Eigenvalue harmonic oscillator

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

For obvious reasons, and a are known as step-up and step-down operators, respectively. They are also called ladder operators since they take us up and down the ladder of harmonic-oscillator eigenvalues. In the context of radiation theory, and a are called creation and annihilation operators, respectively, since their action is to create or annihilate a quantum of energy. 

Since E — hco = E -i for harmonic-oscillator eigenvalues, it follows that... 

Equation (13.25) is the same as the Schrodinger equation for a one-dimensional harmonic oscillator with coordinate x, mass (i, potential energy kgX, and energy eigenvalues Ejm — U(Rg) — J J + Vfi /2tt.R. [The boundary conditions for (13.25) and (4.34) are not the same, but this difference is unimportant and can be ignored Levine, Molecular Spectroscopy, p. 147).] We can therefore set the terms in brackets in (13.25) equal to the harmonic-oscillator eigenvalues, and we have... 

Spectroscopy, p. 147).] We can therefore set the terms in brackets in (13.25) equal to the harmonic-oscillator eigenvalues, and we have... 

The Z)-dimensional solution for equation (24) is simply the Z)-dimensional harmonic oscillator eigenvalues... 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

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

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

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 differential equation for X(x) is exactly of the form given by (4.13) for a one-dimensional harmonic oscillator. Thus, the eigenvalues Ex are given by equation (4.30)... 

We have already introduced the use of ladder operators in Chapter 4 to find the eigenvalues for the harmonic oscillator. We employ the same technique here to obtain the eigenvalues of and Jz. The requisite ladder operators and J-are defined by the relations... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

The ground-state effective Hamiltonian is diagonal with eigenvalues ha n + 5], whereas the excited state one is that of a driven quantum harmonic oscillator that must lead to coherent states. 

The eigenvalues of this Hamitonian can calculated by numerical diagonalization of the truncated matrix of the quantum system in the basis of the harmonic oscillator wave functions. The matrix elements of Hq and V are... 

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. 

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

The eigenvalue equations of the quantum harmonic oscillators Hamiltonians Hpree and H° given by Eqs. (21) and (15) are, respectively,... 

If the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function 1/(7 ), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,... 

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