• The average value of many measurements (expectation) of a physical quantity $A$ of a particle of wavefunction $\Psi$ is given by

  $$ \left < \hat A \right > = \int \Psi^* \hat A \Psi\,\text d \vec r\quad \text{where} \; \text d\vec r=\text dx\,\text dy\,\text dz \; \text{in}\; 3\text{D} $$

• Common operators

  $$ \begin{aligned} \hat x &=x \qquad \qquad \;\;\hat{\underline r}=(x,y,z) \\ \hat p_x&=-i\hbar \frac{\partial}{\partial x} \qquad \hat{\underline p}=-i\hbar \left ( \frac{\partial }{\partial z}, \frac{\partial }{\partial y}\frac{\partial }{\partial z}\right ) \\ \hat L_x&=\hat y \hat p_z-\hat z \hat p_y \\ \hat H&=\hat T+\hat V=-\frac{\hbar^2}{2m}\nabla^2+V(\vec r) \\ &\quad \text{where}\;\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} \end{aligned} $$

• Time independent Schrodinger equation

  $$ \hat H \psi(\vec r)=E\psi (\vec r) $$

• Simple wavefunction in the case of $V(\vec r,t)=V(\vec r)$

  $$ \psi(\vec r,t)=\psi(\vec r)e^{-iEt/\hbar} $$

💼 Case: the hydrogen atom

• For a hydrogen atom Schrodinger's time independent equation is of the following form:

  $$ \begin{aligned} \hat H\psi(\vec r)&=E\psi(\vec r) \\ \left ( -\frac{\hbar^2}{2m_e}\nabla^2-\frac{Z e^2}{4\pi \epsilon_0 r} \right )\psi(r)&=E\psi(r) \end{aligned} $$

  where $m_e$ is the electron mass, the potential is a Coulomb potential between the electron and $Z$ protons

• We get the following eigen energies

  $$ E_{n,l,m_l}=E_n=-\frac{Z^2 E_R}{n^2} $$

  where $E_R$ is the Rydberg energy

  $$ E_r=\frac{e^2}{8\pi \epsilon_0 a_0}=13.6\; \text{eV} \; \text{and}\; a_0=\frac{4\pi \epsilon_0 \hbar^2}{m_e e^2}=0.53 \AA $$

• The eigenfunctions are products of two parts $\psi_{n,l,m_l}=R_{n,l}(r)Y_{l,m_l}(\theta,\phi)$

  $$ \begin{aligned} R_{n,l}(r)&=\text{constant}\times \left ( \frac{Zr}{a_0} \right )^l e^{-Zr/n a_0} \sum^{n-l-1}{k=0}(-1)^k c_k \left ( \frac{Zr}{a_0}\right )\\ Y{l,m_l}(\theta,\phi)&=P_{l,m_l}(\theta)\Phi_{m_l}(\phi)\\ P_{l,m_l}(\theta)&=\text{Legendre polynomials} \\ \Phi_{m_l}(\phi)&=e^{im\phi} \end{aligned} $$

• The angular eigenfunction are

  $$ \begin{aligned} \hat L^2 Y_{l,m_l} &=l(l+1)\hbar^2 Y_{l,m_l} \\ \hat L_z Y_{l,m_l}&=m_l \hbar Y_{l,m_l}

  \end{aligned} $$

• In spherical coordinates the operator $\hat L_z$ is

  $$ \hat L_z=-i\hbar \frac{\partial}{\partial \phi} $$

• With allowed quantum numbers

  $$ \begin{aligned} n&=1,2,\ldots \\ l&=0,1,2,\ldots,n-1 \\ m_l&=-l,-l+1,\ldots , l-1,l

  \end{aligned} $$

🗒️Note: if we add the spin DOF, we get 4 quantum numbers $(n,l,m_l,m_s)$, with the spin quantum number $m_s=\pm 1/2$. The spin is usually denoted as $\chi_{m_s}$. The eigenequation for the spin is

$$ \hat S_z \chi_{m_s}=m_s\hbar \chi _{m_s} \qquad m_s=\pm \frac 12 $$

• The complete wave function of the electron in a hydrogen atom is

  $$ \Psi_{n,l,m_l,m_s}=\psi {n,l,m_l}(r\theta,\phi)\chi{m_s} $$

  🗒️ Note: we usually denote spin up or spin down states as $\chi_{1/2}~=\chi_\uarr$ and $\chi_{-1/2}=\chi_\darr$

Examples of $R_{n,l}(r)$

the ground state of hydrogen-like atom is described by the wavefunction $\psi_{1,0,0}=\psi_{1s}$ or the 1s-orbital, with the ground state energy

$$ E_1=E_{1s}=-13.6Z^2\; \mathrm{eV} $$

Other states are excited and have higher $E_n$.

We can extend the energy state to multi-electron hydrogen atoms (within the independent-particle approximation) by occupying the energy levels to build up the states of $N$-electron atoms by occupying the $N$ energy within the limitations of the Pauli exclusion principle and spin dof. (Aufbau principle)

Atomic orbitals

💃 Example: $\psi_{1,0,0}$ as $1s$-orbital, $\psi_{2,1,m_l}$ with $m_l=0,\pm 1$ as $2$p-orbitals, $\psi_{3,2,m_l}$ with $m_l=0,\pm1,\pm 2$ as $3d$-orbital, etc. For these orbitals we can write:

$$ p_z=p_0 \qquad p_x=\frac{1}{\sqrt{2}} (p_1+p_{-1}), \quad p_y=\frac{1}{i\sqrt{2}}(p_1-p_{-1}) $$