Need verification: Hamiltonian formulation states that an atom is bounded to an eigenenergy state due to opposing kinetic and potential energy

Question

I received an explanation from someone who said that electrons in an atom are trapped in an eigenenergy state $E_n$ as per Hamiltonian mechanics is because the KE and PE of the atom balances themselves out.

The claim was that as the electron cloud moves, the KE oscillates and the PE oscillates and the summation of that KE and PE oscillation is the eigenenergy

I know about eigenenergy as derivable from Schrodinger's equation, but I have never heard of this explanation. While it sounds plausible, can someone verify whether there is indeed ground for this claim?

Answer 1

The electron in an atom has no defined kinetic energy, because it has no defined velocity. If it had a well-defined velocity $\vec v$ its position would be absolutely undetermined - i.e. all the universe. It would be interesting for you to see the question

Quantum mechanics,and how the law $ΔxΔp≥ℏ/2$ explains the paradox regarding atoms .

However, I can do something else for helping, and that would show that what you were said it not far from the truth. Namely, I shall calculate the average kinetic energy. I will not complicate myself with higher states, I will do the calculus for the ground hydrogen state. This state has a relatively simple wave-function, of the form $ \psi (r) = Ce^{-r/a_0}$, where C is a constant of normalization and $a_0$ is the Bohr radius. So, what I do is

$<E_{K,0}> = C^2 \frac {\hbar ^2}{2m} \int e^{-r/a_0} \frac {d^2}{dr^2} e^{-r/a_0} dr$

$ = \frac {\hbar ^2 }{2m a_0^2} C^2 \int e^{-r/a_0} e^{-r/a_0} dr = \frac {\hbar ^2 }{2m a_0^2}$

Introducing the expression of the Bohr radius, $a_0 = \frac {4\pi \epsilon _0 \hbar ^2}{m_e c}$ we get, the same absolute value as the eigenvalue of the total energy on the ground level, $E_1 = <E_{K,0}>$.

In all, the potential energy is negative, and the total energy is also negative. We have $E_0 = <E_{K,0}> + <E_{P,0}>$, or in another form, $-<E_{P,0}> =-E_0 + <E_{K,0}>$. The kinetic energy balances half of the potential energy.