Quantum Mechanics Solution Manual 

© Leon van Dommelen 

5.2.4.2 Solution hmoldb
Question:
Based on the previous question, how would you think the probability density would look on the axis through the nuclei, again ignoring the existence of positions beyond the axis?
Answer:
For any state, the probability of finding electron 1 near a given , regardless of where electron 2 is, is found by setting equal to and integrating over all possible positions for electron 2. For the twodimensional state shown in the left column of figure 5.2, you are then integrating over vertical lines in the top picture; imagine moving all blank ink vertically towards the axis and then setting . The resulting curve is shown immediately below. As expected, electron 1 is in this state most likely to be found somewhere around , the negative position of the left proton. Regardless of where electron 2 is.
Figure 5.2:
Probability density functions on the axis through the nuclei. From left to right: , , the symmetric combination , and the antisymmetric one . From top to bottom, the top row of curves show the probability of finding electron 1 near regardless where electron 2 is. The second row shows the probability of finding electron 2 near regardless where electron 1 is. The third row shows the total probability of finding either electron near , the sum of the previous two rows. The fourth row shows the same as the third, but assuming the true threedimensional world rather than just the line through the nuclei.

(Note that all possible positions of electron 2 should really be found by integrating over all possible positions in three dimensions, not just axial ones. The final row in the figure gives the total probabilities when corrected for that. But the idea is the same, just harder to visualize.)
The probability density at a given value of also needs to include the possibility of finding electron 2 there. That probability is found by setting and then integrating over all possible values of . You are now moving the blank ink horizontally towards the axis, and then setting . The resulting curve is shown in the second graph in the left column of figure 5.2. As expected, electron 2 is most likely to be found somewhere around , the positive position of the right proton.
To get the probability density, the chance of finding either proton near , you need to add the two curves together. That is done in the third graph in the left column of figure 5.2. An electron is likely to be somewhere around each proton. This graph looks exactly like the correct threedimensional curve shown in the bottom graph, but that is really just a coincidence.
The states and can be integrated similarly; they are shown in the subsequent columns in figure 5.2. Note how the line of zero wave function in the antisymmetric case disappears during the integrations. Also note that really, the probability density functions of the symmetric and antisymmetric states are quite different, though they look qualitatively the same.