Hydrogen Radial Probability Densities

and their derivatives


rmax =   P(r)max =

n = 1, l = 0   P(r) = r2 { 2 e-r }2  
n = 2, l = 0   P(r) = r2 { (1 - r/2) e-r/2 / sqrt(2) }2 
n = 2, l = 1   P(r) = r2 { r e-r/2 / sqrt(24) }2
n = 3, l = 0   P(r) = r2 { 2/sqrt(27) (1 - 2r/3 + 2r2/27) e-r/3 }2
n = 3, l = 1   P(r) = r2 { 8/(27 sqrt(6)) r (1 - r/6) e-r/3 }2
n = 3, l = 2   P(r) = r2 { 4/(81 sqrt(30)) r2 e-r/3 }2 

Instructions: First click on Plot P10. Then plot any other wavefunction. If you click on P10 before clicking another button you will see all plots simultaneously. Use "Set r Scale" to see the entire plot of the n = 2 and 3 states. Use "Set P(r) Scale" to expand the graph as needed. Place cursor on graph and press left mouse button to read coordinates. Press right mouse button to copy graph to a new window.
Note:  All distances are in units of Bohr radii,
a0 = 0.0529 nm

Question 1:  Find the radius at which it is most probable to find the electron in each of the six states. You will find the plot of the derivative of the radial probability density useful.

Question 2:  Do all zeros of the plot of the derivative of the radial probability density indicate a maximum in the radial probability density? Explain.

Question 3:  For states with more than one value of l, in which state does the most probable radius equal the radius predicted by the Bohr model.

Question 4:  Explain qualitatively why the maximum value of the probability density is smaller for states with higher values of n.

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