Abstract. The one-dimensional, non-relativistic, quantum Morse oscillator is studied at the classical limit. In fact, near the classical limit, the energy eigenvalues relative to the eigenstates of the nonrelativistic, time-independent, Schrödinger equation with Morse potential are negative and approximately proportional to the square of the corresponding vibrational quantum number. Within this framework, the mass of the oscillator in question is found to be negative. This can take place in certain particle phenomena and, in fact, occurs, for instance, in semiconductor superlattices. These cases are outlined very briefly in the present paper.Keywords: Morse oscillator; classical limit; negative mass; non-relativistic quantum particles.
IntroductionThe significant role of the Morse potential in Nuclear Physics and Molecular Physics is well-known. Really, the above potential has a great relevance within Particle Physics in its broad sense. Indeed, studying the behaviour of both relativistic and non-relativistic quantum particles under the Morse potential presents a wide variety of issues of much interest. It is well-known that the above (nonrelativistic) quantum anharmonic oscillators have a finite number of bound states and infinite number of unbound states. As a matter of fact, considering the one-dimensional case, the energy eigenvalues relative to the unbound states are negative and roughly proportional to the square of the involved vibrational quantum number. This corresponds to the anharmonic oscillator in question close to the classical limit so energy tends to minus infinity as the quantum number tends to infinity. But, unfortunately, in a certain part of the current literature, the existence of negative energy eigenvalues by, say, extrapolation to them of the general formula for the energy levels, is not understood.In the following, we will show that the mass of the aforementioned Morse oscillator must be negative near the classical limit. The one-dimensional case will be regarded. In addition, as notorious and interesting examples, we will discuss the role of electrons of negative mass in semiconductor superlattices [1][2][3] and, on the other hand, we will comment on cosmological as well as on Bose condensates and various elementary-particle questions dealing with the possibility of negative rest-mass [4,5]. In particular, with respect to semiconducting superlattices, we note that the fact that there are electrons with negative rest-mass mass in these structures comes from the existence of allowed and forbidden minibands formed from allowed and forbidden energy bands [1][2][3]. In this context, there are negative differential and absolute electrical conductivities [1][2][3]. Really, the above mentioned facts have great importance and will be discussed very briefly.
Theoretical FormulationThe energy eigenvalues of a (non-relativistic) quantum-mechanical particle under a one-dimensional Morse potential read: 2
