- #26

- 17,150

- 8,950

$$\int_{\frac L 2}^L \psi_1(x)^2 dx = \int_{\frac L 2}^L \psi_2(x)^2 dx = \frac 1 2$$

$$\int_{\frac L 2}^L\psi_1(x)\psi_2(x) dx = -\frac 4 {3\pi}$$

And we can see that the probability that the particles are found in the same half of the well is:

$$p(\text{same}) = \frac 1 2 + X = 2(\frac 1 4 + \frac X 2) $$

And, in different halves:

$$p(\text{different}) = \frac 1 2 - X = 2(\frac 1 4 - \frac X 2) $$

Where ##X## is some term involving ##\pi^2##.

Note that I was able to see that by focussing on the properties of the eigenfunctions and leaving the normalisation constants alone!

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