3 ms·

I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400

I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?

MatteoFrigo 5y ago
Sorry, I didn't mean to imply that my comment should apply literally to all cases. I am just pointing out to parent that particles can be indistinguishable in principle, and not just as a technological limitation of our measurements. Moreover, there is an experimental way to tell the difference between distinguishable and indistinguishable, roughly based on the difference between probabilities and amplitudes. To dig deeper one must look at the details, e.g. in Feynman's lectures vol. III. The statement that protons are indistinguishable is not strictly correct either, because protons have a spin. Protons with the same spin are indistinguishable, but you can tell apart protons with different spin. The spin of protons can only assume two values, so effectively there are two classes of protons, indistinguishable within the class. In your specific case, it is clearly false that the probability of having one particle in one place is 200%. However, my statement still holds for expectations, and you end up with an expected two particles in one place. In the indistinguishable case, you must compute expectations based on amplitudes, not probabilities.
- db48x 5y ago
  Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a proton can also change over time (usually due to interactions with other particles, such as stray radio waves passing through your experiment). Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected! Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.
db48x 5y ago
In addition to what has already been said, I want to point out that there is a normalization step as well. The amplitudes are always calculated in such a way that the probabilities would never add up to more than 100%. This normalization is generally baked into the wave function of the system you are studying, but it can also be done separately. Incidentally, amplitudes are actually complex numbers. You can think of them as little arrows, like this: →, or this: ↖. In fact, these arrows are also rotating with the passage of time; they trace out little circles. To calculate the probability, we square the absolute value of the complex number. The absolute value of a complex number is equal to the length of the arrow, and squaring a length gives you an area. Thus the probability is essentially the same as the area of the circle traced out by the rotating arrow. Events with high probability correspond to long arrows (big amplitudes), and low probability events have short arrows (small amplitudes). Amplitudes can cancel out when added together if they point in opposite directions. Thus we observe that some sequences of events have very low probability. We sometimes say that these events interfere with each other. Sometimes this interference seems mysterious, as in the double–slit experiment, and other times it seems very mundane. In real life we rarely bother to calculate the probability that the batter will hit the ball before the pitcher throws it, but calculating the correct answers in quantum mechanics requires taking into account many such unusual events.
- whatshisface 5y ago
  Can a horizontally polarized photon destructively interfere with a vertically polarized photon in an experiment that is insensitive to polarization?
  - db48x 5y ago
    Yes. Why do you ask?

I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?

MatteoFrigo 5y ago

Sorry, I didn't mean to imply that my comment should apply literally to all cases. I am just pointing out to parent that particles can be indistinguishable in principle, and not just as a technological limitation of our measurements. Moreover, there is an experimental way to tell the difference between distinguishable and indistinguishable, roughly based on the difference between probabilities and amplitudes. To dig deeper one must look at the details, e.g. in Feynman's lectures vol. III. The statement that protons are indistinguishable is not strictly correct either, because protons have a spin. Protons with the same spin are indistinguishable, but you can tell apart protons with different spin. The spin of protons can only assume two values, so effectively there are two classes of protons, indistinguishable within the class. In your specific case, it is clearly false that the probability of having one particle in one place is 200%. However, my statement still holds for expectations, and you end up with an expected two particles in one place. In the indistinguishable case, you must compute expectations based on amplitudes, not probabilities.

db48x 5y ago
Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a proton can also change over time (usually due to interactions with other particles, such as stray radio waves passing through your experiment). Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected! Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.

db48x 5y ago

In addition to what has already been said, I want to point out that there is a normalization step as well. The amplitudes are always calculated in such a way that the probabilities would never add up to more than 100%. This normalization is generally baked into the wave function of the system you are studying, but it can also be done separately. Incidentally, amplitudes are actually complex numbers. You can think of them as little arrows, like this: →, or this: ↖. In fact, these arrows are also rotating with the passage of time; they trace out little circles. To calculate the probability, we square the absolute value of the complex number. The absolute value of a complex number is equal to the length of the arrow, and squaring a length gives you an area. Thus the probability is essentially the same as the area of the circle traced out by the rotating arrow. Events with high probability correspond to long arrows (big amplitudes), and low probability events have short arrows (small amplitudes). Amplitudes can cancel out when added together if they point in opposite directions. Thus we observe that some sequences of events have very low probability. We sometimes say that these events interfere with each other. Sometimes this interference seems mysterious, as in the double–slit experiment, and other times it seems very mundane. In real life we rarely bother to calculate the probability that the batter will hit the ball before the pitcher throws it, but calculating the correct answers in quantum mechanics requires taking into account many such unusual events.

whatshisface 5y ago
Can a horizontally polarized photon destructively interfere with a vertically polarized photon in an experiment that is insensitive to polarization?
- db48x 5y ago
  Yes. Why do you ask?