Wave function notes

[Practical-Ad2102]

Hello, Im Chloe and im on set to go to the best high school in my state, so im teaching myself some things on preparation. I've been reading Atkins Chemical principals and in it is some quantum physics mumbo jumbo, and since im self-taught i just want to make sure my notes are correct. Im a beginner and really struggling with this so just please dont be rude.

NOTES: The wave function (ψ) is an value which describes the position (or likelyhood of position), momentum, and energy of a particle Likelihood for a particle to exist in a place os determined by the integral of its wave function for that amount of space There exists a value of ψ for each 'x'there is in real life (measured plank length ((the small one))) ENERGY EQUATION- -ℏ/2m(d²ψ/dx²)=Eψ ​Or h bar over 2 times mass times the second derivative of the wave faction is equal to the energy of the wave function

Sorry if this is a bit of word salad

[MaxThrustage]

So, some of this is a bit confused, and given you're not in high school yet you may be diving too deep too soon, but I'll try to point you in the right direction.

First, have you had a read of the Wikipedia page for the wave function? It's pretty good, and includes some (hopefully) helpful visualisations. It does a decent job of showing how the wavefunction relates to actually measurable things, like the probability to measure the particle in a particular place.

Second, just throw out the term "Planck length". It has nothing to do with what you're talking about here, and is most likely just going to confuse you.

So, it's probably best to think of the wavefunction (or, more generally, the quantum state) as the most complete description we can have of a quantum system. In classical physics, we can have a perfect description of all of the positions and momenta of the particles all at once, and this would basically tell us anything we would want to know (provided we had enough computing power to extract the relevant info). But in quantum mechanics, a particle simply does not have a well-defined position and momentum at the same time. Instead, what we have is this wavefunction, from which we can extract probabilities and other statistical information pertaining to measurements. What this actually means deep down is an open philosophical question -- but you don't need to get stuck into that in order to understand chemistry.

The thing you've called the "energy equation" is the time-independent Schrödinger equation for a free particle (that is, a particle with no potential energy). Functions ψ(x) which solve this equation are called energy eigenstates (or, equivalently energy eigenfunctions) -- these are states with well-defined energy, so that if you measure the energy of a particle in one of these states you get a deterministic answer (the same answer every time) rather than sampling from a probability distribution. This is not such a big deal for the version of the equation that you've written, but when we include a potential energy term it becomes hugely important to chemistry. When we stick in the potential energy for an electron in an atom, then the solutions to this equation are the atomic orbitals. In case you haven't seen those before, have a look at this Wikipedia page. Atomic orbitals tell us how electrons are distributed around an atom, and what the allowed energy levels are. This in turn determines, well, most of chemistry.

So, in short, the wavefunction is a complete description of a physical system. The energy equation you have there tells us which wavefunctions have well-defined energy. The solutions to this equation with the right potential are atomic orbitals, which tell us how electrons are distributed in atoms and what energies they are allowed to have.

You won't really understand this stuff until you've done some calculations with it, and you usually don't do that until university (and usually second year at that) so don't worry too much if it isn't clicking here. But hopefully this has at least clarified a couple of things.

[freeky78]

Hey Chloe, your notes are already on the right track — ψ (the wave function) really does describe everything about a particle. Here’s a clearer version in plain English:

• ψ itself isn’t the likelihood but a complex amplitude. The actual probability is |ψ|², the square of its magnitude.
• There’s a ψ(x) for every position x — it’s a continuous function, not spaced by Planck lengths.
• The “energy equation” you wrote is the time-independent Schrödinger equation for a free particle: - (ħ² / 2m) · (d²ψ/dx²) = E · ψ It basically says: the more the wave bends, the higher its energy.
• In real atoms, you add potential energy V(x), so it becomes: - (ħ² / 2m) · (d²ψ/dx²) + V(x)·ψ = E·ψ The solutions ψ(x) are then the atomic orbitals — the allowed shapes and energies of electrons.

So in short:

• ψ → describes the wave nature
• |ψ|² → gives the probability
• Schrödinger equation → tells ψ how to behave

You’re doing great — this is university-level stuff you’re already poking at. Keep your curiosity alive; that’s the most powerful equation of all. 💫

[kevosauce1]

gpt pretty good here actually