Question

In his paper 'On the theory of quantum mechanics' (1926), Dirac makes the following argument for why electrons should be considered indistinguishable:

Consider now a system that contains two or more similar particles, say, for definiteness, an atom with two electrons. Denote by (mn) that state of the atom in which one electron is in an orbit labelled m, and the other in the orbit n. [...] The two transitions [(mn)→(m′n′) and (mn)→(n′m′)] are, however, physically indistinguishable, and only the sum of the intensities for the two together could be determined experimentally.

I don't understand the final statement. The transition frequency, fmm′=Em−E′m/h can be different from fmn′ and similarly fnn′≠fnm′, so the two transitions give separate pairs of spectral lines, whose intensities can be measured separately. What am I missing here?

Answer 1

Electrons in an atom are considered indistinguishable not because their individual spectral lines cannot be measured, but because the quantum states of identical particles like electrons cannot be distinguished, as enforced by the Pauli exclusion principle and the fundamental nature of quantum mechanics.

Step-by-step explanation:

The confusion arises from a difference in how we can theoretically label transitions versus what can be experimentally distinguished.

Despite different transitions having potentially different transition frequencies, for example fmm' and fmn', quantum mechanics imposes a fundamental indistinguishability on identical particles like electrons.

This is rooted in the concept that swapping two indistinguishable fermions should not produce a new state; therefore, transitions involving such particles can't be labeled in a way that reflects individual particle histories.

When Dirac mentions the summed intensities of the transitions, he is referring to the fact that we cannot distinguish whether electron A went from orbit m to orbit m' and electron B from n to n', or vice versa.

As a result, what is measured is the overall probability of the two-electron system moving from the combined initial states to the final states.

Additionally, the Pauli exclusion principle contributes to this indistinguishability, stating no two electrons can have the same quantum state, further complicating the ability to distinguish between individual electron transitions.

Moreover, the observation of spectral lines as doublets in phenomena such as fine structure emphasizes the role of electron spin quantum numbers and reflects the influence of factors like electron spin and magnetic interactions, which contribute to the energy differences even when electrons are in the same orbital.