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An important problem in quantum mechanics is that of a particle in a spherically symmetric potential, i. 0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases. 0 elsewhere, or a particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot. As the previous case, but with an infinitely high jump in the potential on the surface of the sphere. 2 for the three-dimensional isotropic harmonic oscillator.

We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. V0 and it is zero outside the sphere. A potential with such a finite discontinuity is called a square potential. We first consider bound states, i. Those have an energy E less than the potential outside the sphere, i.

The resolution essentially follows that of the vacuum with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i. The wavefunction must be regular at the origin. The wavefunction and its derivative must be continuous at the potential discontinuity. The wavefunction must converge at infinity. The first constraint comes from the fact that Neumann N and Hankel H functions are singular at the origin. A a constant to be determined later.

Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions. Then we normalize the generalized Laguerre functions to unity. This normalization is with the usual volume element r2 dr. This leads to the condition on l given above. Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that n and l are both of the same parity.

The mass m0, introduced above, is the reduced mass of the system. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. Due to its simplicity, the model allows insight into quantum effects without the need for complicated mathematics. The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. The simplest form of the particle in a box model considers a one-dimensional system. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.

The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy. L is the length of the box, xc is the location of the center of the box and x is the position of the particle within the box. In this sense, it is quite dangerous to call the number k a wavenumber, since it is not related to momentum like “wavenumber” usually is. The rationale for calling k the wavenumber is that it enumerates the number of crests that the wavefunction has inside the box, and in this sense it is a wavenumber. T is the kinetic and V the potential energy. The wavefunction must therefore vanish everywhere beyond the edges of the box.

Also, the amplitude of the wavefunction may not “jump” abruptly from one point to the next. Here one sees that only a discrete set of energy values and wavenumbers k are allowed for the particle. It is expected that the eigenvalues, i. This phase shift has no effect when solving the Schrödinger equation, and therefore does not affect the eigenvalue. The momentum wavefunction is proportional to the Fourier transform of the position wavefunction.

The solution to the finite well particle in a box must be solved numerically, 0 is an arbitrary reference momentum. Quantum dots have a variety of functions including but not limited to fluorescent dyes, the probability density does not go to zero at the nodes if relativistic effects are taken into account via Dirac equation. As the previous case, the quantum well laser is heavily based on the interaction between light and electrons. In classical physics — first we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, and in this sense it is a wavenumber.