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When I try to find probability of a particle I just take the product of the complex conjugate of the wavefunction and wavefunction itself, and find the integral from $- \infty$ to $\infty$ or any given range. What happens if anyone tells me to find the probability at a particular point?

For example a square well has a length L and I want to find the particle at $\frac{L}{4}$, how to do this?

What I put the limit is that, $\frac{L}{4} + \epsilon$ to $\frac{L}{4} - \epsilon$ and i got some result. The problem is that, i can not get away from the $\epsilon$ although i put the limit $\epsilon \to 0$

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3 Answers 3

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So the probability of finding a particle at some single point is zero (unless the probability density includes delta-functions or something like that). If you get a different result, there is an error in your calculations.

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  • $\begingroup$ I get multiplication of $\epsilon$ , so if i put the value of that is to zero then yes, the probability value would be zero. $\endgroup$
    – user58143
    Dec 10, 2016 at 5:14
  • $\begingroup$ @SabbirHasan: So what seems to be the problem? $\endgroup$
    – akhmeteli
    Dec 10, 2016 at 5:16
  • $\begingroup$ I understand right now :) $\endgroup$
    – user58143
    Dec 10, 2016 at 5:40
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Physically, we always ask about the probability of a particle to be located in a range of position values. Recall that the probability to find the particle between points $x_1$ and $x_2$ is

$$\int_{x_1}^{x^2} |\Psi(x)|^2 dx.$$

For $x_1=x_2$, the probability is zero. Therefore, the correct question to ask is about the probability of the particle being in a range of $x$ values. In that case, compute the integral above.

NOTE that your $\Psi$ function must be normalized, that is

$$\int_{-\infty}^{\infty} |\Psi(x)|^2 dx=1.$$

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The probability of finding a particle in a position $x$ ($x=a/4$) is equal to the probability density: put $x$ for example $a/4$ in $\left|\psi\right| ^2$. It is wrong to say it is equal to $0$.

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