This page is for various random notes and subject to change.
Some Basic QM Axioms
The quantum state vector, Ψ (psi), embodies the complete state of the system.
Observables are operators. (Typically, square Hermitian matrices.)
The expectation value of a measurement is the mean of possible outcomes.
The Born Rule rules. Probability is the squared amplitude.
A quantum state evolves deterministically via the Schrödinger or Dirac equation.
The eigen equation. Operator × State = EigenValue × State
For Axioms #3 and #4:
\({P}(\lambda_{n})=\left|\langle{U}_{n}|\Psi\rangle\right|^{2}\)
For Axiom #6:
\(\hat{A}|{U}_{n}\rangle=\lambda_{n}|{U}_{n}\rangle\)
Its basic form:
\({i}\hbar\frac{d}{dt}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle\)
Where Ĥ is the Hamiltonian for the quantum system in question.
For a one-dimensional free particle:
\({i}\hbar\frac{\partial}{\partial{t}}\,\Psi(x,\!t)=\!\left[\!-\frac{\hbar^2}{2m}\,\frac{\partial^2}{\partial{x}^2}+\!V(x,\!t)\right]\Psi(x,\!t)\)
Where V(x, t) defines the potential energy due to the environment.
The main basis states of a qubit:
\(|{0}\rangle=|{+Z}\rangle=\begin{bmatrix}{1}\\{0}\end{bmatrix},\quad|{1}\rangle=|{-Z}\rangle=\begin{bmatrix}{0}\\{1}\end{bmatrix}\)
These give us four superpositions (which can also be basis states):
\(|{+}\rangle=|{+Y}\rangle=\,\frac{1}{\sqrt{2}}\left(|{0}\rangle+|{1}\rangle\right)\,=\,\frac{1}{\sqrt{2}}\begin{bmatrix}{1}\\{1}\end{bmatrix}\)
And:
\(|{-}\rangle=|{-Y}\rangle=\,\frac{1}{\sqrt{2}}\left(|{0}\rangle-|{1}\rangle\right)\,=\,\frac{1}{\sqrt{2}}\begin{bmatrix}{1}\\{-1}\end{bmatrix}\)
Plus:
\(|{+i}\rangle=|{+X}\rangle=\,\frac{1}{\sqrt{2}}\left(|{0}\rangle+{i}|{1}\rangle\right)\,=\,\frac{1}{\sqrt{2}}\begin{bmatrix}{1}\\{i}\end{bmatrix}\)
And:
\(|{-i}\rangle=|{-X}\rangle=\,\frac{1}{\sqrt{2}}\left(|{0}\rangle-{i}|{1}\rangle\right)\,=\,\frac{1}{\sqrt{2}}\begin{bmatrix}{1}\\{-i}\end{bmatrix}\)
\(\sigma_{x}=\frac{\hbar}{2}\begin{bmatrix}{0}&{1}\\{1}&{0}\end{bmatrix},\;\;\sigma_{y}=\frac{\hbar}{2}\begin{bmatrix}{0}&{-i}\\{i}&{0}\end{bmatrix},\;\;\sigma_{z}=\frac{\hbar}{2}\begin{bmatrix}{1}&{1}\\{1}&{1-}\end{bmatrix}\)
These act as operators on the spin state.
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(…more to come…)
