Finite potential well on arvau

Wikipedia / Finite potential well

The finite potential well (also known as the finite square well) is a concept from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a "box", but one which has finite potential "walls". Unlike the infinite potential well, there is a probability associated with the particle being found outside the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than the potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a non-zero probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls (cf quantum tunnelling).

Particle in a one-dimensional potential well

For the one-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as: where

is the reduced Planck constant,
is the mass of the particle,
is the potential energy at each point x,
is the (complex valued) wavefunction, or "eigenfunction", and
is the energy, a real number, sometimes called eigenenergy.

For the case of the particle in a one-dimensional box of length L outside the box, and zero for x. The wavefunction is composed of different wavefunctions; depending on whether x is inside or outside of the box, such that:

Inside the box

For the region inside the box, V(x) = 0 and Equation 1 reduces to resembling the time-independent free schrödinger equation where A and B can be any complex numbers, and k can be any real number.

Outside the box

and equation

There are two possible families of solutions, depending on whether E (the particle is in a bound state) or E (the particle is in an unbounded state).

, i.e., the non-normalizable states still contribute to the continuous part of the spectrum as generalized eigenfunctions of an unbounded operator.

Now in order to find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A, B, F, G, H and I that satisfy those conditions.

Finding wavefunctions for the bound state

Solutions to the Schrödinger equation must be continuous, and continuously differentiable. These requirements are boundary conditions on the differential equations previously derived, that is, the matching conditions between the solutions inside and outside the well.

In this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations.

function must be continuous and differentiable. In other words, the values of the functions and their derivatives must match up at the dividing points: so taking the ratio gives

depend on the energy. What we have found is that the continuity conditions cannot are discrete; the corresponding eigenfunctions are bound states are continuous.)

The energy equations cannot be solved analytically. Nevertheless, we will see that in the symmetric case, there always exists at least one bound state, even if the well is very shallow..

and using the floor or ceiling functions:

. ).

is (however shallow or narrow the well), there is always at least one bound state.

, and we recover the case of the infinite square well.

. But this is just the energy of the bound state of a Delta function potential, as it should be.

as is reported in the figure.

Asymmetric well

Consider a one-dimensional asymmetric potential well given by the potential

Particle in a spherical potential well

is the radius from the origin. The solution for the wavefunction with zero angular momentum is given by satisfying the condition

is given by

, the ground state case) is given by

Spherically symmetric annular well

The results above can be used to show that, as to the one-dimensional case, there is two bound states in a spherical cavity, as spherical coordinates make equivalent the radius at any direction.

The ground state (n is the radial part of the wave function. Notice that for (n = 1) angular part is constant (ℓ = 0).

Different differential equation

are Bessel, Newman and Hankel spherical functions respectively, and could be rewritten as function of standard Bessel function.

Existence of roots to above equations are always guaranteed. The results are always with spherical symmetry.