In various explanations I've read, from what I've gathered all particles have a wavefunction $\Psi(\mathbf{r},t)$ where $\mathbf{r}$ is the cartesian coordinates in however many dimensions you're working in. So in regular 3d space the wavefunction is $\Psi(x,y,z,t)$.
My first question is under what conditions is the wavefunction of a particle a pure quantum state? If I take an electron and completely isolate it from the rest of the universe will it be in a pure quantum state? Is this even possible and how does it relate to the eigenfunctions that appear when you act with some operator on the wavefunction? For example when the momentum operator or the position operator act on a wavefunction we get:
$$\hat{p}š¹ = p_1š_1 + p_2š_2 … + p_nš_n$$
and
$$\hat{x}š¹ = x_1š_1 + x_2š_2 … + x_nš_n$$
Are these functions $š_n$ the same for momentum and position? Does $š_1$ in the momentum equation $= š_1$ in the position equation? And what form do these functions have, I understand that a plane wave can be written in the form $e^{i(kx-\omega t)}$, but which of these functions are in this form? Are the eigenfunctions in this form or is the total wavefunction $\Psi$ in this form or are neither of them?
Sorry if this is a very convoluted or badly worded question, I'll happily clarify anything that doesn't make sense. I just have a very hard time connecting things in my head and unless I can make sense of this I don't feel I'll ever have a satisfactory understanding of this topic.
For reference I'm a 3rd year chemistry student so my math/physics understanding is nothing impressive but it's not totally non-existent.
Thanks for any responses in advance, I really do appreciate it.
Best Answer
I'll try to answer your questions in the same order they were asked.
Absolute purity is impossible in real world. However, if you limit the domain of your experiment, and look for the properties of particle(s) at specific time- and distance- scale, it is possible to create a state that behaves similarly to the pure state. (In other words, if your experiment lasts very short amount of time, and happens in limited amount of space - you can minimize the effects of the external world on your particle).
Functions $\psi_i$ are different for $\hat{x}$ and $\hat{p}$ because of the uncertainty principle. These wavefunctions are eigenstates of these operators, and since they don't commute - they can't be the same.
Momentum eigenstates have a plane wave form indeed, while position eigenstates have a form of a delta-function. This is very natural when the wavefunction is viewed as a probability distribution. The eigenstate of $\hat{x}$ is a particle that sits at fixed position. Thus there is a sharp peak at that specific position in the wavefunction and zeroes everywhere else.
As for the rest, what you're looking for is called Completeness Relation. One can decompose a wavefunction into a sum of finite (or infinite) number of eigenstates of a particular operator. The operator could be $\hat{x}$, $\hat{p}$, or even the Hamiltonian $\hat{H}$ itself.