Quantum Mechanics Solution Manual 

© Leon van Dommelen 

5.2.4.1 Solution hmolda
Question:
Obviously, the visual difference between the various states is minor. It may even seem counterintuitive that there is any difference at all: the states and are exactly the same physically, with one electron around each proton. So why would their combinations be any different?
The quantum difference would be much more clear if you could see the full sixdimensional wave function, but visualizing sixdimensional space just does not work. However, if you restrict yourself to only looking on the axis through the nuclei, you get a drawable plane describing near what axial combinations of positions you are most likely to find the two electrons. In other words: what would be the chances of finding electron 1 near some axial position and electron 2 at the same time near some other axial position ?
Try to guess these probabilities in the plane as grey tones, (darker if more likely), and then compare with the answer.
Answer:
Here are the pictures, assuming the origin is halfway in between the protons:
Figure 5.1:
Wave functions on the axis through the nuclei. From left to right: , , the symmetric combination , and the antisymmetric one .

These results can be explained as follows: For the state , electron 1 is around the left proton, so its likely positions are clustered around the position of that proton, indicated by a tick mark on the negative axis in figure 5.1. Similarly electron 2 is around the right proton, so its positions are clustered around the positive value indicated by the tick mark on the positive axis. This means the wave function, , will look as shown in the left picture of figure 5.1. It will be mostly in the quadrant of negative and positive .
Similarly will look as the second picture. Here the positions of electron 1 cluster around the positive position of the right proton and those of electron 2 around the negative position of the left proton.
When you average the two states symmetrically, you get a twoblob picture like the third picture. Now it is electron 1 around the left proton and electron 2 around the right one or viceversa. But there is still almost no probability of finding both protons in the first quadrant, both near the right proton. Nor are you likely to find them in the third quadrant, both near the left proton.
If you average the first two states antisymmetrically, you get the fourth picture. In the antisymmetric combination, the wave function is zero on the symmetry line between the blobs.
You see that the states are really different when looked at in the full sixdimensional space.