Ionisation energy of hydrogen: Why are kinetic and potential energy connected by $2 T = -V$ in the derivation?

Question

The following is a translation from the German Wikipedia:

The ionisation energy or binding energy B is the sum of potential energy V and kinetic energy T of the electron. $B = V+T$.
Since the following always applies in a stable orbit: $2 T = -V$, the ionisation energy is always equal in magnitude to the kinetic energy or half the potential energy: $B = T+V = -T = V/2$.

Apart from the fact that the 13.6 eV for the ionisation energy of hydrogen was obtained from the assumption of an orbiting electron, I do not understand why $2 T = -V$?

Answer 1

Classically, this is called the virial theorem, which states that for a bound system of particles under the influence of a potential of the form $V(|\mathbf x_i-\mathbf x_j|)\propto |\mathbf x_i - \mathbf x_j|^n$, then $$2\langle T \rangle = n\langle V\rangle$$

where $T$ and $V$ are the total kinetic and potential energies of the system, and the angle brackets denote the time-average.

Essentially the same relationship holds in quantum mechanics with $T$ and $V$ replaced by the corresponding operators and the time-averaging brackets replaced by expectation values.

Answer 2

This refers to the Bohr model of the hydrogen atom. The potential energy of the electron is taken to be zero at infinity. This means that the potential energy must become negative as the electron gets closer to the hydrogen atom. But kinetic energy is always positive and according to this model, the potential energy of the ground state is $-27.2 \ eV$. The kinetic energy is $+13.6 \ eV$, so when we add the two together the total energy is $-13.6 \ eV$. In other words

$$2T = - V$$

(In the case of any inverse square force law, this link between kinetic energy and potential energy always holds, and is called the Virial Theorem ).

The total energy is negative because we have to put in energy to liberate the electron. Specifically, we have to put in $+13.6 \ eV$ which is the well known ionisation energy of the hydrogen atom. And because the ground state has an energy of $-13.6 \ eV$, when we put in $+13.6 \ eV$ gives a total energy of zero, which is also consistent with what was stated earlier that the electron has zero energy at infinity or when it’s completely liberated from the atom.