tiagovtristao said:
What do you mean by modern quantum mechanics?
Ok here is the outline of the history.
In 1924 De-Broglie had the idea if waves can sometimes act like particles as in light then maybe the converse can happen - particles like electrons can sometimes act like waves. He wrote it up as his PhD thesis but the examiners didn't know what to make of it. Well a copy made its way to Einstein - this book gives the full details:
https://www.amazon.com/dp/1491531045/?tag=pfamazon01-20
He recognized immediately it was an important step in solving the quantum puzzle, but knew it was not the answer - basically it was wrong - but his intuition, correctly, told him it was a step in the right direction. He gave it his enthusiastic support. The examiners were still not convinced despite Einsteins approval, but did like the math, so he got his PhD.
Then things moved quickly. Schrodinger in giving a lecture on it was challenged - if matter was waves then it should obey a wave equation. He found one - but while he got the correct answer he made an error in doing it:
https://arxiv.org/abs/1204.0653
These days we can easily derive the so called Klein-Gordon equation from very simple considerations and the Schrodinger equation as the relativistic limit - but that is another story requiring another thread. Start one if you like.
At about the same time Heisenberg came up with his matrix mechanics. They looked entirely different. Then an upstart called Dirac came up with a still different approach called q numbers that Heisenberg recognized as better than what he did. Now things were in a really bad way. But very quickly and to everyone's surprise, even Einsteins, everything fell into place:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf
Dirac's 1926 transformation theory paper is basically QM as we know it today. But mathematically it used things that were 'dubious'. It was elegant beyond belief but mathematicians like Von-Neumann were concerned that its mathematics was not valid. He came up with a version that was mathematically correct, but physicists spoke with their feet and chose Dirac.
Well mathematicians took this as a challenge and thanks to the efforts of three of the greatest mathematicians of the 20th century, Grothendieck, Gelfland, and Schwartz the issues were all resolved and two very useful branches of math were invented - Rigged Hilbert Spaces and Distribution theory. Distribution theory especially is so useful every applied mathematician (and that very much includes electrical engineers) should know about it. It for example makes Fourier transforms a snap, otherwise it becomes bogged down in issues of convergence. If you are into Electrical Engineering (and applied math in general) I strongly recommend the following book:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20
Einstein was proved right - De-Broglie's idea was wrong - but an important and necessary step to the right answer.
Thanks
Bill