A hydrogen atom consists of proton and electron. The electron is orbiting round the nucleus and the nucleus also in general is in motion. So we assign coordinates for both electron and nucleus separately.

The radial part of the wave equation often appears in problems involving spherically symmetric systems, such as the hydrogen atom.

We have Schrodinger wave equation

V depends upon relative coordinates x_1 - X_2, y_1 - y_2, z_1 - z_2.

Let x_1 - X_2 = x

y_1 - y_2 = y

z_1 - z_2 = z

We introduce centre of mass coordinates

Then the above equation appears

𝜇 is known as reduced mass

1/𝜇 = 1/m_e + 1/m_p

When we put the expression for wave function in the above equation we obtain

The equations ii tells us that the centre of the mass of the system of two particles move like a particle of mass M and the first equation tells that a particle having mass equal equal to reduced mass is in motion in potential field V(x, y, z) about a fixed centre

Thus a particle having mass equal to reduced mass 𝜇 and having charge equal to charge of electron is revolving round a fixed positive charge +e

Now consider the time independent Schrodinger equation

Since the particle is in motion in central symmetric potential we can express 𝜓(r,𝜃, 𝜙) = R(r)𝜓(𝜃, 𝜙) we obtain

solution of radial part of the wave equation

where:

R is the radial part of the wavefunction,
r is the radial distance from the origin,
ℏ is the reduced Planck constant,
E is the total energy of the system,
V(r) is the radial component of the potential energy,
l is the orbital angular momentum quantum number.

The solution of radial part of the wave equation

The solutions to this equation depend on the specific form of the potential V(r) and boundary conditions of the problem. In the case of the hydrogen atom, for example, the potential V(r) is the Coulomb potential due to the nucleus, and solutions to the radial part of the wave equation lead to the radial wavefunctions associated with different energy levels and angular momentum states. These solutions are usually expressed in terms of special functions like the associated Laguerre polynomials.

This note is a part of the Physics Repository.

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