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  • Lecture Notes PH 411/511 ECE 598 A. La Rosa

    INTRODUCTION TO QUANTUM MECHANICS ______________________________________________________________________

    CHAPTER-10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS 10.1 Representations in the spatial and momentum spaces

    10.1.A Representation of the wavefunctionin the space- coordinates basis

    { position x} 10.2.A1 The Delta Dirac 10.2.A2 Compatibility between the physical concept of amplitude probability and

    the notation used for the inner product.

    10.1.B Representation of the wavefunction in the momentum coordinates

    basis { momentum p }

    10.2.B1 Representation of the momentum pstate in space-coordinates basis {

    position x }

    10.2.B2 Identifyingthe amplitude probabilitymomentum pas the Fourier

    transform of the function (x) 10.1.C Tensor Product of State Spaces

    10.2 The Schrödinger equation as a postulate

    10.2.A The Hamiltonian equations expressed in the continuum spatial coordinates. The Schrodinger Equation.

    10.2.B Interpretation of the wavefunction Einstein’s view on the granularity nature of the electromagnetic radiation. Max Born’s probabilistic interpretation of the wavefunction. Deterministic evolution of the wavefunction Ensemble

    10.2.C Normalization condition of the wavefunction Hilbert space Conservation of probability

    10.2.D The Philosophy of Quantum Theory

    10.3 Expectation values

    10.3.A Expectation value of a particle’s position 10.3.B Expectation value of the particle’s momentum 10.3.C Expectation (average) values are calculated in an ensemble of identically prepared

    systems

    10.4 Operators associated to observables

    10.4.A Observables, eigenvalues and eigenstates

    10.4.B Definition of the quantum mechanics operator F~ to be associated with the observable physical quantity f

  • 10.4.C Definition of the Position Operator X ~

    10.4.D Definition of the Linear Momentum Operator P ~

    10.4.D.1 Representation of the linear momentum operator in the

    momentum basis { momentum p } 10.4.D.2 Representation of the linear momentum operator in the spatial

    coordinates basis { position x }

    10.4.D3 Construction of the operators P ~

    , 2 ~ P , 3

    ~ P , …

    10.4.E The Hamiltonian operator 10.4.E.1 Evaluation of the mean energy in terms of the Hamiltonian operator 10.4.E.2 Representation of the Hamiltonian operator in the spatial coordinate

    basis

    10.5 Properties of Operators

    10.5.A Correspondence between bras and kets 10.5.B Adjoint operators 10.5.C Hermitian or self-adjoint operators

    Properties of Hermitian (or self-adjoint) operators: - Operators associated to mean values are Hermitian (or self-adjoint) - Eigenvalues are real - Eigenvectors with different eigenvalues are orthogonal

    10.5.D Observable Operators 10.5.E Operators that are not associated to mean values

    10.6 The commutator

    10.6.A The Heisenberg uncertainty relation 10.6.B Conjugate observables

    Standard deviation of two conjugate observables 10.6.C Properties of observable operators that do commute 10.6.D How to uniquely identify a basis of eigenfunctions?

    Complete set of commuting observables

    10.7 Simultaneous measurement of observables

    10.7.A Definition of compatible (or simultaneously measurable) operators

    10.7.B Condition for observables A ~

    and B ~

    to be compatible

    10.8 How to prepare the initial quantum states

    10.8.A Knowing what can we predict about eventual outcomes from measurement?

    10.8.B After a measurement, what can we say about the state ? References:

    Feynman Lectures Vol. III; Chapter 16, 20 Claude Cohen-Tannoudji, B. Diu, F. Laloe, “Quantum Mechanics”, Wiley. "Introduction to Quantum Mechanics" by David Griffiths; Chapter 3. B. H. Bransden & C. J. Joachin, “Quantum Mechanics”, Prentice Hall, 2nd Ed. 2000.

  • CHAPTER-10

    WAVEFUNCTIONS, OBSERVABLES and OPERATORS Quantum theory is based on two mathematical items: wavefunctions and operators.

     The state of a system is represented by a wavefunction.

    An exact knowledge of the wavefunction is the maximum information one can have of the system: all possible information about the system can be calculated from this wavefunction.

     Quantities such as position, momentum, or energy, which one measures experimentally, are called observables.

    In classical physics, observables are represented by ordinary variables (x, p, E).

    In quantum mechanics, observables are represented by operators ( X ~

    , P ~

    , 𝐻 );

    An operator�̃� acting on a wavefunction  gives another wavefunction  �̃� =.

    In bracket notation:�̃� = .

    Notice also that by definition �̃� =�̃� 

    In this chapter we address three main subjects:

     Description of the spatial-coordinates basis and the momentum-coordinates basis. (A base may be more convenient than another base to represent the same quantum

    state.)

     How to build the quantum mechanics operator corresponding to a given observable.

     How to build a quantum state from a given set of experimental results. The key concept used here is the complete set of commuting operators.

    10.1 Representation of the wavefunctions in the spatial and momentum spaces

    An arbitrary state can be expanded in terms of base states that conveniently fit the particular problem under study. For general descriptions, two bases are frequently used: the spatial coordinate basis and the linear momentum basis. These two basis are addressed in this section.

    10.1.A Representation of the wavefunction in the spatial coordinates basis

    { x , x } Chapter 9 helped to provide some clues on the proper interpretation of the wavefunction

    (the solutions of the Schrodinger equation.) This came through the analysis of the particular case of an electron moving across a discrete lattice:

  • the wavefunction is pictured as a wave of amplitude probabilities (the latter are complex numbers whose magnitude is interpreted as probabilities).

    Notice, however, that when taking the limiting case of the lattice spacing tending to zero, one ends up with a situation in which the electron is propagating through a continuum line space. Thus, this limiting case takes us to the study of a particle moving in a continuum space.

    In analogy to the discrete lattice, where the location of the atoms guided the selection of the

    state basis { n }, in the continuum space we consider the following continuum set,

    { x , x } Continuum spatial-coordinates base (1)

     xstands for a state in which a particle is located around the coordinate x.

    ________________________________________

    x

    For every value x along the line one conceives a corresponding state. If one includes all the points on the line, a complete basis set results as indicated in (1). Such a base will be used to describe a general quantum state.

    A given state specifies the particular way in which the amplitude probability of a particle is distributed along a line.

    One way of specifying this state  is by specifying all the amplitude probabilities that the particle

    will be found at each base state x;

    We write each of these amplitudes as x. We must give then an infinite set of amplitudes, one for each value of x. Thus,

     =  

    -

    x xdx Representation of the wavefunction  (2)

     in the spatial coordinates basis

    x Amplitude probability that the particle

    initially in the state be found (immediately after a position measurement)

    at the state x.

    Sometimes the following notation is preferred:

     x (x) (3) Summary,

  • =  

    

    x xdx =  

    

    x (x) dx

    Number

    Representation of the wave-function in the spatial coordinates basis

    (4)

    Number

    Note: Multiplying the expression (4) by a particular bra x ’, we should obtain,

    x ’= ( x’) (This result will be justified a bit more rigorously in the next section below;

    see the section on “the delta Dirac”).

    Caution: xdoes not mean  [x]*