MathFormalism - 1 PHYS 3143 QUANTUM MECHANICS I REMARKS ON...

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1 PHYS 3143 QUANTUM MECHANICS I: REMARKS ON THE VECTOR SPACE FORMALISM Brian Kennedy, School of Physics, Georgia Tech The notes below are not intended to be complete, but I hope you will find them useful. They are supposed to supplement the class notes and other reading. A. Euclidean and abstract vector spaces. We are used to dealing with vectors in one, two or three dimensional Euclidean space. Consider then a set of abstract vectors S = { vectoru,vectorv,vectorw,... } . The algebra of vectors (due to W.R. Hamilton) involves the operation of vector addition vectoru + vectorv - by the parallelogram rule in the case of Euclidean vectors - giving another vector in the same space (this is called closure under addition). The operation of addition is commutative and associative ( vectoru + vectorv ) + vectorw = vectoru + ( vectorv + vectorw ) . There exists a zero vector vector 0 and for every vector vectoru there is an inverse vector u also in the set: vectoru + vector u = vector 0. The set of vectors thus forms what is called an Abelian group under vector addition (Abelian means that addition is commutative). We get a vector space when we give the set of vectors an additional structure, namely multiplication by (the field of) real numbers, λ R : λvectoru is in the set of vectors. In quantum theory we consider vector spaces over the field of complex numbers rather than the reals. The scalar multiplication properties include λ ( vectoru + vectorv ) = λvectoru + λvectorv , 0 vectoru = vector 0 and ( 1) vectoru = vector u . In summary, for a vector space think of a set of geometrical objects, the vectors, and operations of vector addition and multiplication by scalars. We will henceforth omit the arrow symbol from the abstract vectors, but retain it when we refer to Euclidean vectors, in order to simplify the notation. B. Inner product of vectors: the inner product space To introduce an idea of length, we need a norm for the vectors. This in turn follows from the inner (scalar) product of two vectors. Abstractly an inner product is an operation which maps two vectors to a complex scalar: ( u,v ) C , satisfying ( i ) ( u,v + w ) = ( u,v ) + ( u,w ) , ( ii ) ( u,λv ) = λ ( u,v ) , ( iii ) ( u,v ) = ( v,u ) and ( iv ) ( u,u ) 0 . The equality in (iv) is true if and only if u = 0. The norm of a vector bardbl u bardbl = radicalbig ( u,u ). You can readily verify that the familiar real scalar product of Euclidean vector algebra vectoru · vectorv R satisfies all these properties. We derived in class two important inequalities that follow from (i)-(iv), the Cauchy-Schwarz or Schwarz inequality | ( u,v ) | ≤ bardbl u bardblbardbl v bardbl , and the triangle inequality bardbl u + v bardbl ≤ bardbl u bardbl + bardbl v bardbl . The latter was derived with the help of the former. A complete vector space with an inner product is called a Hilbert space, and it may have finite or infinite dimension. The Schwarz inequality is used in the derivation of the famous Heisenberg Uncertainty principle. The triangle inequality provides a test that a sequence of vectors (called a Cauchy sequence) converges to a limit without having to know what the limit vector is !
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