The discussion revolves around calculating the probability of finding a free particle at a different location after it has been observed at a certain position. Participants explore the implications of the particle's velocity and the vagueness of the initial question, as well as the mathematical framework involved in quantum mechanics, particularly the Schrödinger equation.
Participants express varying levels of agreement on the need for additional information to formulate a probability calculation. There is no consensus on a specific formula or method due to the complexity and vagueness of the initial question.
The discussion highlights limitations related to the vagueness of the initial question, the need for specific details about the particle's properties, and the mathematical complexities involved in quantum mechanics.
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 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?
But is there a formula for it?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.
I think the question as posted is too vague for there to be a formula.Anonymous1212144 said:But is there a formula for it?
What other information do you need?phinds said:I think the question as posted is too vague for there to be a formula.
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 said:What other information do you need?
Couldn't the velocity and the kind of particle be two of the variables in the equation?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.
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.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?
What's the difference between this and the "particle in a box"?Nugatory said:snip
When you go to solve the time-independent Schrödinger 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.Anonymous1212144 said:What's the difference between this and the "particle in a box"?
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?Nugatory said:But with that said, the basic idea is:
1) Solve the time-independent Schrödinger'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 Schrödinger 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.