diff --git a/src/5_atoms_and_lcao_solutions.md b/src/5_atoms_and_lcao_solutions.md index 1838bff34d9951a3c64aa545fa7310b5683bef9f..063a1ce4479b395081a506484456b982f756dcb4 100644 --- a/src/5_atoms_and_lcao_solutions.md +++ b/src/5_atoms_and_lcao_solutions.md @@ -24,18 +24,13 @@ &\sqrt{κ}e^{-κ(x-x_1)}, x>x_1 \end{cases} $$ - - Where $κ = \sqrt{\frac{-2mE}{ħ^2}} = \frac{mV_0}{ħ^2}$. - - The energy is given by $ϵ_1 = ϵ_2 = -\frac{mV_0^2}{2ħ^2}$ - - The wave function of a single delta peak is given by - - $$ + Where $κ = \sqrt{\frac{-2mE}{ħ^2}} = \frac{mV_0}{ħ^2}$. + The energy is given by $ϵ_1 = ϵ_2 = -\frac{mV_0^2}{2ħ^2}$ + The wave function of a single delta peak is given by + $$ \psi_1(x) = \frac{\sqrt{mV_0}}{ħ}e^{-\frac{mV_0}{ħ^2}|x-x_1|} - $$ - - $\psi_2(x)$ can be found by replacing $x_1$ by $x_2$ + $$ + $\psi_2(x)$ can be found by replacing $x_1$ by $x_2$ 2. $$ H = -\frac{mV_0^2}{ħ^2}\begin{pmatrix} @@ -49,7 +44,6 @@ $$ 3. $$ ϵ_{\pm} = \beta(1/2+\exp(-2\alpha) \pm \exp(-\alpha)) $$ - Where $\beta = -\frac{mV_0^2}{ħ^2}$ and $α = \frac{mV_0}{ħ^2}|x_2-x_1|$ ### Question 3