Mathematical structure of quantum mechanics

• A quantum description consists of a Hilbert space of states
• Observables are self adjoint operators on the space of states
• Time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states
• Physical symmetries are realized by unitary transformations

Postulates of quantum mechanics

The mathematical framework can be traced back to the Dirac-von Neumann axioms

Tensor Calculus

• vector (contra-variant vector) - arrow in space
• covector (co-variant vector) - gradient 

Eigenvalues and Eigenvectors

• \(\lambda\in \lambda(A) \Leftrightarrow A-\lambda I \text{ is singular} \Leftrightarrow \text{det}(A-\lambda I)=0\)
• \(\{x\neq0|x\in \mathcal{N}(A-\lambda I)\}\) is the set of all eigenvectors associated with \(\lambda\).  
• \(\mathcal{N}(A-\lambda I)\) is the eigenspace for A.

Diagonalizability of a matrix

• A nilpotent matrix \(A=\{A\in M_n |A^2=0\}\) is not diagonalizable. 
• Two matrices \(A\) and \(B\) are similar whenever these exists a nonsingular matrix \(P\) such that \(P^{-1} AP=B\)
• A matrix can be diagonalized if it is similar to a diagonal matrix \(D\), i.e. \(P^{-1}AP=D\)
• Or equivalently, \(AP_{*,j}=\lambda_j P_{*,j}\)
• \(A\) is diagonalizable if and only if \(A\) possesses a complete set of eigenvectors.
• Or equivalently, the geometric multiplicity of \(\lambda_i\) is equal to the algebraic multiplicity of \(\lambda_i\) for each \(\lambda_i\in \lambda(A)\)