Hydrogen Radial Probability Densities

and their Integrals

Plot Function and Do Integral

Start of definite integral = End of definite integral = Graph: rmax=

Hydrogen Wave Functions

1s 2s 2p 3s 3p 3d
Instructions: Select a wave function, then click on "Plot Function and Do Integral." You can choose the limits of integration by changing the values of "Start" and "End." Place cursor on graph and press left mouse button to read coordinates. 
Note:  All distances are in units of Bohr radii, a0 = 0.0529 nm.

Section 2 Exercises:

  1. What is the probability that the electron in the n = 1, l = 0 (1s) state is at a distance greater than 1ao?
  2. The n = 3, l = 0 (3s) state has three regions in which the electron may be located.  Find the probabilities of finding the electron in each of the three regions.
  3. Consider the n = 2, l = 0 (2s) and l = 1 (2p)states.  Compare the probabilities for the electron to be within 2ao in the two states and the probabilities that it is outside 5ao. Interpret this result in light of the angular momentum of the two states.
  4. For the three n = 3 states, find the radii at which the electron has a 50% probability of being inside and 50% outside.
  5. Plot r2R(r)2 for n = 1 and l = 0.  What is the most probable radius for the electron.  Do the same for n = 2 and l = 1 and n = 3 and l = 2.  Do these values agree with the corresponding radii predicted by the Bohr model?  (Remember that r is in units of a0.)
  6. Find the expectation value <r> for the three states mentioned above in Exercise 5.  (This will require integration by hand, calculator, or computer.)  How do your answers compare to those of Exercise 5.  Explain your conclusions.
  7. Calculate the most probable radii and the expectation value of r for the 2s state and compare it to the values for the 2p state calculated in Exercises 5 & 6.  Explain your conclusions.

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