Lecture 5

Eigenvalues are orthonormal

\(\int\psi_{nlm}\psi_{n'l'm'}\)

\(\left\{\begin{align}n=n',l=l',m=m'\rightarrow =1\\=0\end{align}\right.\)

Spatial Spin

\(\left\{\begin{align}\alpha \,\text{for spin up}\\\beta\,\text{for spin down}\end{align}\right.\)

\(\int\alpha\alpha^\star=\int\beta\beta^\star=1\)

\(\int\alpha\beta=\int\alpha^\star\beta^\star=0\)

Variational method

Assume each \(\psi_i\) is larger than the ground state

Trail \(\psi=\sum_iC_i\psi_i\) (linear combination of eigenvectors)

\(\left\{\begin{align}E-\epsilon_0=\frac{\sum|C_i|^2(\epsilon-\epsilon_0)}{\sum|C_i|^2}\\\epsilon_i\ge\epsilon_0\\|C_i|^2\ge 0\end{align}\right.\)\(\longrightarrow\) \(E-\epsilon_0\ge 0\)

1.  Multi-electron wavefunctions

Two-electron wave function must be anti-symmetric

\[ \psi_\text{total}(x_1,x_2)=\frac{1}{\sqrt{2}}(\psi_1(x_1)\psi_2(x_2)-\psi_1(x_2)\psi_2(x_1)) \]

For muti-electrons $$ \frac{1}{\sqrt{6}}

(1)\[\begin{vmatrix} \psi_1(x_1) & \psi_2(x_1) & \psi_3(x_1) \\ \psi_1(x_2) & \psi_2(x_2) & \psi_3(x_2) \\ \psi_1(x_3) & \psi_2(x_3) & \psi_3(x_3) \end{vmatrix}\]

\[## Born-Oppenheimer Approximation\]

\hat{H} = -\frac{1}{2m}\sum_i^n\grad_1^2-\frac{1}{4\pi\epsilon_0}\sum_A\sum_i\frac{Z_Ae^2}{|r_{Ai}|}+\frac{1}{4\pi\epsilon_0}\sum_i\sum_j\frac{1}{2}\frac{1}{|r_{ij}|} $$