Question

Figure 40.27a modeled a hydrogen atom as a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a one-dimensional model, would be the electron + proton electrostatic potential energy in one dimension:

$\mathrm{U}\left(\mathrm{x}\right)=-\frac{{\mathrm{e}}^2}{4{\mathrm{\pi \epsilon }}_0\left|\mathrm{x}\right|}$

a. Draw a graph of U(x) versus x. Center your graph at $\mathrm{x}=0$.

b. Despite the divergence at $\mathrm{x}=0$, the Schrödinger equation can be solved to find energy levels and wave functions for the electron in this potential. Draw a horizontal line across your graph of part a about one-third of the way from the bottom to the top. Label this line ${\mathrm{E}}_2$, then, on this line, sketch a plausible graph of the $\mathrm{n}=2$wave function.

c. Redraw your graph of part a and add a horizontal line about two-thirds of the way from the bottom to the top. Label this line ${\mathrm{E}}_3$, then, on this line, sketch a plausible graph of the $\mathrm{n}=3$ wave function.

Short answer

The graph is shown below in steps.

Step-by-step solution

Part (a) Step 1: Given information

We have given,

The potential of hydrogen atom is

$\mathrm{U}\left(\mathrm{x}\right)=-\frac{{\mathrm{e}}^2}{4{\mathrm{\pi \epsilon }}_0\left|\mathrm{x}\right|}$

We have to draw the graph between the potential and the x.

Explanation

From the formula we can say that when the value of x will increases in both side positive or negative then the potential energy will decreasing nd the negative sign indicates that the graph will plot bottom in the graph.

since at the value $\mathrm{x}=0$the potential energy will tends to negative infinity.

Then, the plot is

Part (b) Step 1: Given information

We have to draw the line line from bottom to the one third distance and plot the wave function for$\mathrm{n}=2.$

Graph

Part (c) Step 1: Given information

We have to draw again the wave function for$\mathrm{n}=3.$

Graph