QUANTUM MECHANICS - Chapter 3 Operator methods in quantum...

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Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be represented through a wave-like description. For example, the electron spin degree of freedom does not translate to the action of a gradient operator. It is therefore useful to reformulate quantum mechanics in a framework that involves only operators. Before discussing properties of operators, it is helpful to introduce a further simplification of notation. One advantage of the operator algebra is that it does not rely upon a particular basis. For example, when one writes , where the hat denotes an operator, we can equally represent the momentum operator in the spatial coordinate basis, when it is described by the differential operator, p ˆ = p . Similarly, p ˆ = − i it! would x , or inbetheusefulmomentumto work withbasis,a basiswhenforit isthejustwavefunctiona number which is coordinate independent. Such a representation was developed by Dirac early in the formulation of quantum mechanics. In the parlons of mathematics, square integrable functions (such as wavefunctions) are said form a vector space, much like the familiar three-dimensional vector spaces. In the Dirac notation , a state vector or wavefunction, ψ , is represented as a “ket”, | ψ ". Just as we can express any three-dimensional vector in terms of the basis vectors, r = x e ˆ1 + y e ˆ2 + z e ˆ3, so we can expand any wavefunction as a superposition of basis state vectors, | ψ " = λ 1 | ψ 1 " + λ 2 | ψ 2 " + ··· . Alongside the ket, we can define the “bra”, # ψ |. Together, the bra and ket define the scalar product , from which follows the identity, # φ | ψ " = #is ψ | φ recovered". In thisfromformulation,the innertheprod-real space representation of the wavefunction Advanced Quantum Physics
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3.1. OPERATORS 20 ψ uct ψ ( x ) = #. x |As ψ " withwhileathethree-dimensionalmomentum spacevectorwavefunctionspace whereis obtained a ·the b ≤vectors,| a from|| b |, ( p ) = # p | ψ " the magnitude of the scalar product is limited by the magnitude of # ψ | φ " ≤ "# ψ | ψ "# φ | φ " , a relation known as the Schwartz inequality . 3.1 Operators intoAn operatoranother, A φ "is, i.e.a “mathematical A ˆ| ψ " = | φ ". Ifobject” that maps one state vector, | ψ ", A ˆ| ψ " = a | ψ " , . anwitheigenvalueeigenstate a real, a thenForof |the ψ example," is momentum said tothebeplanean operatoreigenstate wave state, p ˆ(=or ψ p eigenfunction −( x i )! = x of ,# x with|the ψ p "Hamiltonian,eigenvalue=) Ae of ipx/A ˆ with! p is. For a free particle, the plane wave is also an eigenstate with eigenvalue . In quantum mechanics, for any observable A , there is an operator A ˆ which theacts on the wavefunctionvalue of soisthat, if a system is in a state described by | ψ ", expectation A (3.1) Every operator corresponding to an observable is both linear and Hermitian: That is, for any two wavefunctions | ψ " and | φ ", and any two complex numbers α and β , linearity implies that A ˆ( α | ψ " + β | φ ") = α ( A ˆ| ψ ") + β ( A ˆ| φ ") .
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