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This chapter delves into Hilbert space fundamentals and its relation to wave function space. It explores concepts like scalar products, orthonormal bases, and Dirac notation in quantum physics. Discover how state vectors and state space play a crucial role in characterizing systems in quantum mechanics.
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Chapter 3 Formalism
3.1 Hilbert space • Let’s recall for Cartesian 3D space: • A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors (basis) • The basis spans the vector space • Inner (dot, scalar) product of 2 vectors is defined as: • Length (norm) of a vector
3.1 Hilbert space
3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • Its elements are functions (vectors of Hilbert space) • The space is linear: if φ and ψ belong to the space then φ + ψ, as well as aφ (a – constant) also belong to the space
3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • Inner (dot, scalar) product of 2 vectors is defined as: • Length (norm) of a vector is related to the inner product as:
3.1 David Hilbert (1862 – 1943) Hilbert space • Hilbert space: • The space is complete, i.e. it contains all its limit points (we will see later) • Example of a Hilbert space: L 2, set of square-integrable functions defined on the whole interval
3.1 Wave function space • Recall: • Thus we should retain only such functions Ψthat are well-defined everywhere, continuous, and infinitely differentiable • Let us call such set of functions F • F is a subspace of L 2 • For two complex numbers λ1 and λ2 it can be shown that if
3.1 Scalar product • In F the scalar product is defined as: • Properties of the scalar product: • φ and ψ are orthogonal if • Norm is defined as
3.1 Scalar product • Schwarz inequality Karl Hermann Amandus Schwarz (1843 – 1921)
Orthonormal bases • A countable set of functions • is called orthonormal if: • It constitutes a basis if every function in F can be expanded in one and only one way: • Recall for 3D vectors:
Orthonormal bases • For two functions • a scalar product is: • Recall for 3D vectors:
Orthonormal bases • This means that • Closure relation
Orthonormal bases • A set of functions labelled by a continuous index α • is called orthonormal if: • It constitutes a basis if every function in F can be expanded in one and only one way:
Orthonormal bases • For two functions • a scalar product is:
Orthonormal bases • This means that • Closure relation
Example of an orthonormal basis • Let us consider a set of functions: • The set is orthonormal: • Functions in F can be expanded:
Example of an orthonormal basis • For two functions • a scalar product is:
Example of an orthonormal basis • This means that • Closure relation
State vectors and state space • The same function ψ can be represented by a multiplicity of different sets of components, corresponding to the choice of a basis • These sets characterize the state of the system as well as the wave function itself • Moreover, the ψfunction appears on the same footing as other sets of components
State vectors and state space • Each state of the system is thus characterized by a state vector, belonging to state space of the system Er • As F is a subspace of L 2, Er is a subspace of the Hilbert space
3.6 Dirac notation • Bracket = “bra” x “ket” • < > = < | > = “< |” x “| >” Paul Adrien Maurice Dirac (1902 – 1984)
3.6 Dirac notation • We will be working in the Er space • Any vector element of this space we will call a ket vector • Notation: • We associate kets with wave functions: • F and Er are isomporphic • r is an index labelling components Paul Adrien Maurice Dirac (1902 – 1984)
3.6 Dirac notation • With each pair of kets we associate their scalar product – a complex number • We define a linear functional (not the same as a linear operator!) on kets as a linear operation associating a complex number with a ket: • Such functionals form a vector space • We will call it a dual space Er* Paul Adrien Maurice Dirac (1902 – 1984)
3.6 Dirac notation • Any element of the dual space we will call a bra vector • Ket | φ > enables us to define a linear functional that associates (linearly) with each ket | ψ > a complex number equal to the scalar product: • For every ket in Er there is a bra in Er* Paul Adrien Maurice Dirac (1902 – 1984)
3.6 Dirac notation • Some properties: Paul Adrien Maurice Dirac (1902 – 1984)
Linear operators • Linear operatorA is defined as: • Product of operators: • In general: • Commutator: • Matrix element of operator A:
Linear operators • Example: • What is ? • It is an operator – it converts one ket into another
Linear operators • Example: • Let us assume that • Projector operator • It projects one ket onto another
Linear operators • Example: • Let us assume that • These kets span space Eq, a subspace of E • Subspace projector operator • It projects a ket onto a subspace of kets
Linear operators • Recall matrix element of a linear operator A: • Since a scalar product depends linearly on the ket, the matrix element depends linearly on the ket • Thus for a given bra and a given operator we can associate a number that will depend linearly on the ket • So there is a new linear functional on the kets in space E,i.e., a bra in space of E *, which we will denote • Therefore
Linear operators • Operator A associates with a given bra a new bra • Let’s show that this correspondence is linear
Linear operators • For each ket there is a bra associated with it • Hermitianconjugate (adjoint) operator: • This operator is linear (can be shown) Charles Hermite (1822 – 1901)
Linear operators • Some properties: Charles Hermite (1822 – 1901)
Hermitian conjugation • To obtain Hermitian conjugation of an expression: • Replace constants with their complex conjugates • Replace operators with their Hermitian conjugates • Replace kets with bras • Replace bras with kets • Reverse order of factors Charles Hermite (1822 – 1901)
3.2 Hermitian operators • For a Hermitian operator: • Hermitian operators play a fundamental role in quantum mechanics (we’ll see later) • E.g., projector operator is Hermitian: • If: Charles Hermite (1822 – 1901)
Representations in state space • In a certain basis, vectors and operators are represented by numbers (components and matrix elements) • Thus vector calculus becomes matrix calculus • A choice of a specific representation is dictated by the simplicity of calculations • We will rewrite expressions obtained above for orthonormal bases using Dirac notation
Orthonormal bases • A countable set of kets • is called orthonormal if: • It constitutes a basis if every vector in E can be expanded in one and only one way:
Orthonormal bases • Closure relation • 1 – identity operator
Orthonormal bases • For two kets • a scalar product is:
Orthonormal bases • A set of kets labelled by a continuous index α • is called orthonormal if: • It constitutes a basis if every vector in E can be expanded in one and only one way:
Orthonormal bases • Closure relation • 1 – identity operator
Orthonormal bases • For two kets • a scalar product is:
Representation of kets and bras • In a certain basis, a ket is represented by its components • These components could be arranged as a column-vector:
Representation of kets and bras • In a certain basis, a bra is also represented by its components • These components could be arranged as a row-vector:
Representation of operators • In a certain basis, an operator is represented by matrix components:
Representation of operators • For Hermitian operators: • Diagonal elements of Hermitian operators are always real