Catherine Therese J. Quiñones

The virial theorem is a general theorem relating the potential energy (V) and the kinetic energy (T) in a bound system.  A simple physical example is a small object orbiting around another object  bound by a force as in the case of a hydrogen atom. The average kinetic energy and potential energies of a system of particle that interact by Coulomb forces are related by

                                                                                                                  (1)

Since, the Hamiltonian H of the given system is

                                                                                                   (2)

Thus substituting Eqn (1) to Eqn (2) yields,

                                                                                                       (3)

                                                                                                                        (4)

Now, we will derive the expectation value of  in the unperturbed state of the of a hydrogen atom. We can use the virial theorem to easily solve the expectation value since the system can be considered a bound system with the electron orbiting around the proton which is bound by the Coulombic force.

For a hydrogen atom, the potential energy is expressed as

                                                                                                                  (5)

where is the charge of the electron and the proton, represents the separation distance between the two charges and is the permittivity of free space . The negative sign indicates that the force is attractive.

The allowed energies is given by

                                                                                     (6)

where is the mass of the particle, is Planck’s constant over and which indicates the quantization of the energy level. The solution is very straight forward. All we need is to plug in eqn (5) and (6) to eqn (4).  Hence,

                                                                                                                                           (7)

                                                                                                        (8)

Note that the term inside the parenthesis is just , where is the Bohr radius . Hence we can write the expectation value of as,

                                                                                                                      (9)

Thus we have derived the expectation value, , of the hydrogen atom in the unperturbed state using the virial theorem.

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