Complex Vector Spaces and Quantum Mechanics

Jochen Pade

Mikio Nakahara

The set of natural numbers {1, 2, 3, . . .} is denoted by N. The set of integers {. . . ,−2,−1, 0, 1, 2, . . .} is denoted by Z. Q denotes the set of rational numbers. Finally R and C denote the sets of real numbers and complex numbers, respectively. Observe that N ⊂ Z ⊂ Q ⊂ R ⊂ C The vector spaces encountered in physics are mostly real vector spaces and complex vector spaces. Classical mechanics and electrodynamics are formulated mainly in real vector spaces while quantum mechanics (and hence this book) is founded on complex vector spaces. In the rest of this chapter, we briefly summarize vector spaces and matrices (linear maps), taking applications to quantum mechanics into account.

Didik Nur Huda

The study of quantum mechanics has been built on a Hilbert space, the complex vector space with inner product <.|.>:VxV--> C with C is a complex vector space dan is complex numbers. Hilbert module is a generalization of Hilbert space, by changing the scalar set become ring with certain conditions so that a Hilbert module. This ring is called the number ring. Hilbert modules that will be used to build the modular foundations of quantum mechanics. Keywords : Scalar, Rings, Hilbert Module, Quantum Mechanics

Kundeti Muralidhar

Yakir Aharonov, Daniel Rohrlich

Srinivasa Mugeraya

Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.