The Harmonic Oscillator Wavefunctions

We return to the wavefunction itself It has already been established that the wave-function is an exponential times a power series that has been argued to have a limited, not an infinite, number of terms. The final term in the sum is determined 

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It should be pointed out that the Cq constant in does not have the same value as the Cq in 2 or 4, or other s. This is also true for the values of Cp C2, and so on, in the expansions of the summations. The first wavefunction, I o consists only of the exponential term multiplied by the constant Cq. 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 wavefunctions consist of the exponential term multiplied by a power series in x that 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 is easiest to normalize because it has only a single term in its polynomial. The range of the one-dimensional harmonic oscillator is — o° to +o°, because there is no restriction on the possible change in position. To normalize, the wavefunction must be multiplied by some constant N such that 

Because N and Cq are both constants, it is customary to combine them into a single constant N. The complex conjugate of the exponential does not change the form of the exponential, because it does not contain the imaginary root i. The integral becomes 

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The absorption spectrum consists of sequences of transitions from v" = 0, 1, 2 to various v levels in the upper state, and the relative intensities of the vibration-rotation bands are given primarily by the product of the FCF value and a Boltzmann term, which can be taken to be exp — hcv v /kT). Common choices for the i/r s are harmonic oscillator and Morse wavefunctions, whose mathematical form can be found in Refs. 7 and 9 and in other books on quantum mechanics. The harmonic oscillator wavefunctions are defined in terms of the Hermite functions, while the Morse counterparts are usually written in terms of hypergeometric or Laguerre functions. All three types of functions are polynomial series defined with a single statement in Mathematica, and they can be easily manipulated even though they become quite complicated for higher v values. 

This has the form of the harmonic-oscillator wavefunction (eqn 9.24). The difference in adjacent energy levels is ... 

Question. The next term in the Taylor series for the potential energy is — —0 . Treat this as a perturbation to the harmonic oscillator wavefunction and compute the first-order cmrection to the energy. 

Fig. 8.3. The zero-temperature Pranck-Condon factor, as a function of v for different values of the Huang-Rhys parameter, S. S = 0 (circles), S = 1.5 (squares), S = 4.5 (triangles), and S = 9.5 (diamonds). The overlap of the harmonic oscillator wavefunctions, shown in Fig. 8.2, ensures that the 0 — v (or vertical) transition is the largest.

TABLE 8.1 The harmonic oscillator wavefunctions. The Hermite polynomials and normalization constants are given for the first six harmonic oscillator wavefunctions rjvly) =, where the... 

The harmonic oscillator wavefunctions i7v(J ) in this equation are identical to the wavefunctions rj,(x) described previously, except for a change in coordinate from X to R. Previously, the potential energy was zero at x = 0. Now the potential energy is zero atR = R, so our expression for x is... 

This is a fairly clumsy procedure for producing polynomials, but Eq. (3-65) is useful in determining general mathematical properties of these polynomials. For instance, Eq. (3-65) will be used in showing that the harmonic oscillator wavefunctions are orthogonal. 

We will now show that the harmonic oscillator wavefunctions are orthogonal, i.e., that... 

It turns out that the set of harmonic oscillator wavefunctions were already known. This is because differential equations like those of equation 11.6, the rewritten Schrodinger equation, had been studied and solved mathematically before quantum mechanics was developed. The polynomial parts of the harmonic oscillator wavefunctions are called Hermite polynomials after Charles Hermite, the nineteenth-century French mathematician who studied their properties. For convenience, if we define (where is the Greek letter xi, pronounced... 

The normalization constants for the harmonic oscillator wavefunctions follow a certain pattern (because the formulas for the integrals involve Hermite polynomials) and so can be expressed as a formula. The general formula for the harmonic oscillator wavefunctions given below includes an expression for the normalization constant in terms of the quantum number n ... 

As we discussed in Chap. 2, the solutions to the Schrodinger equation for a quadratic potential energy function of coordinate x are the harmonic-oscillator wavefunctions,... 

This demonstrates that the wavefunctions x /o and Vi are orthogonal. This is tme for aU of the harmonic oscillator wavefunctions. 

The first several wavefunctions for the harmonic oscillator are shown in Figure 5-1 and should be compared to the Particle-in-a-Box wavefunctions shown in Figure 2-2. Note that the wavefunctions for the Particle-in-a-Box and the harmonic oscillator have similar shapes for each corresponding energy level. The principal difference is that the harmonic oscillator wavefunctions asymptotically approach zero as x approaches infinity (as the potential approaches infinity). Because the wavefunctions must... 

The points of maximum and minimum amplitude for the harmonic oscillator wavefunctions (indicative of the greatest probability of the mass) can be found by taking the first derivative of the wavefunction and setting it equal to zero. 

Equation 6-86 must be analyzed term by term. The 5 s refer to the orthonormality integrals of the harmonic oscillator wavefunctions. The first term is non-zero when the vibrational quantum state does not change but only when the molecule has a non-zero equilibrium dipole moment (such as HCl and HE). The second term is non-zero either when the vibrational quantum state increases or decreases by one if the first derivative of the dipole moment of the molecule is non-zero. The third term is non-zero when the vibrational quantum state increases or decreases by two if the second derivative of the dipole moment of the molecule is non-zero. Further expansion terms reveal that any change in vibrational states is allowed for a vibrational transition for a molecule with a non-zero equilibrium dipole moment. The vibrational transition observed in an infrared spectrum must be determined using the Boltzmann distribution that was described previously. 

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