Harmonic oscillator energy eigenfunctions

The harmonic-oscillator wave functions are given by a Hermite polynomial times an exponential factor (Problem 4.19b). By virtue of the expansion postulate, any well-behaved function/(x) can be expanded as a linear combination of harmonic-oscillator energy eigenfunctions ... 

When the solution is carried out, the factors i/f i (1), V 2(2), V 3 (3), and V 4(4) are harmonic oscillator energy eigenfunctions with quantum numbers vi, V2, vs, and V4. Since the oscillators are distinguishable from each other, the system wave function does not have to be symmetrized or antisymmetrized, and there are no restrictions on the values of the quantum numbers. The system energy eigenvalues corresponding to Eq. (25.1-2) are... 

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. 

Since equation (10.43) with F = 0 is already solved, we may treat V as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S H q) are the eigenkets n) for the harmonic oscillator. The first-order perturbation correction to the energy as given by equation (9.24) is... 

The total wave function can, accordingly, be written as a product of wave functions corresponding to each mode. The energy eigenfunctions corresponding to each mode are, in particular, just the well-known eigenfunctions for a harmonic oscillator. 

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... 

The operator between the brackets is the harmonic oscillator Hamiltonian, which has the basis functions (112) as its eigenfunctions the remaining term is taken into account via Eq. (114). The rotational kinetic energy operator L(cuP) [Eq. (26)] can be written in terms of the shift operators J = J, + iJi, and the operator J, which act on the basis as... 

Readers are invited to do their own reading of Ref. [1], including the effects of the confinement by elliptical cones on the energy spectra and eigenfunctions of the familiar free particle and harmonic oscillator. 

The choice of parabolic coordinates in [35] and Equation (56) is motivated by our interest in exploiting the connection between the superintegrable harmonic-oscillator and atomic-hydrogen systems [33-35]. For instance, the well-known eigenfunctions and energy eigenvalues for the two-dimensional harmonic oscillators can be written immediately by borrowing them from [33] ... 

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.

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