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Unformatted text preview: An extension of Dirac notation and its consequences Spyridon Koutandos email@example.com MANIAKIOU 17, AGIA PARASKEVI , ATTICA 15343, GREECE(EU) www.geocities.com/skoutandos In this article we observe first that there is an extension of Dirac notation to include the so called “external interchangeability” of the hermitian operators when taking the real part of the bracket in the bra-ket notation. Then we conclude that this, together with the usual “internal interchangeability” of the hermitian operators concerning the real part of the bracket leads to the commutability of operators as to both cross and dot product therefore (by making use of the fact that operators are linear) all the vector algebra is valid when taking the real part of the bracket and we have hermitian operators. We deal with one particle situations and bound states. Finally we find a new way for calculating the square of the Runge-Lenz vector and we make speculations about hidden variable. First principles are used for the deduction of our conclusions,like the independence of the phase angle of ψ from its absolute value and the commutation relationships which are valid for any function. Integration as to volume is meant from zero to infinity and units are omitted for ease of calculations. I. INTERCHANGEABILITY OF HERMITIAN OPERATORS Let us begin by giving some definitions. The so called in this article “external interchangeability” that is to be proved is the following: (1.1) [ ] [ ] ψ ψ ψ ψ | ˆ ˆ | Re ˆ | | ˆ Re B A A B × = × − The so called, on the other hand “internal interchangeability” is: (1.2) ψ ψ ψ ψ A B AB | Re | | Re = Both are valid when A,B are hermitian. We are going to start by proving that the operator ΑΒ-ΒΑ is antihermitian when Α,Β are hermitian. First we notice that : * | | | | | | ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ B A B A A B B A BA AB − = = − = − (1.3) The last part of the equality containing the two terms is obviously purely imaginary, thus: (1.4) | | Re = − ψ ψ BA AB The validity of identity (1.4) by taking in mind that A,B are hermitian and can be interchanged readily proves (1.2) that is the internal interchangeability of the operators concerning the real part of the bracket. Next we are going to prove that : (1.5) [ ] [ ] ) Re( | ˆ ˆ | Re ˆ | | ˆ Re C B A A B v = × = × − ψ ψ ψ ψ If (1.5) is to be valid obviously then we also have: (1.6) [ ] | ˆ ˆ | ˆ | | ˆ Re = × − × − ψ ψ ψ ψ B A A B For proving (1.5) we decompose into cartesian components. The x component of (1.5) is written as: (1.7) [ ] [ ] [ ] ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ | | Re | | | | Re | | Re ) Re( y z z y y z z y y z z y x A B A B A B A B A B A B C − − = − − = = − − = The vector operators A,B are hermitian and so are their components. Therefore the x component of the last part of (1.7) is written as: (1.8) [ ] [ ] ψ ψ ψ ψ | | Re | | Re ) Re( y z z y y z z y x B A B A A B A B C − = − − = By subtracting the second part of (1.8) from the By subtracting the second part of (1....
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This note was uploaded on 12/02/2009 for the course NONE None taught by Professor Spyridonkoutandos during the Spring '09 term at 東京国際大学.
- Spring '09