# Position Vector in Wave Functions

• Rosie135
In summary, to create a 2-D electron energy density plot in Mathematica, you would first need to calculate the superposition of the symmetric and anti-symmetric wave functions, and then square it to get the energy density plot. You would then need to input the position vector r to Mathematica to plot the energy density plot.
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.

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.

Rosie135
vanhees71 said:
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.

Thank you, that makes much more sense!

## What is a position vector in wave functions?

A position vector in wave functions is a mathematical tool used to describe the position of a particle in a three-dimensional space. It is represented by the symbol r and is typically measured from the origin of a coordinate system to the location of the particle.

## How is a position vector used in wave functions?

In wave functions, a position vector is used to describe the location of a particle in space at a specific time. It is an essential component in the Schrödinger equation, which is used to determine the behavior and properties of particles in quantum mechanics.

## Can a position vector be negative in wave functions?

Yes, a position vector can have negative values in wave functions. This indicates that the particle is located in a direction opposite to the origin of the coordinate system. In quantum mechanics, it is common for particles to exist in negative positions due to the nature of their wave-like behavior.

## How is a position vector related to probability in wave functions?

In wave functions, the square of the wave function amplitude at a specific point is proportional to the probability of finding a particle at that point. Therefore, the position vector is used to calculate the probability of finding a particle at a given location in space.

## Can a position vector determine the momentum of a particle in wave functions?

Yes, a position vector can be used to determine the momentum of a particle in wave functions. This is achieved through the use of the momentum operator, which is related to the position vector in the Schrödinger equation. The position vector and momentum are closely linked in quantum mechanics and are used to describe the behavior of particles.

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