Dirac Notation.pdf - Chapter 4 The Hilbert Space of States...

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Chapter 4 The Hilbert Space of States and the Dirac Notation The single defining relation for a wave packet is its “square integrability”, i.e., its finite energy. The space of complex functions with finite energy denoted by L 2 ( R 3 ) in three dimensions, and by L 2 ( R ) in one dimension, 1 , is the home of all quantum mechanical wave functions; it is known as a Hilbert space denoted by H and it is technically defined as a complete inner-product space. We shall now describe the Hilbert space of states H , in one spatial dimension (generalization to three spatial dimensions is straightforward), whose defining property is their finite energy, namely, Z -∞ | ψ ( x ) | 2 dx < . 4.1 Inner-product and Normed Vector Spaces The most basic algebraic structure on H is the vector space property that we summarize in the following form 2 : if ψ 1 , ψ 2 H , then so is the arbitrary 1 R is the real line, i.e., { x : -∞ < x < + ∞} , and R 3 is the three dimen- sional Euclidean space, i.e., { ( x, y, z ) : -∞ < x, y, z < + ∞} . The complex plane { x + iy : -∞ < x, y < + ∞} is denoted by C . 2 For the more mathematically inclined here is the full definition of a vector space V , whose members are denoted by v , defined over a field F with a binary operation called addition, and a scalar multiplication by the members of the field satisfying the following axioms: to every pair v 1 and v 2 of vectors corresponds a vector v - 1 + v 2 V that is called their sum. Addition of vectors is commutative and associative. There is a unique vector 0 such that v + 0 = v for all v V . For every vector v V there is a unique vector - v such that v + ( - x ) = 0 . For every vector v V and every scalar α F there corresponds a vector α v V
46 Chapter 4. The Hilbert Space of States and the Dirac Notation linear combination c 1 ψ 1 + c 2 ψ 2 for complex constants c 1 , c 2 C . To define convergence in a vector space we must introduce a norm. A normed vector space is equipped with a non-negative real valued function known as the norm such that k ψ k ≥ 0 ψ H k ψ k = 0 ψ = 0 k k = | c | k ψ k k ψ 1 + ψ 2 k ≤ k ψ 1 k + k ψ 2 k , ψ 1 , ψ 2 H An inner product can be used to project vectors (functions) in arbitrary directions, in addition to introducing the concept of orthogonality between vectors (functions). For any two functions ψ 1 ( x ) and ψ 2 ( x ) (we are special- izing to one spatial dimension for brevity) in H we define the inner product h ψ 1 , ψ 2 i ≡ Z -∞ ψ * 1 ( x ) ψ 2 ( x ) dx. The inner product can be defined more abstractly as a function that maps any two members ψ 1 and ψ 2 of H to a complex number denoted by h ψ 1 , ψ 2 i such that: h ψ 1 , ψ 2 i = h ψ 2 , ψ 1 i * h ψ 1 , cψ 2 i = c h ψ 1 , ψ 2 i h ψ, ψ i is non-negative and is 0 if and only if ψ = 0.
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