²l_n²

this is where l_n is x and so the outcomes of this are

the classical system cannot be found beyound l_n
the quantum system has a finite probability for where x>l_n

so for a high n classical and quantum probability distributions do become broadly similar.

16.3 Vibrational States of Diatomic Molecules

The interatomic potential in a diatomic molecule is approximately parabolic

Figure 15.3 - Parabolic Approximation of Diatomic Molecules

so the potential is

V(R)=

mw²(R-R₀) (16.2)

strictly speaking m is replaced by the reduced mass µ . So

V(x)=

mw²x²

we find a zero point vibrational energy state at

E₀=

and so the energies of the excited states are

E_n=

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The study of spectra comfirms this and gives values for w . eg.

HCl®w =0.36eV

CO®w =0.27eV

if n is high the atoms are likely to be formed near the turning point (this is simple harmonic motion)

16.4 The Vibrational Modes of a Crystal Lattice

We are interested in

small amplitude oscillations in a crystal lattice
the analysis of normal modes of oscillation

Figure 15.4 - Types of Normal Mode Oscillations
each normal mode is a Simple Harmonic Oscillation with a characteristic frequency w_q
each mode of q can be quantised

E_m= æ
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è m+

1

2
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ø w_q (16.3)
each quantum of energy w_q is called a photon

The mean energy per mode is determined by the Boltzmann distribution

<E>_q=

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w_qe

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w_q

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w_q

(16.4)

Þ <E>_q=

w_q

-1

w_q (16.5)

A similar proceedure for the quantisation of energy for the electromagnetic modes of a cavity leads to photons

E_n=

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where this is the energy of the n^th excited state of mode and the w is the photon energy.