diff --git a/src/5_atoms_and_lcao_solutions.md b/src/5_atoms_and_lcao_solutions.md index 7b82c37862365a42e97332c56f85b224a46a366f..e9b467795daf76cb595421186b10f741fbe09324 100644 --- a/src/5_atoms_and_lcao_solutions.md +++ b/src/5_atoms_and_lcao_solutions.md @@ -7,22 +7,24 @@ 2. The atomic number of Tungsten is 74: $$ -1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^66s^24f^{14}5d^4 +1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^66s^24f^{14}5d^4 $$ 3. $$ - Cu &= [Ar]4s^23d^9 - Pd &= [Kr]5s^24d^8 - Ag &= [Kr]5s^24d^9 - Au &= [Xe]6s^24f^145d^9 +\begin{align} +\textrm{Cu} &= [\textrm{Ar}]4s^23d^9 +\textrm{Pd} &= [\textrm{Kr}]5s^24d^8 +\textrm{Ag} &= [\textrm{Kr}]5s^24d^9 +\textrm{Au} &= [\textrm{Xe}]6s^24f^145d^9 +\end{align} $$ ### Question 2 -1. +1. $$ - \psi(x) = + \psi(x) = \begin{cases} &\sqrt{\kappa}e^{\kappa(x-x_1)}, x<x_1\\ &\sqrt{\kappa}e^{-\kappa(x-x_1)}, x>x_1 @@ -31,9 +33,9 @@ $$ Where $\kappa = \sqrt{\frac{-2mE}{\hbar^2}} = \frac{mV_0}{\hbar^2}$. -The energy is given by $\epsilon1 = \epsilon2 = -\frac{mV_0}{\hbar^2}$ +The energy is given by $\epsilon_1 = \epsilon_2 = -\frac{mV_0}{\hbar^2}$ -The wavefunction of a single delta peak is given by +The wave function of a single delta peak is given by $$ \psi_1(x) = \frac{\sqrt{mV_0}}{\hbar}e^{-\frac{mV_0}{\hbar^2}|x-x_1|} @@ -41,18 +43,18 @@ $$ $\psi_2(x)$ can be found by replacing $x_1$ by $x_2$ -2. +2. $$ H = -\frac{mV_0^2}{\hbar^2}\begin{pmatrix} - 1/2+\exp(-\frac{mV_0}{\hbar^2}(x_2-x_1)) & + 1/2+\exp(-\frac{mV_0}{\hbar^2}(x_2-x_1)) & \exp(\frac{mV_0}{\hbar^2}(x_2-x_1))\\ - \exp(-\frac{mV_0}{\hbar^2}(x_2-x_1)) & + \exp(-\frac{mV_0}{\hbar^2}(x_2-x_1)) & 1/2+\exp(+\frac{mV_0}{\hbar^2}(x_2-x_1)) \end{pmatrix} $$ -3. +3. $$ \epsilon_{\pm} = \beta(3/2+\cosh{2\alpha}+2\cosh{\alpha}\pm \cosh{\alpha}) @@ -104,4 +106,3 @@ $$ $$ P = -\frac{2\gamma^2}{\mathcal{E}}(\frac{1}{\sqrt{\gamma^2+t^2}}) $$ -