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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.

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# Position Vector in Wave Functions

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