How can this possible tricky situation be explained?

[james1234567890]

This situation is related to the one I asked in the previous post a few days back.

There is an apparatus which is made of partly mirror and partly lens. A particle P1 is passed through the apparatus. The wave function of the particle is such that there is a 75% probability of the particle hitting the mirror and 25% probability of the particle hitting the lens. Thus after the particle interacts with the apparatus, the wave function of the particle P1 transforms such that there is 75% probability of the particle landing on the same side of apparatus after reflection from the mirror and 25% probability of the particle landing on the opposite side of apparatus after refraction from the lens. Till this point, there is no confusion.

But suppose let us assume that the particle P1 is composed of constituent particles P2, just as a proton is composed of quarks. I think the wave function of particle P2 need not be the same as P1. Suppose the wave function of each P2 composing P1 is such that each P2 has a 60% chance of hitting the mirror and 40% chance of hitting the lens. Then after the interaction with the apparatus, the wave function of each P2 transforms such that it has a 60% chance of landing on the same side of apparatus having reflected from mirror and 40% chance of landing on the opposite side having refracted from the mirror.

But the confusion is that when we consider particle P1 as a whole without considering P2 individually as in the first case, the wave function of P1 splits on either sides of apparatus in the ratio 75%:25% which means each P2 composing P1 has a 75% - 25% chance of landing up on either sides of apparatus since the position of each P2 should be the same as the position of P1 since P2 is a constituent of P1.

But if we take the constituent P2 individually without considering P1 as a whole as in the second case, then the wave function of each P2 splits on either sides of the apparatus in the ratio 60%:40% thereby giving a 60-40 chance of each P2 landing on either sides of apapratus.

Thus when we approach our situation through two different angles i.e. one with respect to the whole particle and the other with respect to the constituents, we get two different probabilities of outcomes. I want to know what would happen in the real scenario, i.e. whether the wave function will transform in accordance to the first case or in accordance to the second case. Is there really some ambiguity here or is there is any flaw in the above reasoning and my understanding of quantum mechanics? Kindly clarify my doubt as soon as possible. Thanks in advance.

 

[james1234567890]

Someone please shed some light on my doubt.

Let us assume that the particle P1 is composed of 100 particles of P2. Suppose if we consider particle P1 as a whole, there is 75% chance that all the 100 P2 particles that compose P1 will land on the same side of apparatus after interaction and 25% chance that all the P2 particles making up P1 will land on opposite side in accordance to the wave function of P1.

But if we consider the constituent particles P2 individually, then of the 100 P2 particles, around 60 P2 particles are likely to land on the same side of the apparatus while 40 P2 particles are likely to land on the opposite side of apparatus in accordance to the wave function of P2. Thus it's like the entire particle P1 splits up in the ratio 6:4.

The above two different angles of approaching the situation gives two different outcomes. What is the ambiguity in this and what will be the real outcome in such a situation? Is there any misconception in my understanding and reasoning? I am confused about this. Someone please clarify me.

 

[jostpuur]

I've already learned a lesson about answering, so I'll start with a question. Do you know basic things about quantum mechanics. Have you solved some simple problems with plane waves and step potentials and so on?

I cannot promise that I can answer your problem, but it seems a good problem, and I already came up with a good exercise related to it, although I'm not yet sure if I'm able to solve it myself... I'll return to this if your answer to my previous questions is positive.

 

[jostpuur]

But if we take the constituent P2 individually without considering P1 as a whole as in the second case, then the wave function of each P2 splits on either sides of the apparatus in the ratio 60%:40% thereby giving a 60-40 chance of each P2 landing on either sides of apapratus.

You are certainly doing something wrong, if you calculate probabilities of particles P2 separately, since you are ignoring their interactions. The big question is of course, that how is that done correctly then. I'm not sure yet.

 

[james1234567890]

I've already learned a lesson about answering, so I'll start with a question. Do you know basic things about quantum mechanics. Have you solved some simple problems with plane waves and step potentials and so on?

I cannot promise that I can answer your problem, but it seems a good problem, and I already came up with a good exercise related to it, although I'm not yet sure if I'm able to solve it myself... I'll return to this if your answer to my previous questions is positive.

I am very new to quantum mechanics and I have not solved any problems in it. But when I start thinking about it from whatever little and basics of QM I know, such confusing and ambiguous questions arise to me. So I am posting my doubts here to obtain a more clear picture of quantum mechanics.
This question that I posted here just occurred to me by chance today when I was thinking about wave functions and probabilities.

 

[jostpuur]

Suppose you have an object that is made of two particles, and the object is shot onto a half reflecting mirror. One way or another, there is going to be a probability for the object to bounce back, probablity to go through, and a probability to break into two separate particles, of which other one goes through and other one does not. These probabilities are not kind of things you are going to be able to solve with some easy ideas. Your problems start when you make too much assumptions about how these probabilities behave. I got some idea how this calculation could be done with some rough approximations in one dimension, but I guess there's no point in getting into detail now, because there are simpler exercises you would have to do first before understanding this one. I'm currently busy with my courses, but I think I'm going to return to this problem later. Maybe in summer. Anyway, such an exercise should be a somekind of answer to your original problem.

 

[james1234567890]

Suppose you have an object that is made of two particles, and the object is shot onto a half reflecting mirror. One way or another, there is going to be a probability for the object to bounce back, probablity to go through, and a probability to break into two separate particles, of which other one goes through and other one does not. These probabilities are not kind of things you are going to be able to solve with some easy ideas. Your problems start when you make too much assumptions about how these probabilities behave. I got some idea how this calculation could be done with some rough approximations in one dimension, but I guess there's no point in getting into detail now, because there are simpler exercises you would have to do first before understanding this one. I'm currently busy with my courses, but I think I'm going to return to this problem later. Maybe in summer. Anyway, such an exercise should be a somekind of answer to your original problem.

Thanks a lot for your kind assistance Sir. Please do post your solutions and ideas whenever you find time so that I can get a better picture.

 

[reilly]

With a few changes, this is a very standard issue; sometimes known by the fancy name: multichannel scattering. As jostpuur notes: there are multiple outcomes possible. In your case, the possibilities are: 1. composite particle goes through, or bounces back, 2. each constituant can go through or bounce back, alone or with company(If you want real trouble, there is also the possibility that the composite particle will emit radiation. In fact radiative corrections -- taking the radiation into account -- can have a significant effect on the scattering outcomes. This is true, for example, for electron-deuteron scattering, which allowed Hofstadter(at Stanford in the 1960s) to measure the electric and magnetic properties of the neutron.) The whole deal required electron energies much greater than the binding energy of the deuteron, in which the probability of the deuteron remaining a deuteron is close to zero.)

The problem of scattering of composite particles goes way back -- Mott and
Massey in 1933, in the Theory of Atomic Collisions, discuss the scattering of composite particles in great detail. More recently Goldberger and Watson in their Collision Theory discuss multichannel scattering. Most any text on nuclear physics will deal with scattering of composite systems -- like nucleii --, as will texts on particle physics.

In other words, there is a huge literature on the scattering of composite systems. It's an old and very important topic. Good question indeed.
Regards,
Reilly Atkinson