The Harmonic Oscillator Problem

Harmonic oscillator model employing a mass m affixed to a spring having a force constant k. [Attributed to Svjo, reproduced from http //en.wikipedia.org /wiki/Hooke s law (accessed December 1, 2013).] 

Substituting the classical potential energy into the Schrodinger equation in one-dimension gives Equation (3.72). Following rearrangement and substitution of 

The probability function for the harmonic oscillator with v= 10, showing how the potential energy for the quantum mechanical harmonic oscillator approaches that of the classical harmonic oscillator for very large values of v. [ Michael D Payer, Elements of Quantum Mechanics, 2001, by permission of Oxford University Press, USA.] 

Fluorescent light bulbs often consist of the line spectrum of Hg. Mercury has two strong atomic emission lines in the UV at 254 and 366 nm. (a) Calculate the frequency corresponding with each of these two wavelengths, (b) Calculate the energy of each line in units of joules. 

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Relate characterization of stationary points via the eigenvalues of the Hessian to the corresponding matrix under the harmonic oscillator problem. 

Substituting the values for < /, and respectively, in the harmonic oscillator problem,... 

J.O. Nordling, J.S. Faulkner, Approximate Linear Dependence, Scaling and Operator Convergence in the Harmonic Oscillator Problem, J. Mol. Spectros. 12 (1964) 171. 

To develop the quantum theory of electromagnetic radiation, it is useful to reformulate the harmonic-oscillator problem in terms of creation and annihilation operators, following a derivation due to Dirac. The Schrodinger equation (5.9) can be written... 

It turns out, very fortunately, that this asymptotic approximation is also an exact solution of the Schrddinger equation Eq (7.29) with = 0, Just what happened for the harmonic-oscillator problem in Chapter 5. The solutions are designated Rnlit), where the label n is known as the principal quantum number, as well as by the angular momentum i, which is a parameter in the radial equation. The solution (7.32) corresponds to Rioir)- This should be normalized according to the condition... 

To show that the functions we have defined above are the same as those used in the solution of the harmonic oscillator problem, we look for the differential equation satisfied by // ( ). It is first convenient to derive certain relations between successive Hermite polynomials and their derivatives. We note that since S = its partial derivative with respect... 

This is just the equation, 11-6, which we obtained from the harmonic oscillator problem, if we put 2n in place of - — 1, as... 

ATI.123), equation (A 1.1.124) and equation (AT 1.125) is necessary. Placing an equal number oiV= 2q harmonic oscillator functions at the position of each minimum (these correspond to solutions of the harmonic oscillator problems with V = Hq — VTtil-and V = 2 q + -/lO)-, respectively) yields the eigenvalues given below for the four lowest states (in each case, there are Nil functions centred at each point). 

Example 4 The analogous treatment for the harmonic oscillator problem in Example... 

In this section we describe all of these formulations and others reported in the literature and discuss them critically. First, however, the validity of treating librations in terms of the harmonic oscillator problem will be considered. 

Hamilton s formulation, applied to the harmonic oscillator problem, has yielded a differential equation. Equation 7.11, that relates the position function to the second time derivative of the position function. This differential equation has a general solution that can be expressed in either of two ways. They are mathematically equivalent ... 

Apply the variational method to the harmonic oscillator problem using the following as the trial wavefunction ... 

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