# Position Vector in Wave Functions

1. Sep 25, 2015

### Rosie135

Hello,
I need to create a 2-D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the superposition of the symmetric and anti-symmetric wave functions,

$$\Psi_s(\textbf{r}) = \sum_{i=1}^{3}a_icos(\textbf{k}_i\cdot\textbf{r})$$
$$\Psi_a(\textbf{r}) = \sum_{i=1}^{3}b_isin(\textbf{k}_i\cdot\textbf{r})$$
and squaring it to get this,
$$\Psi_s^2(\textbf{r})+\Psi_a^2(\textbf{r})=\Big[\sum_{i=1}^{3}a_icos(\textbf{k}_i\cdot\textbf{r})\Big]^2+\Big[\sum_{i=1}^{3}b_isin(\textbf{k}_i\cdot\textbf{r})\Big]^2$$
Where the k's are equal to,
$$\textbf{k}_1=\frac{4\pi}{3d_b}\hat{y}$$
$$\textbf{k}_2=\frac{4\pi}{3d_b}(-\frac{1}{2}\sqrt{3}\hat{x}-\frac{1}{2}\hat{y})$$
$$\textbf{k}_3=\frac{4\pi}{3d_b}(\frac{1}{2}\sqrt{3}\hat{x}-\frac{1}{2}\hat{y})$$
Although this seems trivial, my question is what do I set position vector r to when I am inputting to Mathematica? Could I just set it to $$\hat{x}$$ or $$\hat{y}$$ or even $$\hat{x}+\hat{y}$$
And also, since the dot product of the argument Sin and Cos would contain a scalar, I would need to attach a variable to that in order to plot it right? Would that just be variables x and y? The result should just be a uniform distribution of "bright spots" but my results have been way off. My inputs to Mathematica may be wrong, but before I start figuring out where I went wrong in Mathematica, I want to make sure my concept of how to represent these quantities is correct.

2. Sep 26, 2015

### vanhees71

I'd work with the components of the vectors and express them as lists. Mathematica has pretty intuitive operations with vectors and vector calculus. So for any vector $\vec{v}$, you map
$$\vec{v}=v_x \hat{x} + v_y \hat{y} \mapsto \{v_x,v_y \}$$
in Mathematica. The dot product is then just written as {a,b}.{c,d}=ac+bd in Mathematica.

3. Sep 30, 2015

### Rosie135

Thank you, that makes much more sense!