The One-dimensional Schrodinger Equation

The simple de Broglie relation, X = hip, proved to be adequate to account for the properties of a particle moving with constant momentum in one dimension, but it cannot be applied to more complex systems where the momentum of the particle varies with position, or there is more than one variable. The equation which describes the wave behaviour of such systems is the celebrated Schrodinger wave equation, developed by Erwin Schrodinger in 1926. For one-dimensional systems with constant potential energy, this equation produces exactly the same results as those obtained from application of the de Broglie relation. 

Because the Schrodinger wave equation is an expression of some of the most basic principles of quantum mechanics, it cannot be derived from a more fundamental equation. However, it is possible to arrive at the equation by considering a general wave equation and applying the de Broglie relation to it. This is done below. 

All the wavefunctions that have been discussed so far (e sin x and cosA jc) are solutions of the differential equation  

When differentiated twice, al these functions return to their original form, multiplied by a oonstant. 

From equation (1.14) we know that the kinetic energy of the particle, T, is equal to (kk)y 2m). Thus, we can write equation (4.1) in the form  

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The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

One can expect that the electron density corresponding to the electronic state of lowest energy is roughly constant in the interior of the metal and decreases to zero outside the metal. This means that the potential seen by an electron, due to the ion cores and the other electrons, is roughly constant inside the metal, with a value significantly lower than the potential outside. The simplest model for electrons in a metal, the Sommerfeld38 model, takes this potential as -V0 inside and 0 outside. One is then led to consider the one-dimensional Schrodinger equation... 

Between collisions individual particles are described by the one-dimensional Schrodinger equation... 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

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...

When this net potential energy function is substituted into the one-dimensional Schrodinger equation and the suitable mathematical operations are carried out, the allowed energies E are found to be... 

The vibrationally adiabatic approximation should be made for the fast HCl vibrational coordinate r. Solving the one-dimensional Schrodinger equation with (R,9) fixed leads to the parameterized form

rotationally averaged potential is then used to compute the one-dimensional potential... 

J/ = amplitude of the particle/wave at a distance x from some chosen origin The one-dimensional Schrodinger equation is easily elevated to three-dimensional status by replacing the one-dimensional operator d2/dx2 by its three-dimensional analogue... 

In the present chapter we have presented a new family of exponentially-fitted four-step methods for the numerical solution of the one-dimensional Schrodinger equation. For these methods we have examined the stability properties. The new methods satisfy the property of P-stability only in the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation (i.e. they are singularly P-stable methods). The new methods integrate also exactly every linear combination of the functions... 

The relevant equations for the derivative Numerov-Cooley (DNC) method closely follow Cooley s [111] presentation. Let R be the radial coordinate or bond displacement coordinate, P R) a radial eigenfunction, and U R) the potential function. The one-dimensional Schrodinger equation is then... 

The methods are called multiderivative since uses derivatives of order two, four or six. The parameters of the method are computed in order to have eighth algebraic order and minimal phase-lag. Finally, a family of eighth algebraic order multiderivative methods with phase-lag of order ten is developed. Numerical application of the new obtained methods to the resonance problem of the one-dimensional Schrodinger equation shows their efficiency compared with other similar well known methods of the literature. 

The formulation of the problem begins with the one-dimensional Schrodinger equation. Note that the Schrodinger equation is always valid, the nature of the problem will change the potential energy function, V(x)... 

Our preliminary exercise will state the problem in one dimension, with the generalization to follow shortly. The one-dimensional Schrodinger equation may be written... 

While the reader may find recurrently that key, we present the basic idea of the WKBJ method, which consists in inserting into the one-dimensional Schrodinger equation... 

The idea of using functions of the form frjr for approximate solutions of the boundary value problems has been used in a large number of papers. For example, this idea was used in [70] for the one-dimensional Schrodinger equation to prove the Hull-Julius relation (4.21) and to study the confined IlJ molecule. The direct use of the Kirkwood-Buckingham relation for variational calculations of the hydrogen atom in a half space was... 

U. Blukis and J. M. Flowell,/. Chem. Educ., 60,207 (1983). Numerical Solution of the One-Dimensional Schrodinger Equation. 

Often, the boundary conditions imposed on a differential equation determine significant aspects of its solutions. We consider two examples involving the one-dimensional Schrodinger equation in quantum mechanics ... 

TABLE 4.1 BASIC Program for Numerov Solution of the One-Dimensional Schrodinger Equation... 

Use the one-dimensional Schrodinger equation to calculate the energies of particles with simple wavefunctions... 

In quantum mechanics, however, there is a finite probability of finding the bond extended beyond this classical limit. In such a region the kinetic energy would appear to be negative because the total energy E is now less than the potential energy V. An examination of the one-dimensional Schrodinger equation ... 

Show that the wavefunction y/ = y4sin(fcx) + Bcos(kx), where A, B and k are constants, is a solution of the one-dimensional Schrodinger equation, and hence derive an expression for the energy. 

The one-dimensional Schrodinger equation can still be used to describe the motion of the particle if the Cartesian coordinate x is replaced by s, the distance moved by the particle along the circumference of the circle from its starting point. Although this variable describes a curved path, its use in place of x can be justified because the particle is constrained to move along this path. With the potential energy K put equal to zero, the Schrodinger equation can then be written as ... 

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