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I am trying to learn a little bit on my own about how we talk about physical systems on a high level of precision, partially because I'm getting into chemistry and want to understand what is considered a proper formal description of any given molecule.
I will explain my current understanding, which may be quite wrong, and I am looking for revision and further explanation.
I think, up until the advent of quantum mechanics in the early 20th century, the background ontology of physics was basically, 3 dimensions of space, 1 dimension of time, and they are continuously valued sets, so they can be modeled by the real numbers, except for all we know there is no "origin", so maybe we can say it is an affine space. The smallest existing thing anybody knew of was particles, which have fixed, spherical volumes and occupy positions in space mutually exclusively. They also have certain essential properties like mass and electric charge, and hence exert certain forces on each other, governing their movement and so on.
Through what course of events I know not, but eventually, the picture changed a bit, where the fundamental description of the smallest elements of physical systems (i.e., still particles) was to be given bywavefunctions. Wavefunctions are functions which contain all the important specifying information about the properties and state of something, i.e., its momentum, position, energy, angular momentum, etc.
Wavefunctions do not directly tell you this information, but you can apply further operations to them to derive certain pieces of information from them, like taking the magnitude squared.
The most common presentation of a wave function I have seen so far is$\Psi(x, t)$. In an elementary introduction, we can work in a one-dimensional space, so$x$ is a one-dimensional position value, not a 3-dimensional vector or something. This is a complex-valued function; i.e., it will return a complex number as its output.
I guess here is my first question.
Looking atWikipedia, it seems like it dives straight into telling you what you can do with a wave function, like interpret the "square modulus" as the "probability density of the particle's position". But it does not tell us what the specific wavefunction is, i.e. what the body of$\Psi$ is.
Is the idea here that different kinds of particles have different wavefunctions? Are we supposed to work backwards from observation and measurement to figure out what the wavefunction of a given system is?
An AI told me one example of a wavefunction, say for an electron, could be$\Psi(x, t) = Ae^{i(kx - \omega t)}$. The Wikipedia article seems to be missing a lot of basics.
My question is basically, how do we figure out the wavefunction for any given subatomic particle, like an electron, a proton, or a neutron?
- 9$\begingroup$The question in your final sentence (v1) is the subject of the first semester of an introductory course on quantum mechanics. It sounds to me like you would do better to invest in a well-written text versus cobbling together reference material from Wikipedia and from chatbots. An introductory course on partial differential equations will explain why $e^{i(kx-\omega t)}$ isa solution tomany wave equations.$\endgroup$2025-10-13 15:09:11 +00:00CommentedOct 13 at 15:09
- 3$\begingroup$To understand quantum mechanics read a textbook on quantum mechanics. Gasiorowicz should do.$\endgroup$my2cts– my2cts2025-10-13 16:05:34 +00:00CommentedOct 13 at 16:05
- $\begingroup$The Wikipedia article seems to be missing a lot of basics. Such as? Did you see that it gives the energy eigenfunctions for a hydrogen atom? It also explains that “This is the only atom for which the Schrödinger equation has been solved exactly. Multi-electron atoms require approximative methods.” Molecules also require approximation methods, but in principle one is trying to solve a multi-particle (i.e., nuclei and electrons) Schrödinger equation.$\endgroup$Ghoster– Ghoster2025-10-13 19:44:19 +00:00CommentedOct 13 at 19:44
- $\begingroup$Figuring out the wave function for a particle in real-world applications can be enormously difficult, and in practise highly simplified models are often used. Crudely speaking, the nature of the wave function depends upon the potential acting on the particle, and that could depend on the combined wave functions for large numbers of other particles (eg think of all the charged particles in a complicated molecule, and how they would each be interacting with each other). You have to use numerical techniques and oodles of computer power.$\endgroup$Professor Sushing– Professor Sushing2025-10-13 21:42:18 +00:00CommentedOct 13 at 21:42
1 Answer1
My question is basically, how do we figure out the wavefunction for any given subatomic particle, like an electron, a proton, or a neutron?
Typically you try to solve theSchrodinger equation.
For example, the "hydrogenic orbitals" are solutions to the single-particle Schrodinger equation for a Coulomb potential. We use them to describe the probability amplitude of an electron in a hydrogenic atom.
Different potential energy functions give rise to different solutions.
An AI told me one example of a wavefunction, say for an electron, could be$\Psi(x, t) = Ae^{i(kx - \omega t)}$.
An AI chatbot provided you with output based on your input prompt.
The provided wavefunction is a solution to the Schrodinger equation when the potential is zero (i.e., the free-particle Schrodinger equation).
Unfortunately, that solution is not normalizable in free space, so it does not describe a physically realizable situation (at least not without additional context).
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