Question about uncertainty principle

The answer is a little complex.The theory of relativity and the quantum mechanics are always being used in the same situations.Relativity effects are obvious when we observe the cosmic ray or the particle in particle accelerators,while quantum mechanics principles,like uncertainty principle and the principle of superposition of states are also effective in those situations.What I really mean is relativity and quantum principle(not include Schrodinger equation) describe different sides of one object.There's no connection and constradiction between them.

 

How, exactly?

 

Just reasking the original question: whether relativistic effects modify the uncertainty principle. One still has [x,p] = iħ, so my first guess would be no. Although this may be related to the issue of whether x is really the position operator.

 

One question .. in the twi split experiment ... do scientist explain the paradox with heisenbergs uncertainity principle .. ?

 

As opposed to x being...? Please clarify.

 

Relativistic quantum mechanics is a multiparticle theory. To talk about single-particle quantities you need to restrict it. In particular to exclude the antiparticles you have to use just the positive energy modes.

But it turns out to be impossible, using only positive energy modes, to construct a state which is entirely localized at a point (δ-function). Also, the operator in momentum space that one would normally call the position, i∇_p, is not even Hermitian. So for both these reasons, to answer the pertinent question, "what is the probability of finding the particle at a particular point", you can't use x to specify the point.

Newton and Wigner defined a different position operator X to be the Hermitian part of i∇_p, namely X = i∇_p -ip/(p² + (mc)²). X obeys the right commutation relations, [X, p] = iħ. And its eigenstates are little Hankel-function-shaped things about a Compton wavelength across.

 

Hmmm...this alternative position operator X is then a differential operator by including p in its definition, isn't it?

 

Sure it is, so is x = iħ∇_p!

I know it's backwards from the usual situation where p is represented by a differential operator and x is not. But it works either way, and in quantum field theory the momentum representation is used all the time and for many purposes is a lot simpler.

For the standard x, the eigenvalue equation is iħ∇_pφ = x₀φ, with solution φ(p) = e^{-(i/ħ)p·x₀}. To get the corresponding eigenfunction ψ(x) in coordinate space, Fourier transform it. The Fourier transform of a plane wave of course is δ(x - x₀), a wavefunction completely localized at x₀.

For the Newton-Wigner X, the same procedure leads to a φ(p) which is just slightly different: φ(p)= √p₀ e^{-(i/ħ)p·x₀}. But the Fourier transform of this back to x space is no longer a delta function, it's slightly spread out.

 

I think you probably misunderstand the host's question.Your answers may be a little stray from the point and trap in complexity.The host has doubt that whether uncertainty principle should be modified in relativistic situation.The answer is absolutely no.

 

You needn't.If you moving fast and watch an object,you would still find its velocity(or say momentum) and location satisfy the uncertainty principle.Because generally there's no exact value of velocity as well as location to describe the particle,the relativistic transformations are powerlessness.In some case,we regard the quantal particle as classic one,and relativistic transformations work.For example, the particles in the accelerator.

 

I think your argument confuse something.You follow the particle and don't see any radiation,sound like different from the result in another frame.But these different phenomenons
are still unconcerned with HUP.The particle stays in the ground state and has no extra kinetic energy,but it doesn't matter.Because the radiation comes from the potential energy.We already understood this business in classic views.

 

i mean that in the two slit experiment .. the electrons first behave like a wave .. but when scientist put an detector to see form which slit each electrons go (or both) the electron behaved like a particle ....
how did they explain this... ?