First, we know for every wave function(adsbygoogle = window.adsbygoogle || []).push({});

$$p(x)=\psi(x)^*\psi(x)$$ indicates the probability density of a particle appearing at the point x.

So if we calculate $$P=\int _M p \text{d}x$$ this gives the probability of the particle appearing in the range M.

On the other side, I was thinking about superposition:

$$\Psi=\frac{1}{2}(\psi_A+\sqrt{3}\psi_B)$$

So it means that as long as we measure the system, at 25% the wave will collapses to state A, 75% to B.

And e.g. in state A, the particle has probability P_{A}to appear in range M; while in B, has P_{B}.

So there would be $$\frac{P_A+3P_B}{4}$$ to appear in range M. Right??

However, if we calculate in this way:

$$P(x)= \int _M \Psi^*\Psi\text{d}x=\frac{1}{4}\int _M (\psi_A^*+3\psi_B^*)(\psi_A+3\psi_B)\text{d}x$$

$$=\frac{1}{4}\int _M \psi_A^*\psi_A+3\psi_B^*\psi_B+\psi_A^*\psi_B+\psi_A\psi_B^*\text{d}x$$

First two terms can be eliminated to probability,

$$=\frac{1}{4}\left[P_A+3P_B+\int _M \psi_A^*\psi_B+\psi_A\psi_B^*\text{d}x\right]$$

We know: if M is the whole space, then the last two terms are 0, but they can NOT vanish for every function and every interval. So the calculation would be the same!!! Anything wrong???

Look forward to your help, thanks!

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# Wave equation indicates the probability density

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