Schrodinger equation harmonic oscillator

The harmonic-oscillator Schrodinger equation (5.9) leads to a second-order differential equation with nonconstant coefficients of the form (5.12) ... 

The harmonic oscillator has V = and the harmonic-oscillator Schrodinger equation contains the three constants k, m, and h. We seek to find a dimensionless reduced energy , and a dimensionless reduced x coordinate x, that are defined by... 

Computer Program for the Numerov Method. Table 4.1 contains a BASIC computer program that applies the Numerov method to the harmonic-oscillator Schrodinger equation. The character in the names of variables makes these variables double precision. M is the number of intervals between and and equals (x,niax r,o)/ S- Lines 55 and 75 contain two times the potential-energy function, which must be modified if the problem is not the harmonic-oscillator. If there is a node between two successive values of x then the values at these two points will have opposite signs (see Problem 4.43) and statement 90 will increase the nodes counter NN by 1. 

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. 

We can multiply both sides by R, leaving the harmonic oscillator Schrodinger equation,... 

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

The operators have now been dealt with and the wavefunction will cancel on both sides. This shows that we have found the value for jS that makes our trial function into a working solution for the harmonic oscillator Schrodinger equation. 

Figure 11.10. First three eigenfunctions for harmonic oscillator Schrodinger equation.

In this section we shall apply the realizations of so(2, 1) to physical systems, such as the nonrelativistic Coulomb problem, the three-dimensional isotropic harmonic oscillator, Schrodinger s relativistic equation (Klein-Gordon... 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

Tlris is the Schrodinger equation for a simple harmonic oscillator. The energies of the system are given by E = (i + ) x liw and the zero-point energy is Hlj. 

We now solve the Schrodinger eigenvalue equation for the harmonic oscillator by the so-called factoring method using ladder operators. We introduce the two ladder operators d and a by the definitions... 

The Schrodinger equation for this three-dimensional harmonic oscillator is... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrodinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian... 

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 first derivative vanishes since U(R) has a minimum at Re. Within this approximation the nuclear Schrodinger equation reduces to that of a harmonic oscillator, whose frequency to is given by ... 

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. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

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