Schrodinger equation harmonic oscillator potential

Vibrational energy levels can be estimated by inserting the Hooke s law potential energy, U(r) = 0.5k(r— r f, in the Schrodinger equation (harmonic oscillator approximation). This yields eigenvalues, Ey, for the permissible energy levels, of... 

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. 

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...

Therefore the following question arises in spite of the fact that the 3-dimension harmonic-oscillator potential is nonlocal and has specific properties, is it possible to derive potential forms which, put into the standard Schrodinger equation, will give about the same spectrum as the QZO model This is a standard question in inverse scattering problems [33]. Classical potentials giving approximately the same spectrum as the g-deformed, one-dimensional harmonic oscillator have been obtained either through the use of standard perturbation theory [34], or within the WKB approximation [35]. 

It is not easy to find a quantum many-body system for which the Schrodinger equation may be solved analytically. However, a useful example is provided by the problem of two electrons in an external harmonic-oscillator potential, called Hooke s atom. The Hamiltonian for this system is... 

Introducing the potential of the harmonic oscillator (eq. 3.2) in the monodimensional equivalent of equation 3.9 (i.e., the Schrodinger equation for onedimensional stationary states see eq. 1.9), we obtain... 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

The potential energy function for a chemical bond is far more complex than a harmonic potential at high energies, as discussed in Chapter 3. However, near the bottom of the well, the potential does not look much different from the potential for a harmonic oscillator we can then define an effective force constant for the chemical bond. This turns out to be another problem that can be solved exactly by Schrodinger s equation. Vibrational energy is also quantized the correct formula for the allowed energies of a harmonic oscillator turns out to be ... 

In section 6.8.2 we described and solved the Schrodinger equation for a harmonic oscillator, equation (6.178). The potential energy was expressed in terms of a vibrational coordinate q which was equal to R - Re, Re being the equilibrium bond length. The dependence of the electric dipole moment on the internuclear distance may be expressed as a Taylor series,... 

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. 

To obtain the quantum version, we substitute the potential energy function for the harmonic oscillator into the Schrodinger equation to get... 

In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

In 35 the numerical solution of the two-dimensional time-independent Schrodinger equation is studied using the method of partial discretization. The discretized problem is treated as a problem of the numerical solution of a system of ordinary differential equations and Numerov type methods are used to solve it. More specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe el al. and the minimum phase-lag method of Rao et al. are applied to this problem. The methods are applied for the calculation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heils potential. The results are compared with the results produced by full discretization. Conclusions are presented. 

The above classification of asymmetric potential functions is convenient for comparison of different molecules or as a systematic basis for making an initial fit to experimental data. However, when the Schrodinger equation is being solved by the linear variation method with harmonic-oscillator basis functions, it may not provide the best choice of origin for the basis function. For example, a better choice in the case of an asymmetric double-minimum oscillator, where accurate solutions are required in both wells, would be somewhere between the two wells. Systematic variation of the parameters may still be made as outlined above, but the origin should be translated before the Hamiltonian matrix is set up. The equations given earlier... 

Equation (10.38) is recognized as the Schrodinger equation (4.13) for the one-dimensional harmonic oscillator. In order for equation (10.38) to have the same eigenfunctions and eigenvalues as equation (4.13), the function Slq) must have the same asymptotic behavior as in (4.13). As the intemuclear distance R approaches infinity, the relative distance variable q also approaches infinity and the functions F(R) and S(q) = RF(R) must approach zero in order for the nuclear wave functions to be well-behaved. As 7 —> 0, which is equivalent to q —Re, the potential U(q becomes infinitely large, so that F(R) and S(q rapidly approach zero. Thus, the function S(q) approaches zero as q -Re and as Roo. The harmonic-oscillator eigenfunctions V W decrease rapidly in value as x increases from x = 0 and approach zero as X —> oo. They have essentially vanished at the value of x corresponding to q = —Re. Consequently, the functions S(iq in equation (10.38) and V ( ) in... 

For the case of the one-dimensional harmonic oscillator the potential energy is given by V (x) = kx, which is parameterized in terms of a stiffness constant k. After some manipulation, the Schrodinger equation associated with this potential may be written as... 

Both Equations (64) and (65) have the same form and they can be interpreted as Schrodinger equations in circular-like coordinates for harmonic oscillators [33], as indicated by their respective kinetic energy and quadratic potential energy terms. The identification and interpretation are even more convincing if we parametrize the negative energy of the bound states of the hydrogen atom as... 

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... 

We have shown how to derive potentials which, when put into the standard Schrodinger equation, provide approximately the same spectrum as the q-deformed, 3-dimensional harmonic oscillator. In the present work, we have also found prescriptions for choosing the model parameters r and A, thus closing the procedure for obtaining magic numbers. This could be very useful in further investigations on other properties of metal clusters. 

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