8 - Hydrogen Atom

Summary

The hydrogen atom is the simplest atom, which is a system of two particles, namely, a proton and an electron. In this system, a single electron moves in the field of a nucleus of unit positive charge. This system is bound by electrostatic force of attraction. In other words, we can say that the electron of the hydrogen atom revolves around the nucleus in the field of the force of the nucleus. Since hydrogen is the simplest atom, it formsthe basis for the theoretical treatment of more complex atomic systems. The Schrödinger equation for hydrogen atom can be solved in closed form. When the next atom in the order of increasing complexity is considered, for example, Helium, which consists of two electrons moving in the field of the nucleus with charge two, the Schrödinger equation can no longer be solved in closed form. For the solution of such systems, the approximate methods are employed to get the wave function.

The hydrogen atom (simple solution of Schrödinger equation)

As already stated that the hydrogen atom is one electron system with a charge ‘−e’ rotating around a central proton (p) with charge ‘+e’. The potential energy of the system is expressed as

Putting the value of V in the Schrödinger equation, we have

To solve the Schrödinger equation for hydrogen atom, let us consider

where, r is the distance of electron from the nucleus.

Differentiating Eq. (8.3) with respect to x, we shall get

Differentiating the above equation again with respect to x, we obtain

Putting the value of or r²ψ in Eq. (8.2), we shall get

The simplest solution of Eq. (8.5) may be given by

Differentiating this equation with respect to r twice, we shall get

Putting these values in Eq. (8.5), we have

Since Eq. (8.8) is true for any value of r, the sum of the first two terms, which is independent of r and the third term involving r should be separately zero because the equation should also be satisfied when r → ∞,

We can equate Eqs (8.9) and (8.10) and shall get

This is the value of energy of hydrogen atom. It is clear that E is negative, which indicates that the electron is bound.