Monthly Archives: September 2025

Problem: State the form of the singular value decomposition (SVD) of an arbitrary linear operator \(X:\mathbf C^m\to\mathbf C^n\). Solution: The SVD of \(X\) is given by: \[X=U_2\Sigma U^{\dagger}_1\] where \(U_1\in U(m)\) and \(U_2\in U(n)\) are unitary operators and \(\Sigma:\mathbf C^m\to\mathbf … Continue reading →

Problem: The Hamiltonian of an atom may be written in the perturbative form: \[H=H_0+V\] what are \(H_0\) and \(V\)? Solution: If the atom has \(N\) electrons and atomic number \(Z\) (if neutral then \(Z=N\) but here one can also allow … Continue reading →

Problem: Given a function \(f(x,p)\) defined on a single-particle classical phase space \((x,p)\) in \(1\) dimension, define the Weyl transform \(\hat f\) of \(f(x,p)\). Solution: \[\hat f=\int\frac{dx’dp’}{h}e^{i(p’X+x’P)/\hbar}\tilde f(x’, p’)\] where the phase-space Fourier transform \(\tilde f(x’,p’)\) of \(f(x,p)\) is given … Continue reading →