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I'm starting a series on "Quantum Mechanics Basics". Feel free to comment with any questions about the posted material. Be sure to stay tuned and enjoy!pic.twitter.com/NHmeD3kR4H
Quantum Mechanics Basics #2: Completeness. Here I go over the notion of "completeness", using the familiar standard basis for ℝ³, and making use of our new QM vector notation introduced in QM Basics #1. Equation (2) is extremely important and we will use it time and time again.pic.twitter.com/JfI3SKgUOC
Quantum Mechanics Basics #3: Wave Functions & Probabilities.pic.twitter.com/vMcSNzlKoJ
It should be noted that, for pedagogical purposes, I say that |<x|ψ>|^2 is the probability that the particle *at* position x. Since x is a continuous variable, the gory truth is actually that "|<x|ψ>|^2*dx is the probability that the particle lies in the interval [x, x+dx]"
Quantum Mechanics Basics #3(a): Example and Application of Wave Functions & Probabilitiespic.twitter.com/xYQuYIQb12
Quantum Mechanics Basics #4: The Schrödinger Equation & Time Evolutionpic.twitter.com/vNDomHkrew
What is an “observable” or should I just say “observer” in QM?
Another good question. An observable is simply a dynamical value that one can measure (e.g. momentum, position, spin, angular momentum etc.). These are represented by hermitian operators because that enforces that the eigenvalues (what we measure) always be real
Is Euclidean space a part of the Hilbert space?
Good question. To answer your question I would have to follow up with "which Hilbert space?". That is, every particle has different degrees of freedom and these are reflected in the nature of the Hilbert space.
are you assuming each eigenvalue has multiplicity one? How do you choose one eigenvector |a> in the eigenspace of the eigenvalue a?
no the eigenvalues are allowed to be degenerate, in which case an orthonormal basis can still be found by Graham Schmidt Orthogonalization and eigenvector is usually not chosen in QM, rather the representation of the operator O, and the eigenstates are the EVs
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