A simple formula is presented for the change in energy of a quantum system in a box when the box is infinitesimally deformed. The result is applied to rederive a formula given by Barton et al., [Am. J. Phys. 58, 751–755 (1990)] for the difference in energy of a bound state calculated with and without box boundary conditions. The formula can also be used to discuss pressure in a quantum system. A generalization of the result has been used for calculating scattering amplitudes for waves scattered by a rough extended surface. Here, a similar manifestly reciprocal result is derived for scattering by bounded objects.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
Wave mechanics of a particle in 1-D box (size
=d) is critically analyzed to reveal its untouched aspects. When the particle rests in its ground state, its zero-point force (
Fo) produces non-zero strain by modifying the box size from
d to
d′=d+Δd in all practical situations where the force (
Fa) restoring
d is not infinitely strong. Assuming that
Fa originates from a potential
∝x2 (
x being a small change in
d), we find that: (i) the particle and strained box assume a mutually bound state (under the equilibrium between
Fo and
Fa) with binding energy
ΔE=−εo′Δd/d′ (with
εo′=h2/8md′2 being the ground state energy of the particle in the strained box), (ii) the box size oscillates around
d′ when the said equilibrium is disturbed, (iii) an exchange of energy between the particle and the strained box occurs during such oscillations, and (iv) the particle, having collisional motion in its excited states, assumes collisionless motion in its ground state. These aspects have desired experimental support and proven relevance for understanding the physics of widely different systems such as quantum dots, quantum wires, trapped single particle/ion, clusters of particles, superconductors, superfluids, {\it etc.} It is emphasized that the physics of such a system in its low energy states can be truly revealed if the theory incorporates
Fo and related aspects.
A free particle is shown to possess any non-negative energy value. Its position is described by a localized wave function called a wave packet. The solution to the particle in infinite potential well problem displays how the concept of energy quantization appears in quantum mechanics. Like the particle in an infinite well, the particle in a finite well shows discretization of energy. But unlike the particle in an infinite well, the particle in a finite well exhibits a non-zero probability of being found outside the well leading to the concept of quantum-mechanical tunneling.
A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
A particle moves in one spatial dimension in an attractive symmetric potential vf(x) and obeys nonrelativistic quantum mechanics. It is supposed that the trajectory function F(v), which tells us how the ground-state energy depends on the coupling parameter v, is known. We determine the minimum f(0) of the potential and the mean kinetic energy s(v) from F(v), and we prove that the potential shape f(x) has the constant value f(0) on an interval [-a,a] if and only if s is bounded for all v>0. Moreover, if sK, then a>(\ensuremath{\pi}/2)[\ensuremath{\Elzxh}/(2mK)
]1/2. These are partial results towards geometric spectral inversion, the determination of the potential shape f from the corresponding ground-state energy trajectory F.