Solution of the Harmonic Oscillator Schrodinger Equation

The quantities a and p have units of m . We will assume that p is the positive root of P. The quantity a is necessarily positive. The Schrodinger equation now can be written 

At very large values of x, the quantity a (which is a constant since is a constant) becomes negligible compared to p x. That is, the Schrodinger equation (3-18) approaches more and more closely the asymptotic form 

What we need, then, are solutions x r x) that approach the solutions of Eq. (3-19) at large values of x. The solutions for Eq. (3-19) can be figured out from the general rule for differentiating exponentials  

We want the term in square brackets to become equal io p x in the limit of large x. We can arrange for this to happen by setting 

At large values of x, is negligible compared to 0 x, and so exp( x /2) are asymptotic solutions for Eq. (3-19). As x increases, the positive exponential increases rapidly whereas the negative exponential dies away. We have seen that, for the wave-function to be physically meaningful, we must reject the solution that blows up at large x. On the basis of these considerations, we can say that, if contains exp(- x /2), it will have the correct asymptotic behavior if no other term is present that dominates at large x. Therefore, 

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Section 3-4 Solution of the Harmonic Oscillator Schrodinger Equation... 

FIGURE 4.2 Plots of the harmonic-oscillator Schrodinger-equation solution containing only even powers of x for = 0.499hi, f = O.BOOhv, and = 0.501 hr . In the region around X = Othe three curves nearly coincide. For a x > 3 the = O.SOOhv curve nearly coincides with the xaxis. 

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

The GTFs are directly related to the solutions of the one-particle Schrodinger equation for the simple harmonic oscillator and so have well-defined completeness properties. 

The use of wave groups or wave packets in physics, and certainly in chemistry, was limited to a few theoretical examples in the applications of quantum mechanics. The solution of the time-dependent Schrodinger equation for a particle in a box, or for a harmonic oscillator, and the elucidation of the uncertainty principle by superposition of waves are two of these examples. However, essentially all theoretical problems are presented as solutions in the time-independent frame picture. In part, this practice is due to the desire to start from a quantum-state description. But, more importantly, it was due to the lack of experimental ability to synthesize wave packets. 

This must now be walked into a quantum-mechanical formalism. What we have learned above permits us to write a Schrodinger equation similar to Eq. (3.39.14), whose solutions will be of the harmonic oscillator type ... 

We simply list the solutions, which you can verify by substituting them into the Schrodinger equation. The first four wave functions for the quantum harmonic oscillator are listed in Table 4.2 and plotted in Figure 4.31. The energy levels of the harmonic oscillator are given by... 

We have found the ground state for the oscillator, but there are also other functions that can satisfy the Schrodinger equation. These will represent the excited states of the oscillator and become important when the molecule absorbs energy from the light used in IR spectroscopy. The complete set of solutions for the wavefunctions of the harmonic oscillator are actually a product of the Gaussian function discussed above and a Hermite polynomial which ensures that the cancellation of the function we forced by one choice of fi in Equations (A6.21)-(A6.23) also occurs for the excited states. The general solution for state n of the harmonic oscillator is then... 

Some simple models for V(r) are shown in Fig. 2.1. Two crude approximations, the infinite square well (ISW) and the 3-dimensional harmonic oscillator (3DHO), have the advantage of leading to analytical solutions of the Schrodinger equation which lead to the following energy levels ... 

The solutions of the Schrodinger equation with this potential are related to the representations U(2) 3 U(l). In the case in which the quantum number N characterizing these representations goes to infinity, the cutoff harmonic oscillator potential of Figure 2.1 becomes the usual harmonic oscillator potential. 

Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

The polynomials defined here are different from the Hermite polynomials which occur in the solutions of the Schrodinger equation for the harmonic oscillator. 

In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems the harmonic oscillator, the orbital angular momentum, and the hydrogen atom, respectively. The ladder operator technique is used in each case to solve the resulting differential equation. We present here the solutions of these differential equations using the Frobenius method. 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

In 32 the authors consider the solution of the two-dimensional time-independent Schrodinger equation by partial discretization. The discretized problem is treated as a problem of the numerical solution of system of ordinary differential equations and has been solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. As numerical illustrations the authors have found the eigenvalues of the two-dimensional harmonic oscillator... 

As soon as bound states are considered there are only discrete energy levels. Nevertheless it was shown by Bell [77] that it is possible to employ approximately a continuum of energy levels for the calculations of the tunnel rates, which is adequate for the description of many experimental systems. In the simplest form (see Fig. 21.5) of the Bell model, the potential barrier is an inverted parabola. This allows the use of the known solution of the quantum mechanical harmonic oscillator for the calculation of the transition probability through the barrier. The corresponding Schrodinger equation is... 

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