This is Heisenberg's uncertainty equation.
∆x.∆p≥ ħ
∆E.∆t≥ ħ
what is physical meaning of above equation?
Salam Aliakum Sit Demen
According to QM postulates , Commute operators have simultaneous eignfunction (the same function), in other word , the measurement of a physical observable cannot alter the wave function if the function is an eignfunction. Unfortunately neither position, momentum operator pair nor energy time operator pair are commute, the results is as mentioned above . For observables which have commute operators, like (energy, square of angular momentum pair), we can measure E and L^2 simultaneously.
I hope that the explanation is useful
Regards
In quantum mechanics a microscopic system is specified by it's wavefunction.
Position and momentum are replaced by corresponding operators x̂ and p̂ (x-hat and p-hat). Notion of measurement is replaced by an eigenvalue problem. Now abstract operators have to have their representations. That is how we know their action. Two well known representations are
In coordinate representation, let us look at the commutator bracket ( x̂p̂- p̂x̂ ) ψ = [ x̂, p̂ ]ψ=-iħ x ∂ψ/∂x + iħ ∂(xψ)/∂x = iħ ψ. Because ψ can be an "arbitrary" wave function of a quantum system, [ x̂, p̂ ]= iħ. The same result is also obtained in the momentum representation.
This is the fundamental uncertainty. If we measure position first and them momentum, and we also do it in the reverse order, the product of results in two cases are surprisingly not identical.
Salam St.Diman:
According to my points of view ( we can not measure both simultaneously because of misleading between observable and measurements[in classical physics consider we measure position of the system afterward measure momentum we get the same result if we measure momentum first then position but in quantum world we can not measure momentum or position (above process) directly because of straggling in observables and measurements,to connect both we use an operator or more general commutator relation,by using commutator relation unfortunately as Biswajoy Brahmachari mentioned momentum and position in same direction are not identical .
to find out more about this logical interpretation I guide you read
1- Physics and beyond by werner Heisenberg also translated to Kurdish
2- page 167-169 in Quantum mechanics concept and application by Nouredine Zettili
3-This Pdf maybe help you to reduce your difficulties
https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-007-electromagnetic-energy-from-motors-to-lasers-spring-2011/lecture-notes/MIT6_007S11_lec38.pdf
Regards,
In simple words: The measurement is influencing the measured system - because it's energy is changing the QM-state-relations! P and momentum or time and energy are correlated by quantum physical laws (forces). That's the starting law for Quantum Mechanics.
Out of my knowledge in Information Science I realize that Physics is coming to a border because we lose the abiliy to measure exactly and Mathematics as ought serving science is showing different "algorithms" (particle or wave) wich can't be brought to one realy explaining theory.
So I think Systems Theory will bring better results - but we have to think and wait till we get satisfying answers in a level deeper. Photons, Electrons and Atoms are not definitive researched enough to understand that new relation like Heisenberg wrote down in his uncertainty formula. This is a rough estimation (delta is not defineable exactly). We need a "quantum microscope" which can bring deeper insights in Photons, Electrons and Atoms.
Salam Aliakum Sit Demen
According to QM postulates , Commute operators have simultaneous eignfunction (the same function), in other word , the measurement of a physical observable cannot alter the wave function if the function is an eignfunction. Unfortunately neither position, momentum operator pair nor energy time operator pair are commute, the results is as mentioned above . For observables which have commute operators, like (energy, square of angular momentum pair), we can measure E and L^2 simultaneously.
I hope that the explanation is useful
Regards
عليكم السلام ورحمة الله وبركاته
Thank you very much Mr.Abdulghefar .
all above answers are great and all information of microscopic system related to wave function if we can localize it all problems of Q.M will be solve.
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