Hello! My name is Jiho Jun, and I’m from South Korea. I just created my blog using Github pages, and from now on, I’m going to post about what I studied, daily stories, etc. There are so many thi...

Ground state of the helium atom Let’s approximate the ground state energy of the helium atom using the variational method. The Hamiltonian of the system is; ignoring the motion of the nucleus (Born...

Variational method Variational method is one way of finding approximations to the ground state, and some excited states. This allows for calculating approximate wavefunctions such as molecular orb...

Kramers’ relation Kramers’ relation, named after the Dutch physicist Hans Kramers, is a relationship between expectation values of nearby powers of $r$ for the hydrogen-like atoms: \[\boxed{ \fra...

Introduction The fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation, by usin time-...

Introduction The Schrödinger equation allows one to calculate the stationary states and also the time evolution of quantum systems. Exact analytical answers are available for the non-relativistic ...

Introduction In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded...

Ehrenfest’s theorem Generalized Ehrenfest’s theorem: \[\boxed{ \frac{d}{dt}\expct{\hat{Q}} = \frac{i}{\hbar}\expct{\com{ \hat{\mathcal{H}} }{ \hat{Q} }} + \Expct{\frac{\partial \hat{Q}}{\partial...

Standard deviation of a measurement For an arbitrary Hermitian operator $\hat{L}$ we can associate a standard deviation: \[ \begin{align*} \sigma_L^2 &=\expct{\hat{L}^2}-\expct{\hat{L}}^2 \nl...