Hamiltonian equation harmonic oscillator

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

The case of three and four electrons is more complicated, but the two characteristic features of the energy spectra observed for small coz, i.e., the nearly-degenerate multiplet structure of the energy levels of different spin multiplicities and the harmonic band structure of these levels, can be rationalized in a similar way. In the case of three electrons, for example, the internal space can be defined by the two correlated coordinates Zb and zc defined by Equation (11). The potential function becomes a sum of two harmonic-oscillator Hamiltonians for the Zb and zc coordinates plus three Coulomb-type potentials originating from the three electron-electron... 

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

Since the rotational term in equation (2.157) can also be expanded as a power series in /, the complete perturbation to the harmonic oscillator Hamiltonian is... 

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

The matrix equation (18) is block-diagonal (as shown) only if the coupling modes, a = 2,5e, and 5a, are mutually orthogonal. The two-dimensional harmonic oscillator Hamiltonians for the a = 2 and 5e modes are given by... 

When the quantum mechanical Hamiltonian for vibration is constructed from Eq. (22.4-16) there are 3 — 5 or 3n — 6 terms, each one of which is a harmonic oscillator Hamiltonian operator. The variables can be separated, and the vibrational Schrodinger equation is solved by a vibrational wave function that is a product of 3 - 5 or 3n — 6 factors ... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

The Hamiltonian operator H for the harmonic oscillator is given in equation (4.12). The quantity c) is then determined as follows... 

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... 

Thus, the solution to the classical problem with the Hamiltonian (energy) function given by Equation 3.19 is a set of 3N harmonic oscillators with 3N frequencies Aj = 4tt2v2. The A s result from the diagonalization of the F matrix of Equation 3.15. 

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

Hamiltonian operators, 88, 151, 153, 197 core, 204 commutation with O, 200, 218 invarianoe of, 151. harmonic foroe constants, 165. harmonic oscillator, approximation, 165 equation, 170. 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

Let us now lift this disk D2 into phase space. To do so, one must go back to the sphere equation, Eq. (37). There are several ways of depicting a 3-sphere one is particularly appropriate here [24]. The sphere is dynamically composed of two identical harmonic oscillators without explicit coupling, but whose total energy is a constant, hs3 > 0. Let us thus transform the Hamiltonian (37) in action angle variables, where N,Iy are the actions of the two oscillators and Q, 0, are the two associated angles. Since... 

---