Particle in a centrally symmetric potential field

A particle in a centrally symmetric potential field is a classical and quantum mechanical problem that describes the motion of a particle under the influence of a force that depends only on the distance from a central point. Mathematically, the potential energy V(r) depends only on the radial distance r from the central point, where r is typically the distance from the particle to the origin in a spherical coordinate system.

The classical description of such a system is governed by Newton's laws of motion, where the particle moves under the influence of the force derived from the potential energy V(r). In this case, the particle's motion can often be described using radial and angular coordinates.

In quantum mechanics, the problem of a particle in a centrally symmetric potential field is solved by separating the Schrödinger equation into radial and angular parts. The radial part typically results in the radial wavefunction R(r), which describes the radial probability density of finding the particle at a certain distance from the origin. The angular part is usually solved using spherical harmonics, which describe the angular dependence of the wavefunction.

For example, in the case of the hydrogen atom, where the electron is under the influence of the Coulomb potential created by the proton, the potential energy depends only on the distance r from the nucleus:

V(r)=−4πϵ_o/​r²​

Solving the Schrödinger equation for this potential results in the well-known hydrogen wavefunctions, which describe the probability distributions of finding the electron at different distances from the nucleus and in different angular orientations.

In general, systems with centrally symmetric potential fields are important in many areas of physics, including atomic and molecular physics, nuclear physics, and classical mechanics. They provide valuable insights into the behaviour of particles under the influence of central forces.

∇²𝜓( r, 𝜃, 𝜙) + 2m/ℏ² ( E - V(r)𝜓( r, 𝜃, 𝜙) = 0

Since V depends only on r, we can solve the equation by the method by separation of variable

Since LHS depends only upon r and RHS depends 𝜃 and 𝜙. Each side must be equal to the some constant c. Let's consider

angular part of wave equation. Multiply throughout by Y(𝜃, 𝜙)sin²𝜃

Again using method of separation

Putting the equation for Y( 𝜃, 𝜙) in the equation and dividing by ⊝(𝜃)Φ(𝜙)

LHS depends only upon 𝜃 and RHS depends on 𝜙 each side must be equal to the same constant m². Now considering

The solution is

Since wavefunction must be single value

Now, we consider

Dividing throughout by sin²𝜃

Put cos 𝜃 = x

=> - sin 𝜃 d𝜃 = dx

⊝(𝜃) = P(cos𝜃) = P(x)

First we solve this equation for m = 0

It can be shown that by the method of series, the solution exists only when C = l (l +1) where l is zero or positive integer, otherwise the solution diverges at x= ±

The series is now polynomial P_l(x) of Lth degree known as Legendre Polynomial.

The solution of the above differential equation for m not equal to zero can be shown to be

Where P_l^m is the product of ( 1- x²)^lml/ 2 and the polynomial of ( L - l m l ) th degree.

This note is a part of the Physics Repository.