PX262 - F7 - properties of hydrogen-like atoms

• the solution the schrödinger equations for a single electron in coulomb potential is:

  $${\varphi }_{nlm}=R_{nl}(r)Y_{lm}(\theta ,\phi )$$

• three quantum numbers are needed to label the states:

  • the main quantum number: $n\in \mathbb{Z}>0$
  • the orbital quantum number: $l\in \mathbb{Z}\ge 0$
  • the magnetic quantum number: $m\in [-l,l]$

• the energies if individual eigenstates:

  $$E_1=-\frac{m{e}^4}{2(4\pi {ϵ}_0{)}^2{\hslash }^2}=-\frac{m{e}^4{c}^2}{2(4\pi {ϵ}_0{)}^2{\hslash }^2{c}^2}$$

• defining the fine structure constant:

  $$\alpha =\frac{e^2}{4\pi {ϵ}_0\hslash c^2}=\frac{1}{137}$$

• the energy level becomes:

  $$E_1=-\frac{m{c}^2{\alpha }^2}{2}=-13.6eV$$

• note: the energy is negative as the electron is bound inside the atom

• the energies for other states:

  $$E_n=E_1\frac{1}{n^2}$$

• states get closer in energy as $n$ increases, but they will always stay negative

• for atoms other than hydrogen the energy scales by $Z^2$

• to determine the number of states:

  • if $n=1$, $l=0$, and there is only a single state
  • if $n=2$, $l=\{0,1\}$, $m_{l=0}=\{0\},m_{l=1}=\{-1,0,1\}$

• for a given $l$, there are $2l+1$ states

• an often used notation for orbital quantum numbers, $l=0,1,2,3\dots$ is $s,p,d,f\dots$

• the notation for states:

  • $1s\iff n=1,l=0$
  • $2p\iff n=2,l=1$

• comment: $l<n$ is not a general requirement, and it depends on the potential and there are systems in which it does not hold, eg: spherically symmetric potential wells

shape wavefunctions

• it was found that:
  • the angular part are described by the spherical harmonics: $Y_{lm}(\theta ,\phi )$
  • the radial parts are described by the laguerre polynomials: $R_{nl}(r)=F_{nl}(\rho )\exp (-\frac{\rho }{2})$
• the probability of finding the particle:$$P(r)={\int }_0^{2\pi }{\int }_0^{\pi }{\varphi }_{nlm}^2{r}^2\sin \theta d\theta d\phi$$
• this can be solved numerically