Spatial Wave function of two indistinguishable particles

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  • #26
PeroK
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Here's something else I noticed that helped me check my answer. We know that ##\psi_1, \psi_2## are orthonormal. So, we can write down immediately:
$$\int_{\frac L 2}^L \psi_1(x)^2 dx = \int_{\frac L 2}^L \psi_2(x)^2 dx = \frac 1 2$$
$$\int_{\frac L 2}^L\psi_1(x)\psi_2(x) dx = -\frac 4 {3\pi}$$
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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  • #27
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That all makes sense and good proof of the answer.

I have looked at the approach in #13 and #23 and I can now see where that factor came from.

thanks
 

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