For a Hamiltonian $H$, if the all elements of matrix is non-positive under a set of basis $\{|\phi\rangle\}$:$$\langle\phi|H|\phi'\rangle\leq0$$ then the ground state of $H$ will be the liner combination of basis with non-negative coefficient:$$|G\rangle=\sum_{\{\phi\}}\chi_\phi|\phi\rangle\\\ch...

For a hydrogen atom in a particular state, e.g. $|3^2d_{\frac{3}{2}}(m_j=\frac{-1}{2})\rangle$, how would I decompose this as a superposition of $|n,l,s,m_l,m_s\rangle$ states? For example, I understand that this particular state could correspond to $|3,2,\frac{1}{2},-1,\frac{1}{2}\rangle$ or $|3...

Suppose we have an initial ensemble described by a density matrix $\rho$ and any given member of the ensemble scatters from one of some set of scattering matrices $\{S_g \equiv O_g S O_g^\dagger : g \in G \}$ where $S$ is some unitary matrix and the $O_g$ form a unitary representation of some sym...

When I first saw QM I was confused by the formalism and everything. Then I learned about experiments, such as double slit experiment and Stern-Gerlach experiment. From this it was very clear what the formalism is all about and why it is necessary.Then I started looking into field quantization and...

We consider the motion of a particle without spin submitted by the hamiltonian $H$ defined by the eigen-value equation:$H|n_1,n_2\rangle =(n_1^2+n_2^2)|n_1,n_2\rangle $ the proper kets of $H$ verify the orthonormalization relation. At the initial moment, the particle is in the normed state $ |\p...

Why we are not using Lagrangian instead of Hamilton in quantum mechanics formalism?

In 1-particle non-relativistic QM we have the continuity condition as a per definitionem property for the 1-electron probability current density for an electron in the magnetic field in an eigenstate of the full Hamiltonian (=static problem) and ignoring spin $$ \nabla\cdot{\bf j}=0$$ with the ...

At page 34 of "Relativistic Quantum Mechanics" by Bjorken and Drell the following spin projection operator for spinors "at rest" is considered: $$\frac{1+\sigma_{z}}{2}\quad ,$$ where $\sigma_{z}$ is the 4x4 extension of the usual z Pauli matrix. The operator is then rewritten in "scalar form" a...

Can you write the quantum harmonic oscillator hamiltonian $$H = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}+\dfrac{1}{2}m\omega^2x^2$$ in terms of the parity operator $P$? Thank you for your answers.

Stack limits question titles to 150 character (dumb!) so here is what I originally intended the title to say/ask for: [Opening Question].. "If we could build 'a' a towable spaceship, 'b' a leading, rope-towing spacecraft, along with 'c' a rope long & strong enough to reach the event horizon of...