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Unformatted text preview: S3: THERE ARE HIDDEN VARIABLES IN QUANTUM MECHANICS Dr Spyridon Koutandos [email protected] Abstract: T he first two chapters in this article are dealing with the compatibility of Schrödinger equation with Maxwell equations. Although the reader may find them boring or difficult to understand, they are the continuation of my first two articles S1,S2 as mentioned in the introduction. Of course the essence of the title of this article lies within chapter 3, in which I discuss the fact that there are “hidden variables” in quantum mechanics, as Dr Albert Einstein first named them while he lived. INTRODUCTION In my previous articles S1)“An extension of Dirac notation and its consequences”, S2)“An educative approach on the problem of defining magnetic permeability” I may have caused some confusion on the following matter: We have a magnetic induction made of two parts: B=B 1 +B 2 . φ ∇ × ∇ = C B 1 (1) Since we know, and it is reasonable that the wave function psi is single valued, it must be a perfect differential and one may find after some elementary use of identities that: ( ) φ ψ ψ ψ φ ∇ × ∇ = ∇ × ∇ = = ∇ × ∇ i e i | | (2) Actually, the theorem of the combs informs us that there should be one more point that psi goes to zero as we change the principal quantum number. This should be before considering magnetic field, and in central potential (actually it is always central- but this is another big story) psi is made real through combinations of the proper degenerate (equal in energy) eigenfunctions. So, finally our magnetic field B 1 is different than zero only on specific surfaces. CHAPTER 1 In my article “An extension of Dirac notation and its consequences” I found out that when taking the real part of functions in space of hermitian operators all vector algebra is obeyed. I also found out about the orthogonality of operators in that space. All these directly imply (although I can not give a formal proof right now) that in transforming inside that space an operator to another we have equality. In that paper I found out that a term of magnetic work that should be zero was made of B 1 M, where M is the magnetization .That was the beginning of my investigations. On these (equipotential) surfaces thus, it is that B 1 can be specified on. HYPOTHESIS H1 Either: a) M=0, Or b)M is perpendicular to B 1 , and that should make B 1 have only a radial component. The reader may refer now to a book of electromagnetism such as is the following: “Foundations of Electromagnetic theory” Reitz-Milford-Christy Suppose now for a moment that we could make use of the steps found in this book. In the book psi is referred to as real. The only two “dangerous steps” taken there are a,b: ----------------- HYPOTHESIS H2 ------------ a)Suppose ( ) = × ∇ × ∇ r v ψ , b) = ∇ × ∇ ⋅ ⇒ = ⋅ ∇ ψ r L r r If these two are satisfied we are lead to: ( ) = ∇ × ∇ ψ (i) Except for a spherical shell ....
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- Spring '09
- Magnetic Field