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Position Vector in Wave Functions

  1. Sep 25, 2015 #1
    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. jcsd
  3. Sep 26, 2015 #2

    vanhees71

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    Science Advisor
    2016 Award

    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.
     
  4. Sep 30, 2015 #3
    Thank you, that makes much more sense!
     
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