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When I was reading, the question just popped into my head after noticing that the Greek letter ψ looks kind of like a wave itself. Stylized, they look even more wavy:

$$\Huge \Psi\;\Huge\psi$$

This made me want to ask the question why $\Psi$ was chosen to represent the wave function. I am aware that Erwin Schrödinger is responsible for the Schrödinger equation which calculates the wave function, giving the value for $\Psi$. Also, I read somewhere that I can no longer find, that Schrödinger is the one who chose $\Psi$ to represent the wave function.

I have found a Quora answer that gives quite a reasonable, I am just not sure if it is correct, given that they provide no sources, and I cannot find any supporting documentation to back up it. The answer given there is that the Greek letter $\Psi$ was associated with the Greek god of the sea, Poseidon. The connection is that $\Psi$ looks like the trident that Poseidon was known for. They also mentions how pronouncing $\Psi$ (Psi), sounds like the beginning of his name. I am not sure if they were meaning to suggest that there was an actual connection between the two, but judging from the Greek spelling of Poseidon on Wikipedia there is no $\Psi$ is Poseidon's name. So, the explanation goes, because $\Psi$ is associated with Poseidon, who is king of the waves, and the value this variable was to represent was the wave function, why not make $\Psi$ represent the wave function. This makes perfect sense to me.

So, my questions are, first, was Schrödinger the one who chose $\Psi$ to represent the wave function, and if not, who did? And, second, why was $\Psi$ chosen to represent the wave function (i.e. was the Quora explanation correct)? Lastly, how do you know/ what are you sources (i.e., hopefully a link to a authoritative document, or maybe your professor told you the story that they were told by Schrödinger himself)?The last part is where the Quora answer is lacking.

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    $\begingroup$ Might be able to track this down for you, but in the mean time, I suspect that $\psi$ or $\Psi$ was used simply as a placeholder, the symbol is commonly used in e.g. partial differential equations, mathematical physics, and potential theory. See my answer about electrostatic potential here: hsm.stackexchange.com/questions/14328/… $\endgroup$ Commented Oct 11, 2023 at 17:41
  • $\begingroup$ @Sam Gallagher: the symbol is commonly used in e.g. partial differential equations, mathematical physics, and potential theory --- That would be my guess also (i.e. probably one of the two or three standard symbols used at that time for PDEs in mathematical physics that were not already essentially "fixed" in meaning by heavy use in some sub-area), although I haven't devoted any time towards actually looking into this. $\endgroup$ Commented Oct 11, 2023 at 17:49
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    $\begingroup$ IMO (agreeing with the answer below) there is no "deeper" meaning and the fancy about Poseidon is quite ridiculous. More simply, taking into account that up to early 20th Century math and scientists were familiar with Latin and ancient Greek, we may consider that wave (independent of the language) can point to Latin fluctus.) and this is turn to Ancient Greek φλέω. Thus $\varphi$ can be a good candidate for expressing a magnitude regarding "waves" 1/2 $\endgroup$ Commented Oct 12, 2023 at 12:37
  • $\begingroup$ But presumably $\varphi$ was already used, and thus either $\chi$ or $\psi$ can be a good choice. 2/2 $\endgroup$ Commented Oct 12, 2023 at 12:38
  • $\begingroup$ I concur that any linguistic association of ψ with Poseidon and tridents is crank and unsound. Seek elsewhere. $\endgroup$ Commented Oct 17, 2023 at 21:46

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Prior to Schrodinger

There is no use of the symbol by de Broglie, Schrodinger's predecessor. In the first three short notes from de Broglie on the topic of wave mechanics (1923), there is no use of the symbol. In de Broglie's thesis there is only one brief mention of the symbol that I can see, in chapter 2 section 3 Les deux principes de moindre action dans la dynamique de l'electron, where $\Psi$ is used to denote the scalar potential. So this is unrelated to the wavefunction.

  • De Broglie, L. (1923). Ondes et quanta. Compte Rendus, 177, 507-510. (Link)
  • De Broglie, L. (1923). Quanta de lumière, diffraction et interférences. Compte Rendus, 177, 548-550. (Link)
  • De Broglie, L. (1923). Les quanta, la théorie cinétique des gaz et le principe de Fermat. Comptes rendus, 177, 630-632. (Link)
  • De Broglie, L. (1925). PhD Thesis. (Link)

Schrodinger's Wavefunction

This brings us to Schrodinger, and his 1926 papers, which introduce the wavefunction. The first paper is Quantisierung als Eigenwertproblem (Quantization as an Eigenvalue Problem).

  • Schrödinger, E. (1926). Quantisierung als eigenwertproblem. Annalen der physik, 385(13), 437-490. (Link, see also the English Translation translated by Oliver F. Piatetella.)

In this paper, Schrodinger uses the symbol $\psi$ when describing the Hamiltonian differential equation,

$$ H\left(q, \frac{\partial S}{\partial q}\right) = E, $$

We introduce now in place of $S$ a new, unknown function $\psi$ in such a manner that $\psi$ would appear as a product of suitable functions of the single coordinates.

Setting up a variational problem, he arrives at the wave equation

$$ \Delta \psi +\frac{2m}{K^2}\left(E + \frac{e^2}{r}\right)\psi = 0 $$

which demonstrates the wavelike behavior of the scalar field $\psi$. From this we can conclude that there is no deeper meaning to the symbols $\psi$ or $\Psi$ beyond convention when solving problems in mathematical physics.

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