DOCSLIB.ORG

PY 451 Notes — April 12-19, 2018 Mcintyre Chap. 7 Notes — Spectrum of Angular Momentum Professor Kenneth Lane1 A. Preliminar

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PY 451 Notes — April 12-19, 2018 Mcintyre Chap. 7 Notes — Spectrum of Angular Momentum Professor Kenneth Lane1 A. Preliminar PY 451 Notes | April 12-19, 2018 McIntyre Chap. 7 notes | Spectrum of Angular Momentum Professor Kenneth Lane1 A. Preliminaries 1.) In the following I will use J = (Jx;Jy;Jz) = (J1;J2;J3) for a generic angular momentum, L = r × p for the orbital angular momentum of a particle whose position (operator) is r and momentum is p, and S for the spin operator of the particle. (For a massive particle, its spin is defined as its total angular momentum when it is at rest. The spin is therefore an intrinsic fundamental property of the particle, like its mass, electric charge, and other quantities which are not effected by the particle's motion. Defining the spin for a massless particle like the photon is a little tricky, so we won't be concerned with that here.) For a particle in motion, its total angular momentum (operator) is J = L + S. 2.) As for spin, angular momentum is a hermitian operator and its commutation relations with itself are [Jx;Jy] = i~Jz plus cyclic permutations. They are written compactly using the summation convention of summing over repeated indices: 3 X [Ja;Jb] = i~ abcJc ≡ i~ abcJc; (a; b; c = 1; 2; 3): (1) c=1 Here, abc is the completely antisymmetric Levi-Civita tensor introduced in the Chapter 2 notes. 3.) It is convenient to define ladder operators for angular momentum, y J± = Jx ± iJy = J1 ± iJ2 = J∓: (2) These have the commutation relations (from Eq. (1)) [Jz;J±] = ±~J± ; [J+;J−] = 2~Jz : (3) 4.) The square of the angular momentum operator is 2 2 2 2 1 2 J = JaJa ≡ Jx + Jy + Jz = 2 (J+J− + J−J+) + Jz : (4) This is also hermitian and its possible eigenvalues must be non-negative (≥ 0). It is easy to see 2 1 that [Ja; J ] = 0 for a = 1; 2; 3 (just as it was for spin- 2 ). Summing on repeated indices again and using [A; BC] = B[A; C] + [A; B]C, we have (by the antisymmetry of abc = −acb) 2 [Ja; J ] = [Ja;JbJb] = [Ja;Jb]Jb + Jb[Ja;Jb] = i~ abc(JcJb + JbJc) = 0: (5) 2 Thus, one component of the angular momentum, usually taken to be Jz, and J can be simultaneously 2 diagonalized. Considered by themselves, with no other operators, (Jz; J ) constitute a complete set of commuting observables (CSCO); see the Chapter 2 notes again. 1 c Kenneth Lane, 2018. 1 B. The Spectrum of Angular Momentum 2 1.) We denote by jjmi the simultaneous eigenstates of Jz and J , where j and m are real, dimen- sionless numbers we determine below. They label the total angular momentum and its z-component, respectively. The eigenvalue equations are Jzjjmi = m~jjmi; (6) 2 2 J jjmi = j(j + 1) ~ jjmi: (7) Since J 2 is a positive semi-definite operator, j(j +1) ≥ 0. For simplicity in this derivation, I assume 2 that j ≥ 0. For convenience, I set ~ = 1 in this derivation. As usual, the eigenstates of Jz and J form a complete orthonormal set in the space in which these two operators act: 0 0 hjmjj m i = δjj0 δmm0 ; (8) X jjmihjmj = 1 ; (9) j;m where the sum is over all possible values of j and m. 2.) For fixed (j; m), consider the states obtained by applying the ladder operators J± to jjmi. We 2 act on these states with Jz and J to find their m and j-values. Using the commutation relations in Eqs. (3, 5) (fill in the missing steps yourself!) JzJ±jjmi = (J±Jz + [Jz;J±]) jjmi = (m ± 1)J±jjmi ; (10) 2 J J±jjmi = j(j + 1)J±jjmi: (11) Thus, J±jjmi is a state with total angular momentum j and z-component m + 1: ± J± = cjmjj; m + 1i: (12) Hence the name \ladder operators" for J±. ± 3.) Now determine cjm: ± 2 2 2 jcjmj = hjmjJ∓J±jjmi = hjmjJx + Jy ± i[Jx;Jy]jjmi 2 2 = hjmjJ − Jz ∓ Jzjjmi = j(j + 1) − m(m ± 1): (13) ± 2 Since the squared norm of J±jjmi must be non-negative, jcjmj ≥ 0, we must have (for j ≥ 0): j(j + 1) − m(m + 1) ≥ 0 =) m ≤ j; (14) j(j + 1) − m(m − 1) ≥ 0 =) m ≥ −j: (15) (16) Choosing the positive square root (a simple choice of overall phase), we have ± p p cjm = j(j + 1) − m(m ± 1) = (j ∓ m)(j ± m ± 1): (17) This is a useful and important equation. You should try to memorize it. 4.) Finally: For fixed j ≥ 0, each application of the ladder operator J± to the state jjmi raises or lowers m by one unit until m = j (for J+) or m = −j (for J−). The next application of J+ or J− 2 gives zero: J+jjji = J−jj; −ji = 0. Thus, there are 2j + 1 values of m = −j; −j + 1; : : : ; j − 1; j. But, then 2j + 1 must be a non-negative integer!. (Had we assumed j ≤ −1 so that j(j + 1) is still ≥ 0, this shows that we would have obtained a contradiction.) We have now found the eigenvalue 2 2 spectrum of the Jz and J (or of J and any other component of J): Jzjjmi = m~jjmi; (18) 2 2 J jjmi = j(j + 1) ~ jjmi; (19) 1 3 j = 0; 2 ; 1; 2 ;:::; (20) m = −j; −j + 1; : : : ; j + 1; j: (21) More usually, one writes −j ≤ m ≤ j. A question: Can you prove that there are no states between jjmi and j; m ± 1i? Recall the similar question about the states of the harmonic oscillator. 5.) I hope you have enjoyed this derivation of the spectrum of angular momentum. With the ladder operators, it is somewhat similar to determining the spectrum of the harmonic oscillator and, indeed, many other diagonalization problems (including the hydrogen atom!). So, now we 1 know why the spin operators studied in Chapters 1-3 can describe spin- 2 , 1, etc. Everything is determined by the commutation relations of the operators and the fact that they are hermitian! As you know, the spectrum of orbital angular momentum contains only the integers: l = 0; 1; 2;::: with −l ≤ ml ≤ l. I will explain the reason for that when we discuss orbital angular momentum | which is next. C. Orbital Angular Momentum As discussed in some detail in the discussion section on April 11, the orbital angular momentum operators L = r × p in the r representation are @ @ @ @ @ @ L = −i r × r = −i y − z ; z − x ; x − y ; (22) ~ ~ @z @y @x @z @y @x and they are easily (more easily than in McIntyre!) seen to satisfy the commutation relations of angular momentum: [La;Lb] = i~ abcLc: (23) For spherically symmetric potentials V (r) | which we shall focus on almost exclusively for the rest of the semester | the entire Hamiltonian is spherically symmetric (invariant under rotations), and so, as we have discussed many times now, [H; L] = 0; (24) or, what is the same thing (as we have also discussed repeatedly), orbital angular momentum is 2 conserved: dhL(t)i=dt = 0. Thus, we can simultaneously diagonalize H, Lz and L , and the eigenstates of H, states with a particular energy eigenvalue En` and orbital angular momentum ` will be (2` + 1)-fold degenerate. These states are labeled jn`mi, and Hjn`mi = En` jn`mi; 2 2 L jn`mi = `(` + 1)~ jn`mi (25) Lzjn`mi = m~ jn`mi − ` ≤ m ≤ `: 3 The reason for labeling the energy with both a \principal" quantum number n and the orbital angular momentum ` will become clear presently. (Note: We are ignoring the possibility of spin here. In atoms like the hydrogen atom, spin-effects are small and can be ignored to a good approximation.) Now, it turns out that the only allowed values of ` for orbital angular momentum are ` = 0; 1; 2;::: . 2 Why is that? The usual answer is that the wave function (here the eigenfunction of Lz and L imφ must single-valued, i.e. that the eigenfunction e of Lz = −i~ @=@φ must have the same value at φ = 0 and φ = 2π; one complete revolution by 2π around the z-axis should give us back the same wave function. But, since the overall phase of the wave function has no physical meaning, that 1 3 argument cannot be right! Furthermore, it is not correct for spin 2 , 2 , etc. The real argument is more complicated; I'll merely state the result below. The components of L and L2 in the (θ; φ) representation are: @ @ L = i sin φ + cos φ cot θ ; x ~ @θ @φ @ @ L = i − cos φ + sin φ cot θ ; y ~ @θ @φ @ L = −i ; (26) z ~@φ 1 @ @ 1 @2 L2 = − 2 sin θ + : (27) ~ sin θ @θ @θ sin2 θ @φ2 This expression for L2 should look familiar. It appeared in the energy eigenvalue equation for the Hamiltonian in spherical coordinates on page 7-5 of the notes labeled \Chapter-7-prelims". Using it here with our notation in Eqs. (25), that equation is 2 1 @ @ L2 − ~ r2 + + V (r) (r; θ; φ) = E (r; θ; φ) ; (28) 2µ r2 @r @r 2µr2 n`m n` n`m where n`m(r; θ; φ) = hrjn`mi ≡ hr; θ; φjn`mi. Note the separation of the kinetic energy (derivative) terms into a purely radial piece and a purely orbital piece. This is very important, because the 2 orbital term will now become part of the central potential. Since L jn`mi = `(` + 1)~2jn`mi, Eq. (28) becomes | in the spherical coordinate representation of operators and states: 2 1 @ @ `(` + 1) 2 − ~ r2 + ~ + V (r) (r; θ; φ) = E (r; θ; φ) : (29) 2µ r2 @r @r 2µr2 n`m n` n`m Thus the orbital kinetic energy has become part of an effective central potential called the \cen- trifugal barrier": `(` + 1) 2 V (r) = V (r) + ~ : (30) eff: 2µr2 As we shall see, for ` 6= 0, it is a repulsive potential keeping the particle with reduced mass µ away from r = 0.
Load more

Recommended publications
  • Rotation and Spin and Position Operators in Relativistic Gravity and Quantum Electrodynamics
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 December 2019 doi:10.20944/preprints201912.0044.v1 Peer-reviewed version available at Universe 2020, 6, 24; doi:10.3390/universe6020024 1 Review 2 Rotation and Spin and Position Operators in 3 Relativistic Gravity and Quantum Electrodynamics 4 R.F. O’Connell 5 Department of Physics and Astronomy, Louisiana State University; Baton Rouge, LA 70803-4001, USA 6 Correspondence: [email protected]; 225-578-6848 7 Abstract: First, we examine how spin is treated in special relativity and the necessity of introducing 8 spin supplementary conditions (SSC) and how they are related to the choice of a center-of-mass of a 9 spinning particle. Next, we discuss quantum electrodynamics and the Foldy-Wouthuysen 10 transformation which we note is a position operator identical to the Pryce-Newton-Wigner position 11 operator. The classical version of the operators are shown to be essential for the treatment of 12 classical relativistic particles in general relativity, of special interest being the case of binary 13 systems (black holes/neutron stars) which emit gravitational radiation. 14 Keywords: rotation; spin; position operators 15 16 I.Introduction 17 Rotation effects in relativistic systems involve many new concepts not needed in 18 non-relativistic classical physics. Some of these are quantum mechanical (where the emphasis is on 19 “spin”). Thus, our emphasis will be on quantum electrodynamics (QED) and both special and 20 general relativity. Also, as in [1] , we often use “spin” in the generic sense of meaning “internal” spin 21 in the case of an elementary particle and “rotation” in the case of an elementary particle.
    [Show full text]
  • Chapter 5 ANGULAR MOMENTUM and ROTATIONS
    Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another.
    [Show full text]
  • Hamilton Equations, Commutator, and Energy Conservation †
    quantum reports Article Hamilton Equations, Commutator, and Energy Conservation † Weng Cho Chew 1,* , Aiyin Y. Liu 2 , Carlos Salazar-Lazaro 3 , Dong-Yeop Na 1 and Wei E. I. Sha 4 1 College of Engineering, Purdue University, West Lafayette, IN 47907, USA; [email protected] 2 College of Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 3 Physics Department, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA; [email protected] 4 College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] * Correspondence: [email protected] † Based on the talk presented at the 40th Progress In Electromagnetics Research Symposium (PIERS, Toyama, Japan, 1–4 August 2018). Received: 12 September 2019; Accepted: 3 December 2019; Published: 9 December 2019 Abstract: We show that the classical Hamilton equations of motion can be derived from the energy conservation condition. A similar argument is shown to carry to the quantum formulation of Hamiltonian dynamics. Hence, showing a striking similarity between the quantum formulation and the classical formulation. Furthermore, it is shown that the fundamental commutator can be derived from the Heisenberg equations of motion and the quantum Hamilton equations of motion. Also, that the Heisenberg equations of motion can be derived from the Schrödinger equation for the quantum state, which is the fundamental postulate. These results are shown to have important bearing for deriving the quantum Maxwell’s equations. Keywords: quantum mechanics; commutator relations; Heisenberg picture 1. Introduction In quantum theory, a classical observable, which is modeled by a real scalar variable, is replaced by a quantum operator, which is analogous to an infinite-dimensional matrix operator.
    [Show full text]
  • The Position Operator Revisited Annales De L’I
    ANNALES DE L’I. H. P., SECTION A H. BACRY The position operator revisited Annales de l’I. H. P., section A, tome 49, no 2 (1988), p. 245-255 <http://www.numdam.org/item?id=AIHPA_1988__49_2_245_0> © Gauthier-Villars, 1988, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ .
    [Show full text]
  • 22.51 Course Notes, Chapter 9: Harmonic Oscillator
    9. Harmonic Oscillator 9.1 Harmonic Oscillator 9.1.1 Classical harmonic oscillator and h.o. model 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schr¨odinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons 9.4.1 Harmonic oscillator model for a crystal 9.4.2 Phonons as normal modes of the lattice vibration 9.4.3 Thermal energy density and Specific Heat 9.1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h.o.). The quantum h.o. is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is ∆E. The number of levels is infinite, but there must exist a minimum energy, since the energy must always be positive. Given this spectrum, we expect the Hamiltonian will have the form 1 n = n + ~ω n , H | i 2 | i where each level in the ladder is identified by a number n. The name of the model is due to the analogy with characteristics of classical h.o., which we will review first. 9.1.1 Classical harmonic oscillator and h.o.
    [Show full text]
  • Assignment 7: Solution
    Quantum Mechanics I / Quantum Theory I. 14 December, 2015 Assignment 7: Solution 1. In class, we derived that the ground state of the harmonic oscillator j0 is given in the position space as 1=4 m! 2 −m!x =2~ 0(x) = xj0 = e : π~ (a) This question deals with finding the state in the momentum basis. Let's start off by finding the position operatorx ^ in the momentum basis. Show that @ pjx^j = i pj : ~@p This is analogous to the expression of the momentum in the position basis, @ xjp^j = −i~ @x xj . You will use [Beck Eq. 10.56] expressing the momentum eigenstate in the position basis as well as the definition of the Fourier transform. @ (b) Sincea ^j0 = 0 and the position operator pjx^ = i~ @p , construct a differential equation to find the wavefunction in momentum space pj0 = f0(p). Solve this equation to find f0(p). Plot f0(p) in the momentum space. (c) Verify that your f0(p) = pj0 derived above is the Fourier transform of xj0 = 0(x). Answer (a) Lets find the position operator in momentum space pjx^j = pjx^1^j Z = dx pjx^jx xj Z = dx pjx x xj * x^jx = xjx : Now the normalized wavefunction of the momentum eigenstate in the position representation is 1 xjp = p eipx=~ Eq 10.56 in Beck 2π~ Z 1 −ipx= ) pjx^j = p dxe ~x xj : (1) 2π~ Due Date: December. 14, 2015, 05:00 pm 1 Quantum Mechanics I / Quantum Theory I. 14 December, 2015 Now @ −ix e−ipx=~ = e−ipx=~ @p ~ @ i e−ipx=~ = xe−ipx=~: ~ @p Inserting into Eq (1): i Z @ pjx^j = p ~ dx e−ipx=~ xj 2π~ @p Z @ 1 −ipx= = i~ p dxe ~ xj @p 2π~ @ = i pj ; ~@p where e(p) is the Fourier transform of (x).
    [Show full text]
  • Chapter 3 Mathematical Formalism of Quantum Mechanics
    Chapter 3 Mathematical Formalism of Quantum Mechanics 3.1 Hilbert Space To gain a deeper understanding of quantum mechanics, we will need a more solid math- ematical basis for our discussion. This we achieve by studying more thoroughly the structure of the space that underlies our physical objects, which as so often, is a vector space, the Hilbert space . Definition 3.1 A Hilbert space is a finite- or infinite-dimensional, linear vector space with scalar product in C. The Hilbert space provides, so to speak, the playground for our analysis. In analogy to a classical phase space, the elements of the vector space, the vectors, are our possible physical states. But the physical quantities we want to measure, the observables, are now operators acting on the vectors. As mentioned in Definition 3.1, Hilbert spaces can be finite- or infinite-dimensional, as opposed to classical phase spaces (which are always 6n-dimensional, where n is the particle number). We shall therefore investigate those two cases separately. 3.1.1 Finite–Dimensional Hilbert Spaces In finite dimensions the vectors of a Hilbert space, denoted by , and the corresponding scalar products differ from the standard Euklidean case onlyH by the choice of complex quantities C instead of real ones R. It means that for vectors x, y ∈ H x1 y1 . i i x = . , y = . , x , y , C , (3.1) ∈ xn yn 55 56 CHAPTER 3. MATHEMATICAL FORMALISM OF QUANTUM MECHANICS where represents the n-dimensional Hilbert space under consideration, the scalar prod- uct canH be written as 1 y n i .
    [Show full text]
  • An Approach to Quantum Mechanics
    An Approach to Quantum Mechanics Frank Rioux Department of Chemistry St. John=s University | College of St. Benedict The purpose of this tutorial is to introduce the basics of quantum mechanics using Dirac bracket notation while working in one dimension. Dirac notation is a succinct and powerful language for expressing quantum mechanical principles; restricting attention to one-dimensional examples reduces the possibility that mathematical complexity will stand in the way of understanding. A number of texts make extensive use of Dirac notation (1-5). In addition a summary of its main elements is provided at: http://www.users.csbsju.edu/~frioux/dirac/dirac.htm. Wave-particle duality is the essential concept of quantum mechanics. DeBroglie expressed this idea mathematically as λ = h/mv = h/p. On the left is the wave property λ, and on the right the particle property mv, momentum. The most general coordinate space wave function for a free particle with wavelength λ is the complex Euler function shown below. ⎛⎞⎛⎞⎛⎞x xx xiλπ==+exp⎜⎟⎜⎟⎜⎟ 2 cos 2 π i sin 2 π ⎝⎠⎝⎠⎝⎠λ λλ Equation 1 Feynman called this equation “the most remarkable formula in mathematics.” He referred to it as “our jewel.” And indeed it is, because when it is enriched with de Broglie’s relation it serves as the foundation of quantum mechanics. According to de Broglie=s hypothesis, a particle with a well-defined wavelength also has a well-defined momentum. Therefore, we can obtain the momentum wave function (unnormalized) of the particle in coordinate space by substituting the deBroglie relation into equation (1).
    [Show full text]
  • Rotations and Angular Momentum
    rotations.nb:11/3/04::13:48:13 1 RotationsandAngularmomentum Á Intro Thematerialheremaybefoundin SakuraiChap3:1-3,(5-6),7,(9-10) MerzbacherChap11,17. Chapter11ofMerzbacherconcentratesonorbitalangularmomentum.Sakurai,andCh17ofMerzbacherfocuson angularmomentuminrelationtothegroupofrotations.Justaslinearmomentumisrelatedtothetranslationgroup, angularmomentumoperatorsaregeneratorsofrotations.Thegoalistopresentthebasicsin5lecturesfocusingon 1.J asthegeneratorofrotations. 2.RepresentationsofSO 3 3.Additionofangularmomentum+ / 4.OrbitalangularmomentumandYlm ' s 5.Tensoroperators. Á Rotations&SO(3) ü Rotationsofvectors Beginwithadiscussionofrotationsappliedtoa3-dimensionalrealvectorspace.Thevectorsaredescribedbythreereal vx numbers,e.g.v = v .ThetransposeofavectorisvT = v , v , v .Thereisaninnerproductdefinedbetweentwo ML y ]\ x y z M ] M vz ] + / T M T ] vectorsbyu ÿ v =Nv ÿ^u = uv cos f,wherefistheanglebetweenthetwovectors.Underarotationtheinnerproduct betweenanytwovectorsispreserved,i.e.thelengthofanyvectorandtheanglebetweenanytwovectorsdoesn't change.Arotationcanbedescribedbya3ä3realorthogonalmatrixRwhichoperatesonavectorbytheusualrulesof matrixmultiplication v'x vx v' = R v and v' , v' , v' = v , v , v RT ML y ]\ ML y ]\ x y z x y z M ] M ] M v'z ] M vz ] + / + / M ]]] M ]]] N ^ N ^ Topreservetheinnerproduct,itisrequirdthatRT ÿ R = 1 u'ÿ v' = uRT ÿ Rv = u1v = u ÿ v Asanexample,arotationbyfaroundthez-axis(orinthexy-plane)isgivenby rotations.nb:11/3/04::13:48:13 2 cos f -sin f 0 R f = sin f cos f 0 z ML ]\ M ] + / M 0 0 1 ] M ]]] N ^ Thesignconventionsareappropriateforarighthandedcoordinatesystem:putthethumbofrighthandalongz-axis,
    [Show full text]
  • 1 Why Do We Need Position Operator in Quantum Theory?
    A New Look at the Position Operator in Quantum Theory Felix M. Lev Artwork Conversion Software Inc., 1201 Morningside Drive, Manhattan Beach, CA 90266, USA (Email: [email protected]) Abstract: In standard quantum theory the momentum and position operators are on equal footing and wave functions in momentum and coordinate representations are related to each other by the Fourier transform. However, while the momentum operator is unambiguously defined as one of the operators of the symmetry algebra, the position operator has a physical meaning only in semiclassical approximation and should be defined from additional considerations. We show that, as a consequence of the in- consistent definition of standard position operator, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum. This leads to the fundamen- tal paradox that we should see not separate stars but only an almost continuous background from all stars. We propose a new consistent definition of the position op- erator. In this approach WPS in directions perpendicular to the particle momentum is absent regardless of whether the particle is nonrelativistic or relativistic. Hence the above paradox is resolved. Moreover, for an ultrarelativistic particle the effect of WPS is absent at all. Different components of the new position operator do not com- mute with each other and, as a consequence, there is no wave function in coordinate representation. PACS: 11.30.Cp, 03.65.-w, 03.63.Sq, 03.65.Ta Keywords: quantum theory, position operator, semiclassical approximation 1 Why do we need position operator in quantum theory? In standard quantum theory the momentum and position operators are on equal foot- ing and wave functions in the momentum and coordinate representations are related to each other by the Fourier transform.
    [Show full text]
  • Observables and Measurements in Quantum Mechanics
    Chapter 13 Observables and Measurements in Quantum Mechanics ill now, almost all attention has been focussed on discussing the state of a quantum system. T As we have seen, this is most succinctly done by treating the package of information that defines a state as if it were a vector in an abstract Hilbert space. Doing so provides the mathemat- ical machinery that is needed to capture the physically observed properties of quantum systems. A method by which the state space of a physical system can be set up was described in Section 8.4.2 wherein an essential step was to associate a set of basis states of the system with the ex- haustive collection of results obtained when measuring some physical property, or observable, of the system. This linking of particular states with particular measured results provides a way that the observable properties of a quantum system can be described in quantum mechanics, that is in terms of Hermitean operators. It is the way in which this is done that is the main subject of this Chapter. 13.1 Measurements in Quantum Mechanics One of the most difficult and controversial problems in quantum mechanics is the so-called measurement Quantum System problem. Opinions on the significance of this prob- S lem vary widely. At one extreme the attitude is that there is in fact no problem at all, while at the other extreme the view is that the measurement problem Measuring is one of the great unsolved puzzles of quantum me- Apparatus chanics. The issue is that quantum mechanics only M ff provides probabilities for the di erent possible out- Surrounding Environment comes in an experiment – it provides no mechanism E by which the actual, finally observed result, comes Figure 13.1: System interacting with about.
    [Show full text]
  • A Point and Local Position Operator Bernice Black Durand Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1971 A point and local position operator Bernice Black Durand Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Elementary Particles and Fields and String Theory Commons Recommended Citation Durand, Bernice Black, "A point and local position operator " (1971). Retrospective Theses and Dissertations. 4876. https://lib.dr.iastate.edu/rtd/4876 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 71-26,850 DURAND, Bernice Black, 1942- A POINT AND LOCAL POSITION OPERATOR. Iowa State University, Ph.D., 1971 Physics, elementary particles University Microfilms, A XEROX Company, Ann Arbor, Michigan A point and local position operator by Bernice Black Durand A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: High Energy Physics Approved: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. Head of Major Department Signature was redacted for privacy. Iowa State University Of Science and Technology Ames, Iowa 1971 PLEASE NOTE: Some pages have
    [Show full text]