Probability of finding a particle

In summary: 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?
  • #1
Anonymous1212144
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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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  • #2
Anonymous1212144 said:
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.
 
  • #3
phinds said:
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?
 
  • #4
Anonymous1212144 said:
But is there a formula for it?
I think the question as posted is too vague for there to be a formula.
 
  • #5
phinds said:
I think the question as posted is too vague for there to be a formula.
What other information do you need?
 
  • #6
Anonymous1212144 said:
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.
 
  • #7
phinds said:
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?
 
  • #8
Anonymous1212144 said:
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".
 
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  • #9
Nugatory said:
snip
What's the difference between this and the "particle in a box"?
 
  • #10
Anonymous1212144 said:
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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  • #11
Nugatory said:
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?
 

Related to Probability of finding a particle

1. What is the probability of finding a particle in a specific location?

The probability of finding a particle in a specific location is determined by the wave function of the particle, which describes the likelihood of finding the particle at different positions. This probability is represented by the squared magnitude of the wave function at that location.

2. How is the probability of finding a particle calculated?

The probability of finding a particle is calculated using the Schrödinger equation, which takes into account the wave function, the potential energy of the particle, and the mass of the particle. This equation allows us to determine the probability of finding the particle at any given point in space.

3. Can the probability of finding a particle be greater than 1?

No, the probability of finding a particle cannot be greater than 1. This is because the total probability of finding the particle in all possible locations must equal 1. If the probability at a specific location is greater than 1, it would mean that the particle is guaranteed to be found there, which is not possible.

4. How does the uncertainty principle affect the probability of finding a particle?

The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle simultaneously. This means that the more accurately we know the position of a particle, the less accurately we know its momentum, and vice versa. This uncertainty affects the probability of finding the particle, as it is impossible to have a completely precise measurement of its location.

5. Can the probability of finding a particle change over time?

Yes, the probability of finding a particle can change over time. This is because the wave function of the particle evolves over time according to the Schrödinger equation. As the wave function changes, so does the probability of finding the particle at different locations. This is known as wave function collapse.

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