Schrodinger equation oscillator

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... 

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. 

Including higher-order terms leads to anharmonie correetions to the vibration, sueh effects are typically of the order of a few %. The energy levels obtained from the Schrodinger equation for a one-dimensional harmonie oscillator (diatomic system) are given by... 

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. 

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... 

We can compute from first principles all possible vibrational modes for 3iA oscillators in the cell unit, solving the Schrodinger equation with appropriate atomic (and/or molecular) wave functions. 

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

To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation the allowed energy levels are... 

The discretized adiabatic procedure, and its analog with STIRAP, is but one possibility for achieving broadband response of an optical device. An alternative, which we discuss, relies on the analogy between the Jones vector description of an optical beam and the two-state time-dependent Schrodinger equation (TDSE). This equation has two commonly used solutions. One is rapid adiabatic passage (RAP), produced by swept detuning (a chirp), and the other is Rabi oscillations, specifically a pi pulse. The RAP has theoretical connection with STIRAP, but the pi pulses have no such connections. We describe application of a procedure that has been used to extend the traditional pi pulses to broadband excitation. This can accomplish the present goal of PAP, under complementary conditions. 

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...

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. 

Solve the Schrodinger equation for the two-dimensional isotropic harmonic oscillator using plane polar coordinates. First show that... 

These manipulations have brought us to a familiar equation we recognize (4.23) as the Schrodinger equation (1.132) for a one-dimensional harmonic oscillator with force constant ke. Before we can conclude that (4.23) and (1.132) have the same solutions, we must verify that the boundary conditions are the same. For quadratic integrability, we require that S(q) vanish for q = oo. Also, since the radial factor F(R) in the nuclear wave function is... 

The 3/V —6 one-dimensional Schrodinger equations (6.50) are easily solved. The one-dimensional harmonic-oscillator Hamiltonian is... 

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