Enclosed Quantum Mechanical Systems

Abstract

A brief description is given of the eigenvalue problems associated with enclosed quantum mechanical systems and of some attempts to deal with these problems. Another method is developed which leads to a general asymptotic formula for the eigenvalues. This formula yields a simple asymptotic approximation to the eigenvalue in each particular case, once the eigenfunction of the corresponding unrestricted system is known.

Spectroscopic characteristics of simple systems in a spherical cavity

“…If a free system has bonded states (as is the case when λ > 0 in the above problems), energy levels E nᐉ (λ, R) are monotonically reduced as R rises, tending to corresponding limits E nᐉ (λ, ∞). For a wide range of systems, this convergence is exponential [1,3,4,29,30]; matrix elements [3,4,8]. From relations (4), (5), it is easy to find that (10) (11) Oscillator strengths in particular do not depend on λ for a free system.…”

“…The case where R → ∞ is more complicated, since it depends on the ratio of all quantum numbers n, n', and ᐉ. As was shown in [3,30], the electron den sity of a free system at large r determines the rate at which the energy falls within limit R → ∞ and, in par ticular, the rate at which the wave functions approach the limit that corresponds to the free system. The wave functions of the 1s ground state, which fall very rapidly, converge to the limit (for the hydrogen atom, e.g., ϕ 1s (r, R) ≈ ϕ 1s (r, ∞) at R ~ 20) much faster than for the…”

Hydrogen atom in a spherical well: linear approximation

“…[3][4][5][6][7][8][9][10][11] The model has often been used as a test problem for various perturbation methods. Using their boundary perturbation method, Hull and Julius [12] obtained a formula which expresses the change of energy for the eigenstates in the confined system in terms of the corresponding wave functions in free space. This method has been improved and generalized by many authors.…”

The confined diatomic molecule problem

“…For cavity of a larger size, the energy levels of the system with fixed center of mass position are close to the corresponding levels of the free system (see, e.g., 9, 10). Thus, for relatively large A = r max ( R , x ) the energy values E v ( x , R ) = E v ( A ) are close to the vibrational energies E of the free system.…”