# B Probability of finding a particle

#### Anonymous1212144

Let's say that I observed a free particle at a certain location. Is there any way I can calculate the probability of finding that same particle at another location when I look for it again?

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#### phinds

Gold Member
Let's say that I observed a free particle at a certain location. Is there any way I can calculate the probability of finding that same particle at another location when I look for it again?
That's going to depend a great deal on how much you know about the particle's velocity vector. If you KNOW that it headed straight North, you're not likely to find it if you look farther South.

#### Anonymous1212144

That's going to depend a great deal on how much you know about the particle's velocity vector. If you KNOW that it headed straight North, you're not likely to find it if you look farther South.
But is there a formula for it?

#### phinds

Gold Member
But is there a formula for it?
I think the question as posted is too vague for there to be a formula.

#### Anonymous1212144

I think the question as posted is too vague for there to be a formula.
What other information do you need?

#### phinds

Gold Member
What other information do you need?
What kind of particle are you talking about? What is it's velocity vector? I'm not sure that's enough but you definitely need at least that much. I won't be any further help on this since the details are beyond my knowledge but perhaps someone else will chime in, assuming you do provide at least that information.

#### Anonymous1212144

What kind of particle are you talking about? What is it's velocity vector? I'm not sure that's enough but you definitely need at least that much. I won't be any further help on this since the details are beyond my knowledge but perhaps someone else will chime in, assuming you do provide at least that information.
Couldn't the velocity and the kind of particle be two of the variables in the equation?

#### Nugatory

Mentor
Let's say that I observed a free particle at a certain location. Is there any way I can calculate the probability of finding that same particle at another location when I look for it again?
Yes, although there are some caveats. One is that the probability of finding a particle at any particular point is always exactly zero; the best that we can do is say that the probability of finding the particle within a distance $\epsilon$ of that point is $P$. When $\epsilon$ is very small and $P$ is close to unity, we say that the particle is found at that point (that is, $P=1$ and $\epsilon=0$) but that's a simplification, and when we do the math to calculate the future behavior of the particle we'll need the real values. The second problem is that we also need to know the velocity of the particle, and that is subject to the same sort of uncertainty.

But with that said, the basic idea is:
1) Solve the time-independent Schrodinger's equation for a free particle. This will tell you what the possible wave functions are.
2) Select the wave function from #1 that is consistent with your initial observation.
3) Use the time-dependent Schrodinger equation to calculate how that wave function changes over time.
4) Use the solution to #3 above to calculate the probability of finding the particle at a distance $\epsilon$ from the point you're interested in at whatever later time $t$ you're interested in.

A college-level intro QM class will work through this, but it requires a fair amount of calculus and differential equations, not stuff that belongs in a B-level thread. However, if you want to dig further, you can try googling for "free particle gaussian".

• Anonymous1212144

#### Anonymous1212144

What's the difference between this and the "particle in a box"?

#### Nugatory

Mentor
What's the difference between this and the "particle in a box"?
When you go to solve the time-independent Schrodinger equation for the particle in a box you'll find a different set of possible wave functions. This is because the TISE is $H\psi=E\psi$ and $H$ is different in the two cases. So answers we get will start to diverge after step #1 above.

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#### Anonymous1212144

But with that said, the basic idea is:
1) Solve the time-independent Schrodinger's equation for a free particle. This will tell you what the possible wave functions are.
2) Select the wave function from #1 that is consistent with your initial observation.
3) Use the time-dependent Schrodinger equation to calculate how that wave function changes over time.
4) Use the solution to #3 above to calculate the probability of finding the particle at a distance $\epsilon$ from the point you're interested in at whatever later time $t$ you're interested in.
I don't know if this is asking too much, but can you give me a general walkthrough of the processes? Or at least give me equations that I can graph?

"Probability of finding a particle"

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