0.3 Quantum energy levels in atoms  (Page 8/9)

This probability distribution is determined by quantum mechanics. The motion of the electron in a hydrogen atom isdescribed by a function, often called the wave function or the electron orbital and typically designated by the symbol Ψ. Ψ is a function of the position of the electron $r$ , and quantum mechanics tells us that $\left|\Psi \right|^2$ is the probability of observing the electron at the location $r$ .

Each electron orbital has an associated constant value of the electronic energy, $E_n$ , in agreement with our earlier conclusions. In fact, quantum mechanics exactly predictsthe energy shells and the hydrogen atom spectrum we observe. The energy of an electron in an orbital is determined primarily by twocharacteristics of the orbital. The first, rather intuitive, property determines the average potential energy of the electron:an orbital which has substantial probability in regions of low potential energy will have a low total energy. By Coulomb’slaw, the potential energy arising from nucleus-electron attraction is lower when the electron is nearer the nucleus. In atoms withmore than one electron, electron-electron repulsion also contributes to the potential energy, as Coulomb’s lawpredicts an increase in potential energy arising from the repulsion of like charges.

A second orbital characteristic determines the contribution of kinetic energy, via a more subtle effect arisingout of quantum mechanics. As a consequence of the uncertainty principle, quantum mechanics predicts that, the more confined anelectron is to a smaller region of space, the higher must be its average kinetic energy. Since we cannot measure the position ofelectron precisely, we define the uncertainty in the measurement as $\Delta (x)$ . Quantum mechanics also tells us that we cannot measure the momentum of an electron precisely either, so there is anuncertainty $\Delta (p)$ in the momentum. In mathematical detail, the uncertainty principle states that these uncertainties are relatedby an inequality:

where $h$ is Planck’s constant, $6.62E-34Js$ (previously seen in Einstein’s equation for the energy of a photon). This inequality reveals that, when an electron moves in a small area with a correspondingly smalluncertainty $\Delta (x)$ , the uncertainty in the momentum $\Delta (p)$ must be large. For $\Delta (p)$ to be large, the momentum must also be large, and so must be the kinetic energy.

Therefore, the more compact an orbital is, the higher will be the average kinetic energy of an electron in thatorbital. This extra kinetic energy, which can be regarded as the confinement energy , is comparable in magnitude to the average potential energy of electron-nuclear attraction. Therefore,in general, an electron orbital provides a compromise, somewhat localizing the electron in regions of low potential energy butsomewhat delocalizing it to lower its confinement energy.

Electron orbitals and subshell energies

We need to account for the differences in energies of the electrons in different subshells, since we knowthat, in a Hydrogen atom, the orbital energy depends only on the n quantum number. We recall that, in the Hydrogen atom, there is a single electron. The energy of that electron is thus entirely due to its kinetic energy and its attraction to thenucleus. The situation is different in all atoms containing more than one electron, because the energy of the electrons is affectedby their mutual repulsion. This repulsion is very difficult to quantify, but our model must take it into account.