Integral to determine position probability.

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.
  • #1
space-time
218
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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?
 
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  • #2
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 [itex]\int x |\psi|^2 dx[/itex] does not give a probability, it gives the average, or expectation value, for position.
 
  • #3
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 [itex]\int x |\psi|^2 dx[/itex] does not give a probability, it gives the average, or expectation value, for position.

So then the other one gives the probability?
 
  • #4
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.
 

FAQ: Integral to determine position probability.

What is an integral?

An integral is a mathematical concept used to find the area under a curve. It is represented by the symbol ∫ and is often used in calculus.

How is an integral related to determining position probability?

In quantum mechanics, the position probability of a particle is represented by a wave function. An integral is used to calculate the probability of finding the particle within a specific range of positions.

What is the importance of using integrals in determining position probability?

Integrals allow us to calculate precise probabilities for the position of a particle in quantum mechanics. This is essential for understanding the behavior of particles on a microscopic level.

Can you provide an example of how an integral is used to determine position probability?

For example, if we have a wave function that describes the position of a particle as a function of time, we can use an integral to find the probability of finding the particle within a certain range of positions at a specific time.

Are there any limitations to using integrals for determining position probability?

Integrals can only be used for particles with well-defined positions, such as electrons. They cannot be used for particles with uncertain positions, such as photons.

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