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Chapter 5 the Dirac Formalism and Hilbert Spaces Chapter 5 The Dirac Formalism and Hilbert Spaces In the last chapter we introduced quantum mechanics using wave functions defined in position space. We identified the Fourier transform of the wave function in position space as a wave function in the wave vector or momen­ tum space. Expectation values of operators that represent observables of the system can be computed using either representation of the wavefunc­ tion. Obviously, the physics must be independent whether represented in position or wave number space. P.A.M. Dirac was the first to introduce a representation-free notation for the quantum mechanical state of the system and operators representing physical observables. He realized that quantum mechanical expectation values could be rewritten. For example the expected value of the Hamiltonian can be expressed as ψ∗ (x) Hop ψ (x) dx = ψ Hop ψ , (5.1) h | | i Z = ψ ϕ , (5.2) h | i with ϕ = Hop ψ . (5.3) | i | i Here, ψ and ϕ are vectors in a Hilbert-Space, which is yet to be defined. For exampl| i e, c|omi plex functions of one variable, ψ(x), that are square inte­ 241 242 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES grable, i.e. ψ∗ (x) ψ (x) dx < , (5.4) ∞ Z 2 formt heHilbert-Spaceofsquareintegrablefunctionsdenoteda s L. In Dirac notation this is ψ∗ (x) ψ (x) dx = ψ ψ . (5.5) h | i Z Orthogonality relations can be rewritten as ψ∗ (x) ψ (x) dx = ψ ψ = δmn. (5.6) m n h m| ni Z As see aboveexpressions look likeabrackethecalledthe vector ψn aket­ vector and ψ a bra-vector. | i h m| 5.1 Hilbert Space A Hilbert Space is a linear vector space, i.e. if there are two elements ϕ and ψ in this space the sum of the elements must also be an element of t|hei vecto|r ispace ϕ + ψ = ϕ + ψ . (5.7) | i | i | i The sum of two elements is commutative and associative Commutative : ϕ + ψ = ψ + ϕ , (5.8) | i | i | i | i Associative : ϕ + ψ + χ = ϕ + ψ + χ . (5.9) | i | i | i | i The product of the vector with a complex quantity c is again a vector of the Hilbert-Space c ϕ cϕ . (5.10) | i ≡ | i The product between vectors and numbers is distributive Distributive : c ϕ + ψ = c ϕ + c ψ . (5.11) | i | i | i In short every linear combination of vectors in a Hilbert space is again a vector in the Hilbert space. 5.1. HILBERT SPACE 243 5.1.1 Scalar Product and Norm There is a bilinear form defined by two elments of the Hilbert Space ϕ and ψ , which is called a scalar product resulting in a complex number | i | i ϕ ψ = a. (5.12) h | i Ths scalar product obtained by exchanging the role of ϕ and ψ results in the complex conjugate number | i | i ψ ϕ = ϕ ψ ∗= a∗ . (5.13) h | i h | i The scalar product is distributive Distributive : ϕ ψ + ψ = ϕ ψ + ϕ ψ . (5.14) h | 1 2i h | 1i h | 2i ϕ cψ = c ϕ ψ . (5.15) h | i h | i And from Eq.(5.13) follows cψ ϕ = ϕ cψ ∗ = c∗ ψ ϕ . (5.16) h | i h | i h | i Thus if the complex number is pulled out from a bra-vector it becomes its complex conjugate. The bra- and ket-vectors are hermitian, or adjoint, to each other. The adjoint vector is denoted by the symbol+ ( ϕ )+ = ϕ , (5.17) | i h | ( ϕ )+ = ϕ . (5.18) h | | i The vector spaces of bra- and ket-vectors are dual to each other. To transform an arbitrary expression into its adjoint, one has to replace all operators and vectors by the adjoint operator or vector and in addition the order of the elements must be reversed. For example + (c ϕ ) = c∗ ϕ , (5.19) | i h | + ϕ ψ = ϕ ψ ∗ = ψ ϕ . (5.20) h | i h | i h | i This equation demands that a scalar product of a vector with itself is always real. Here, we even demand that it is positive ϕ ϕ 0,real. (5.21) h | i > 244 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES The equal sign in Eq.(5.21) is only fulfilled for the null vector, which is defined by ϕ +0= ϕ . (5.22) | i | i Note, we denote the null vector not with the symbol 0 but rather with the scalar 0.Becauset hes ymbol 0 is reserved for the grou| ind state of a system. If the scalar product of a |veci tor with itself is always positive, Eq.(5.21), then one can derive from the scalar product the norm of a vector according to ϕ = ϕ ϕ . (5.23) k k h | i For vectors that are orthogonal to eacph other we have ϕ ψ =0 (5.24) h | i without having one of them be the null vector. 5.1.2 Vector Bases The dimensions of a Hilbert space are countable, i.e. each dimension can be assigned a whole number and thereby all dimensions are referenced in a unique way with 1, 2, 3, .... A vector space that is a Hilbert space has the following additional properties. Completeness: If there is a sequence of vectors in a Hilbert space ϕ1 , ϕ2 , ϕ3 , ϕ4 , ... that fulfills Cauchy’s convergence criterion then the |limiit| veictor| iϕ | is ialso an element of the Hilbert space. Cauchy’s convergence criterion sta| ites that if ϕn ϕm <ε, for some n, m > N(ε) the sequence converges uniformly [2].k − k Separability: The Hilbert space is separable. This indicates that for every element ϕ in the Hilbert space there is a sequence with ϕ as the limit vector. | i Every vector in the Hilbert space can be| diecomposed into linear indepen­ dent basis vectors ψ . Thenumberofbasis vectorscanb e infinite | ni ϕ = cn ψn . (5.25) | i n | i X 5.2. LINEAR OPERATORS IN HILBERT SPACES 245 The components cn of the vector ϕ with respect to the basis ψn are com­ plex numbers denoted with index| ni. It is advantageous to orth| oinormalize the basis vectors ψ ψ = δnm . (5.26) h n| mi The components of the vector ϕ can then be determined easily by | i ψm ϕ = cn ψm ψn = cnδmn, (5.27) h | i n h | i n or X X cm = ψ ϕ , (5.28) h m| i This leads to ϕ = ψn ψn ϕ . (5.29) | i n | ih | i X 5.2 Linear Operators in Hilbert Spaces 5.2.1 Properties of Linear Operators An operator L is defined as a mapping of a vector ϕ onto another vector ψ of the Hilbert space | i | i L ϕ = ψ . (5.30) | i | i Alinear operator L has the property that it maps a linear combination of input vectors to the linear combination of the correponding maps L (a ϕ + b ϕ )=(a L ϕ + b L ϕ ) | 1i | 2i | 1i | 2i = a ψ1 + b ψ2 , for a, b C. (5.31) | i | i ∈ The sum of two linear operators is defined as (L + M) ϕ = L ϕ +M ϕ . (5.32) | i | i | i And the product of two operators is defined as LM ϕ = L (M ϕ ) . (5.33) | i | i The null element and 1-element of the operators is denoted as 0,and1 . Often we will not bold face these operators, especially in products, where a 246 CHAPTER 5. THE DIRAC FORMALISM AND HILBERT SPACES scalar has the same meaning. The two operators are defined by their action on arbitrary vectors of the Hilbert space 0 ϕ =0, ϕ , (5.34) | i ∀ | i 1 ϕ = ϕ , ϕ (5.35) | i | i ∀ | i In generally, the multiplication of two operators is not commutative LM ϕ = ML ϕ , ϕ , (5.36) | i 6 | i ∀ | i or in short LM = ML. (5.37) 6 The expression [L, M]=LM ML (5.38) − is therefore called the commutator between L and M. If [L, M]=0 , the operators commute. The following rules for commutators apply: [L, M]= [M, L] , (5.39) − [L, L]=0 , (5.40) [L, 1]=0 , (5.41) 1 L, L− = 0 , (5.42) [L,a£M]=a¤ [L, M] , (5.43) [L1 + L2, M]=[L1, M]+[L2, M] , (5.44) [L, M]= [M, L] (5.45) − [L1L2, M]=[L1, M] L2 + L1 [L2, M] , (5.46) [M, L1L2]=[M, L1] L2 + L1 [M, L2] . (5.47) Often the anticommutator is also used. It is defined as [L, M]+ = LM + ML. (5.48) If [L, M]+ = 0, the operators are called anti-commuting. 5.2. LINEAR OPERATORS IN HILBERT SPACES 247 5.2.2 The Dyadic Product Two vectors in the Hilbert space can not only be used to build a scalar product but rather what is called a dyadic product, which is an operator α β . (5.49) | ih | The dyadic product is the formal product between a ket- and a bra-vector. If applied to a vector, it projects the vector onto the state β and generates a new vector in parallel to α with a magnitude equal to t|hei projection | i α β ψ = β ψ α . (5.50) | ih | i h | i| i As we have seen from Eq.(5.29), if ψn built a complete orthonormal basis, then | i ϕ = ψn ψn ϕ , ϕ ,. (5.51) | i n | ih | i ∀ | i X Eq.(5.35) implies 1 = ψn ψn . (5.52) n | ih | X When applied to an operator from the left and right side 1L1 = ψm ψm L ψn ψn (5.53) m | ih | Ã n | ih |! X X = Lmn ψm ψn m n | ih | X X with the matrix elements Lmn = ψ L ψ .
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