Show that for a simple product wave function as in the previous question, the relative probabilities of finding particle 1 near a position
Note: This is the reason that a simple product wave function is called
uncorrelated. For particles that interact with each other, an uncorrelated wave function is often not a good approximation. For example, two electrons repel each other. All else being the same, the electrons would rather be at positions where the other electron is nowhere close. As a result, it really makes a difference for electron 1 where electron 2 is likely to be and vice-versa. To handle such situations, usually sums of product wave functions are used. However, for some cases, like for the helium atom, a single product wave function is a perfectly acceptable first approximation. Real-life electrons are crowded together around attracting nuclei and learn to live with each other.
The probability of finding particle 1 within a vicinity
Taking the ratio of the two probabilities, the chances of finding particle 1 at