# Integral to determine position probability.

• space-time
In summary, there are two integrals that can be used to describe different properties of a particle's wave function. The integral \int_a^b {| \psi(x) |^2 dx} gives the probability of finding the particle between positions a and b, while the integral \int_{-\infty}^{+\infty} {x | \psi(x) |^2 dx} gives the expectation value of the particle's position. Both integrals use the notation of the square of the magnitude of the wave function, but they have different limits of integration.

#### space-time

There is something that I just want to make sure I am understanding.

I read once before that ∫ababs(ψ)2 dx will give you the probability that your particle will appear in region between x=a and x=b. Note: abs(ψ)2 means the square of the magnitude of the wave function. I just couldn't find any absolute value bars in the latex and the notation for magnitude looks like absolute value bars around the function. That is why I typed abs, but I really mean the magnitude.

Anyway, much later I believe I read that the formula was supposed to be:

abxabs(ψ)2 dx

(which is the same integral except the integrand is multiplied by x).

Can anyone tell me which integral is the correct one or if they are both correct and they just describe two different things?

space-time said:
There is something that I just want to make sure I am understanding.

I read once before that ∫ababs(ψ)2 dx will give you the probability that your particle will appear in region between x=a and x=b. Note: abs(ψ)2 means the square of the magnitude of the wave function. I just couldn't find any absolute value bars in the latex and the notation for magnitude looks like absolute value bars around the function. That is why I typed abs, but I really mean the magnitude.

Anyway, much later I believe I read that the formula was supposed to be:

abxabs(ψ)2 dx

(which is the same integral except the integrand is multiplied by x).

Can anyone tell me which integral is the correct one or if they are both correct and they just describe two different things?

There is a vertical bar on your keyboard, by the way. (Assuming you have the standard keyboard that is found in the United States--I don't know about elsewhere)

The integral $\int x |\psi|^2 dx$ does not give a probability, it gives the average, or expectation value, for position.

stevendaryl said:
There is a vertical bar on your keyboard, by the way. (Assuming you have the standard keyboard that is found in the United States--I don't know about elsewhere)

The integral $\int x |\psi|^2 dx$ does not give a probability, it gives the average, or expectation value, for position.

So then the other one gives the probability?

To get the probability of finding the particle between positions a and b: $$\int_a^b {| \psi(x) |^2 dx}$$ To get the expectation value of x: $$\int_{-\infty}^{+\infty} {x | \psi(x) |^2 dx}$$ Note the different limits of integration.