How Does Momentum Measurement Affect Uncertainty in an Infinite Potential Well?

[HumbleStudent]

This may sound very basic, but I've just learned about the potential well with infinite barriers at +a and -a and I had a doubt. If we measure the momentum of a particle inside the well, it collapses to an eigenstate of the momentum operator, so, the uncertainty will be zero. Accordingly, the uncertainty of the position shouldn't be infinite? But how can it be possible if the well is finite (from -a to +a)?

This obviously is just a misinterpretation, but could someone tell me what am I doing wrong?

Thanks,
Humble.

 

[jostpuur]

The usual formulation of QM just introduces the concept of measurement as if it was trivial what it is. But in reality, there is no general devices that would actually allow you to project states onto desired basis vectros, as easily as you could project states with projection operators in mathematics. I belive, that the solution to your problem is simply, that such measurement cannot be carried out. The system just evolves according to SE, and in some situations you can interpret some outcomes as measurements.

 

[HumbleStudent]

That means that the measurement postulate about projection of the states only works in specific situations? I thought it was a basic postulate of quantum mechanics. But this means that such an experimental simple set up like measuring the momentum of a particle inside a box cannot be realized?

 

[jostpuur]

I thought it was a basic postulate of quantum mechanics.

It is a basic postulate of QM, but nobody knows what it really means!

For example, I do not believe that anyone could measure a momentum of a single electron that is on a bound state in hydrogen. If it can be done, somebody may correct me.

Another matter is this: If a measurement accuracy isn't infinite, then the particle is not necessarely collapsing onto an eigenstate of a momentum, but instead onto a wavepacket that is localised around some value of momentum. So finding out a momentum of a particle does spread out its position, but not into inifities in reality.

But this means that such an experimental simple set up like measuring the momentum of a particle inside a box cannot be realized?

I don't dare to say it would be impossible. That is a quite theoretical set up. If in some real situation it can be done so that it avoids paradoxes, then it is probably possible

I mean, if you have a macroscopic box, and a particle wave packet bouncing there, then you probably can measure it's momentum there without problems. But if you instead use the box as an approximation of some microscopic system, then you encounter other kind of problems. That question is slightly too theoretical.

 

[HumbleStudent]

Really interesting indeed. Although I will need to digest it (and maybe put up a little fight with my QM professor) in order to understand better. I thought that this question was just something which I was calculating or interpreting wrong, but seems that it touches something more complicated, right?

And forgot to say, many thanks for the explanation!