A simple proof is given of the fact that the ground state of a central potential is an S state. The ordering of the bound states of a central potential according to the value of the angular momentum and the number of nodes of the solution to the radial Schroedinger equation is also discussed.
The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.
Recent work on states bound by a central potential whose Laplacian is positive is extended to cover a wider class of potentials for which energy level ordering theorems can be proved. This class includes power law potentials in particular. Constraints on the moments of the radial distance are improved and similarly for the kinetic energy. These constraints are shown to provide tight bounds on the energies of ground states for power law potentials, and on their variation with angular momentum. They are also used to bound the position of the maximum of the wavefunction for these states.
In quantum physics,the variational method can be used to study the conditions for the existence of bound states in a central field.We take the following six kinds of potentials,V1(r)=-V0a δ(r-a),V2(r)=-V0e-r/a,V3(r)=-V0ae-r/a/r,V4(r)=-V0e-r2/a2,V5(r)=-e2/r,and V6(r)=-A/r2 as examples to demonstrate the usage of this method.The derived conditions for V1 to V4 are μγa2/h-20.68,0.84,1,1.95 respectively;for V5,the bound states can exist for e with arbitrary value,while for V6,there is no bound states for any value of A.We also get the precise conditions by solving the Schroedinger equation for these potentials.A comparison of the results obtained by these two methods shows that the conditions derived by the variational method are sufficient,but not necessary conditions.
It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r --> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r --> 0 or r --> infinity.