Three Dimensional Virial Theorem for the Hydrogen Atom
Catherine Therese J. Quiñones
The virial theorem is a general theorem relating the potential energy (V) and the kinetic energy (KE) in a bound system. A simple physical example is a small object orbiting around another object bound by a force. The virial theorem states that the potential energy is twice the kinetic energy thus,
(1)
Here, 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 bound by the Coulombic force. Hence, the potential energy is
(2)
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 total energy is given by
![E_n = - \Big[\frac{m}{2\hbar^2}\Big(\frac{e^2}{4\pi\epsilon_0}\Big)^2\Big]\frac{1}{n^2}](http://www.quantumsciencephilippines.com/wp-content/uploads/eq_072153e02a9ced355f471d2a2c54be49.png)
(3)
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 (2) and (3) to eqn (1). Hence,
(4)
![-\frac{e^2}{4\pi\epsilon_0}<\frac{1}{r}> = -2\Big[ \frac{m}{2\hbar^2}\Big(\frac{e^2}{4\pi\epsilon_0}\Big)^2\Big] \frac{1}{n^2}](http://www.quantumsciencephilippines.com/wp-content/uploads/eq_9b8f915af0aa3de4c1782cb00be77307.png)
(5)
(6)
Note that the term inside the parenthesis is just 
, where 
is the Bohr radius . Hence we can write the expectation value of as,
(7)
Thus we have derived the expectation value, 
, of the hydrogen atom in the unperturbed state using the virial theorem.
June 2nd, 2010 at 6:44 pm
In Eq (3), isn’t this the *total* energy (kinetic + potential) of an electron in the n-th energy level? It is negative, which doesn’t to make sense for a kinetic energy…
June 14th, 2010 at 9:55 pm
Gekko, definitely you’re right. Kinetic energy is definitely positive. Equation (3) actually refers to the total energy and this post has been updated accordingly.
Thanks for pointing this out!