Eigenfunctions of the harmonic oscillator

Next, let us compute, for the one-dimensional harmonic oscillator, the "first moment" integral n x n). First, remember that the eigenfunctions of the harmonic oscillator are orthonormal ... 

We studied the critical behavior of the eigenfunctions and resonances of the Hamiltonian equation, Eq. (101), using the FSS method described in Section IV. As a basis function for the finite-size scaling procedure, we used the orthonormalized eigenfunctions of the harmonic oscillator with mass equal to 1 and frequency equal to a ... 

Figure 9 Wigner distribution function of the n = 10 eigenfunction of the harmonic oscillator. The picture shows the extent of the wave function in phase space which has nearly optimal sampling due to the balance between the representation of the kinetic and potential energy.

Show that the first two energy eigenfunctions of the harmonic oscillator are orthogonal to each other. 

A crucial point is to see that these wavefunctions are eigenfunctions of the harmonic oscillator Hamiltonian and to determine the eigenvalues associated with each. For lyo. 

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. 

It is customary to express the eigenfunctions for the stationary states of the harmonic oscillator in terms of the Hermite polynomials. The infinite set of Hermite polynomials // ( ) is defined in Appendix D, which also derives many of the properties of those polynomials. In particular, equation (D.3) relates the Hermite polynomial of order n to the th-order derivative which appears in equation (4.39)... 

From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions n), we find that the matrix elements of a and are... 

The eigenvalues and eigenfunctions of the simple harmonic oscillator are well known. A detailed account of the solution of the wave equation in (2.157) is given by Pauling and Wilson [11], The solution of equation (2.163) using creation and annihilation operators is described in the book by Bunker and Jensen [12]. The energy levels of the harmonic oscillator are given by... 

B, 5i[cm 1], d, d [dimensionless], F4, I-, in [aJ and Angstrom] are variable parameters. The basis functions for the treatment of this mode serve as the eigenfunctions m) of the harmonic oscillator zeroth-order Hamiltonian //o 2 , with frequency... 

We have attempted to take into account this oscillatory motion of the neighboring atoms, by averaging the potential over this motion using an uncoupled isotropic harmonic oscillator approximation. This determined the cell potential of the central atom. We assumed that the motion of the central atom can be described by the eigenfunction of a harmonic oscillator. 

Fig. 12.7. Sadlej relation. The electric field mainly causes a shift of the electronic charge distribution toward the anode (a). A GTO represents the eigenfunction of a harmonic oscillator. Suppose that an electron oscillates in a parabolic potential energy well (with the force constant I ). In this situation, a homogeneous electric field corresponds to the perturbation x, that conserres the hannonicity with unchanged force constant k (b).

In Fig. 3-2a are shown the potential, some eigenvalues, and some eigenfunctions for the harmonic oscillator. The potential function is a parabola [Eq. (3-14)] centered at x = 0 and having a value of zero at its lowest point. For comparison, similar information... 

The leading term in Eq. (6.9), li 0) Xm Xt is zero for because of the orthogonality of the eigenfunctions. The term (dnldx)o xm x x, thus dominates the series, provided that (3/i/3x)o 0. Examination of the eigenfunctions of a harmonic oscillator shows that the integral Xm x Xn) in this term is non-zero only if m = n 1 (Box 6.2). The formal selection mles for excitation of a harmonic vibration are, therefore ... 

The eigenfunction of the molecular pair in the ground state can be constructed immediately from the known functions of the harmonic oscillator and we see that the unperturbed function... 

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