DOCSLIB.ORG

Chapter 7. the Translation Operator and Momentum

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 7. the Translation Operator and Momentum 7. Translation and Momentum, October 9, 2013 1 Chapter 7. The Translation Operator and Momentum §1 Introduction §2 The commutator of position and momentum operators §3 Momentum and translation §4 An outline of what is to follow §5 The translation operator:definition and properties §6 The commutator [ˆx, Gˆ] §7 The evaluation of x pˆ ψ h | | i §8 The energy (Hamiltonian) operator §9 Generalization of translation and momentum operators to three dimensions §10 The energy eigenvalue equation in three dimensions §11 The case of many particles §12 How do we use these? §13 A historical note § 1 Introduction. Our work so far dealt with kinematics. This is very simple in classical mechanics: you define the state by specifying the position and the velocity of each particle. You add to this Newton’s equation and the dynamics is completed. All the rest consists of rewriting Newton equation in a variety of forms. Everything you learned so far has to do with the kinematics of quantum mechanics, which is obviously more complicated than classical kinematics. You know how to describe the state of a system but you do not know how to calculate what the states are and how they evolve in time, because we 7. Translation and Momentum, October 9, 2013 2 do not have explicit, computable expressions for the operators. If we had such expressions we could solve the eigenvalue problems for the operators and obtain the pure states. We are in this situation because we do not have a dynamical law that would do for quantum mechanics what Newton’s law does for classical physics. This chapter introduces such a dynamical law, which consists of an ex- pression for the commutator of the coordinate operator with the momentum operator. This can be used to find a formula for the momentum operator in coordinate representation. We already know the expression for the potential energy operator in coordinate representation. Once we have an expression for momentum we know the kinetic energy and therefore we can write an expression for the total energy (the Hamiltonian). Since all observable can be expressed in terms of position and momentum we can calculate anything we can measure. To derive an expression for the momentum operator we use its connection to the operation of translation. To do this we will spend quite a bit of time defining a translation operator and studying its properties. As a result we derive the relationship h¯ ∂ x pˆ ψ = x ψ (1) h | | i i ∂xh | i This allows us to turn abstract operator eigenvalue problems into eigenvalue problems for explicitly defined differential operators. § 2 The commutator of position and momentum operators. We follow here Dirac and formulate the dynamic law in terms of position and momentum operators. Both quantities are vectors. In what follows we use Cartesian 7. Translation and Momentum, October 9, 2013 3 coordinates, and r1(α) is the x-component of the position vector of particle α, r2(α) is the y-component, and r3(α) is the z-component. The x-, y- and z-components of the momentum operator of the particle α (in Cartesian coordinates) are p1(α), p2(α)and p3(α), respectively. The commutation rules for these operators are given below for a system of N particles. 1. All the components of the position operators, for all particles, commute with each other: [~rˆi(α), ~rˆj (β)]= 0 for i, j = 1, 2, 3 and α, β = 1, 2,...,N (2) 2. All components of the momentum operators, for all particles, commute with each other: ˆ ˆ [~pi(α), ~pj(β)]=0 for i, j = 1, 2, 3 and α, β = 1, 2,...,N (3) 3. Momentum and position operators commute if they refer to different Cartesian components or to different particles. 4. The position and momentum operators do not commute if they refer to the same Cartesian component of the same particle, in which case the commutator is equal to i¯hIˆ where Iˆ is the unit operator. 5. These statements are summarized by the equation [ˆri(α), pˆj (β)] = i¯hδij δαβIˆ for i, j = 1, 2, 3 and α, β = 1, 2,...,N (4) where Iˆ is the unit operator. We can simplify Eq. 4 to state that [ˆri(α), pˆi(α)] = i¯hIˆ for i = 1, 2, 3 and α = 1, 2,...,N (5) 7. Translation and Momentum, October 9, 2013 4 and all other commutators are zero. In this chapter we use Eqs. 2–4 to derive an expression for the momentum operator in the coordinate representation. Once we do this, we can derive expressions for operators representing any observable. § 3 Momentum and translation. The idea for the procedure by which we find an equation for momentum has its origin in classical mechanics. It is based on the obvious observation that if the forces acting on a system do not depend on the location of the system (i.e. the space is uniform), the same experiment performed in two different regions of space must give the same result. Saying that the experiments are identical but they are performed in different places means that one experiment can be replicated by translating the other. One can easily prove, by using Newton’s equation, that this statement leads to momentum conservation. This is a simple incarnation of a general theorem, due to Emmy Noether1, which connects all conserved quantities in mechanics (and other branches of physics) to transformations that change the look of the equations but do not change the physics of the system. The fact that identical experiments performed here and performed there must give identical results, leads to conservation of linear momentum. The fact that in the absence of a torque in an experiment does not change if it is described by a 1Noether’s career is a heart-breaking example of irrational discrimination against a woman of great talent who did excellent mathematics while working unpaid in G¨ottingen. Hilbert and Klein wanted to hire her onto the faculty, but the appointment was vetoed by the humanities professors. The reason: the presence of a woman would be too disruptive. Never mind that she was already present and the university seemed to survive the disrup- tion. After Hitler came to power she emigrated to the United States and worked at Bryn Mawr College where she was much appreciated. 7. Translation and Momentum, October 9, 2013 5 coordinate system rotated with respect to another one, leads to conservation of angular momentum. The fact that an experiment done today must give the same results as the same experiment done tomorrow, leads to energy conservation. In this chapter I use the connection between momentum and translation to find a computable expression for the momentum operator in the coordinate representation. § 4 An outline of what is to follow. Here is an outline of how we use the translation operator to find out how the momentum operator acts on a pure state of position. We define a translation operator Tˆ(λ) x x + λ (6) | i≡ | i that moves a particle from a position x to a position x + λ. I will then prove that, if λ is an infinitesimal distance then iλ Tˆ(λ)= Iˆ Gˆ (λ infinitesimal) (7) − h¯ where Iˆ is the unit operator, i = √ 1, and Gˆ is a Hermitian operator defined − by Eq. 7. Gˆ is called the infinitesimal generator of translation. Next we show that the commutator ofx ˆ with Gˆ is [ˆx, Gˆ]=i¯h (8) This means that Gˆ has the same commutation relation as momentump ˆ and therefore it must be equal to it. It follows that we can write iλ Tˆ(λ)= Iˆ pˆ (λ infinitesimal) (9) − h¯ 7. Translation and Momentum, October 9, 2013 6 which implies h¯ Iˆ Tˆ(λ) pˆ = − (λ infinitesimal) (10) i λ ! Our goal is thus achieved: we know how the operators in the right-hand side of Eq. 10 act on a pure state x (see Eq. 6), therefore we know howp ˆ acts on | i x . This knowledge is exploited to derive explicit expressions forp ˆ x and | i | i for x pˆ ψ , where ψ is an arbitrary ket. Once we know this, we can write h | | i | i explicit expressions for operators representing any classical quantity of the form f(p) where f is a function and p is the momentum. We already know how to represent, in coordinate representation, opera- tors that are functions of position. This makes it possible to derive expres- sions for the operators representing the Hamiltonian H = p2/2m + V (x) or the angular momentum L~ = ~r ~p. × § 5 The translation operator:definition and properties. We already defined the translation operator Tˆ(λ) through Tˆ(λ) x = x + λ (11) | i | i Here x is a pure state of the coordinate. A particle in this state is located | i at x. The translation operator Tˆ(λ) acting on x creates a pure state x+λ | i | i in which we are certain that the particle is located at x + λ. For simplicity of notation, we work with a system consisting of one particle, moving in one dimension. Generalizing to many particles in three dimensions is straightfor- ward. For future use, we need to establish the following properties of Tˆ(λ). 7. Translation and Momentum, October 9, 2013 7 Property 1. Tˆ(λ)−1 = Tˆ( λ) (12) − Property 2. Tˆ(λ) is a unitary operator. Property 3. Tˆ(λ)† = Tˆ( λ) (13) − Property 4. If λ is arbitrarily small then iλ Tˆ(λ)= Iˆ Gˆ + (λ2) (14) − h¯ O where Gˆ is a Hermitian operator.
Load more

Recommended publications
  • 1 Notation for States
    The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to define t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric field F µν(~x, t).
    [Show full text]
  • Motion of the Reduced Density Operator
    Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent.
    [Show full text]
  • An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]
    An S-Matrix for Massless Particles Holmfridur Hannesdottir and Matthew D. Schwartz Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract The traditional S-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an S-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard S-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard S-matrix elements. We present a framework for the perturbative calculation of hard S-matrix elements combining Lorentz-covariant Feyn- man rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancella- tion of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and N = 4 super Yang-Mills theory. arXiv:1911.06821v2 [hep-th] 24 Aug 2020 Contents 1 Introduction1 2 The hard S-matrix6 2.1 SH and dressed states . .9 2.2 Computing observables using SH ........................... 11 2.3 Soft Wilson lines . 14 3 Computing the hard S-matrix 17 3.1 Asymptotic region Feynman rules .
    [Show full text]
  • 5 the Dirac Equation and Spinors
    5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.
    [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]
  • Representations of U(2O) and the Value of the Fine Structure Constant
    Symmetry, Integrability and Geometry: Methods and Applications Vol. 1 (2005), Paper 028, 8 pages Representations of U(2∞) and the Value of the Fine Structure Constant William H. KLINK Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA E-mail: [email protected] Received September 28, 2005, in final form December 17, 2005; Published online December 25, 2005 Original article is available at http://www.emis.de/journals/SIGMA/2005/Paper028/ Abstract. A relativistic quantum mechanics is formulated in which all of the interactions are in the four-momentum operator and Lorentz transformations are kinematic. Interactions are introduced through vertices, which are bilinear in fermion and antifermion creation and annihilation operators, and linear in boson creation and annihilation operators. The fermion- antifermion operators generate a unitary Lie algebra, whose representations are fixed by a first order Casimir operator (corresponding to baryon number or charge). Eigenvectors and eigenvalues of the four-momentum operator are analyzed and exact solutions in the strong coupling limit are sketched. A simple model shows how the fine structure constant might be determined for the QED vertex. Key words: point form relativistic quantum mechanics; antisymmetric representations of infinite unitary groups; semidirect sum of unitary with Heisenberg algebra 2000 Mathematics Subject Classification: 22D10; 81R10; 81T27 1 Introduction With the exception of QCD and gravitational self couplings, all of the fundamental particle interaction Hamiltonians have the form of bilinears in fermion and antifermion creation and annihilation operators times terms linear in boson creation and annihilation operators. For example QED is a theory bilinear in electron and positron creation and annihilation operators and linear in photon creation and annihilation operators.
    [Show full text]
  • Second Quantization
    Chapter 1 Second Quantization 1.1 Creation and Annihilation Operators in Quan- tum Mechanics We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation a,a† = 1 (1.1) where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let α be a state which we will take to be an eigenvector of the Hermitian operators| ia†a with eigenvalue α which is a real number, a†a α = α α (1.2) | i | i Hence, α = α a†a α = a α 2 0 (1.3) h | | i k | ik ≥ where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a†a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C]= A [B, C] + [A, C] B (1.4) we get a†a,a = a (1.5) − † † † a a,a = a (1.6) 1 2 CHAPTER 1. SECOND QUANTIZATION i.e., a and a† are “eigen-operators” of a†a. Hence, a†a a = a a†a 1 (1.7) − † † † † a a a = a a a +1 (1.8) Consequently we find a†a a α = a a†a 1 α = (α 1) a α (1.9) | i − | i − | i Hence the state aα is an eigenstate of a†a with eigenvalue α 1, provided a α = 0.
    [Show full text]
  • Angular Momentum 23.1 Classical Description
    Physics 342 Lecture 23 Angular Momentum Lecture 23 Physics 342 Quantum Mechanics I Monday, March 31st, 2008 We know how to obtain the energy of Hydrogen using the Hamiltonian op- erator { but given a particular En, there is degeneracy { many n`m(r; θ; φ) have the same energy. What we would like is a set of operators that allow us to determine ` and m. The angular momentum operator L^, and in partic- 2 ular the combination L and Lz provide precisely the additional Hermitian observables we need. 23.1 Classical Description Going back to our Hamiltonian for a central potential, we have p · p H = + U(r): (23.1) 2 m It is clear from the dependence of U on the radial distance only, that angular momentum should be conserved in this setting. Remember the definition: L = r × p; (23.2) and we can write this in indexed notation which is slightly easier to work with using the Levi-Civita symbol, defined as (in three dimensions) 8 < 1 for (ijk) an even permutation of (123). ijk = −1 for an odd permutation (23.3) : 0 otherwise Then we can write 3 3 X X Li = ijk rj pk = ijk rj pk (23.4) j=1 k=1 1 of 9 23.2. QUANTUM COMMUTATORS Lecture 23 where the sum over j and k is implied in the second equality (this is Einstein summation notation). There are three numbers here, Li for i = 1; 2; 3, we associate with Lx, Ly, and Lz, the individual components of angular momentum about the three spatial axes.
    [Show full text]
  • 13 the Dirac Equation
    13 The Dirac Equation A two-component spinor a χ = b transforms under rotations as iθn J χ e− · χ; ! with the angular momentum operators, Ji given by: 1 Ji = σi; 2 where σ are the Pauli matrices, n is the unit vector along the axis of rotation and θ is the angle of rotation. For a relativistic description we must also describe Lorentz boosts generated by the operators Ki. Together Ji and Ki form the algebra (set of commutation relations) Ki;Kj = iεi jkJk − ε Ji;Kj = i i jkKk ε Ji;Jj = i i jkJk 1 For a spin- 2 particle Ki are represented as i Ki = σi; 2 giving us two inequivalent representations. 1 χ Starting with a spin- 2 particle at rest, described by a spinor (0), we can boost to give two possible spinors α=2n σ χR(p) = e · χ(0) = (cosh(α=2) + n σsinh(α=2))χ(0) · or α=2n σ χL(p) = e− · χ(0) = (cosh(α=2) n σsinh(α=2))χ(0) − · where p sinh(α) = j j m and Ep cosh(α) = m so that (Ep + m + σ p) χR(p) = · χ(0) 2m(Ep + m) σ (pEp + m p) χL(p) = − · χ(0) 2m(Ep + m) p 57 Under the parity operator the three-moment is reversed p p so that χL χR. Therefore if we 1 $ − $ require a Lorentz description of a spin- 2 particles to be a proper representation of parity, we must include both χL and χR in one spinor (note that for massive particles the transformation p p $ − can be achieved by a Lorentz boost).
    [Show full text]
  • Structure of a Spin ½ B
    Structure of a spin ½ B. C. Sanctuary Department of Chemistry, McGill University Montreal Quebec H3H 1N3 Canada Abstract. The non-hermitian states that lead to separation of the four Bell states are examined. In the absence of interactions, a new quantum state of spin magnitude 1/√2 is predicted. Properties of these states show that an isolated spin is a resonance state with zero net angular momentum, consistent with a point particle, and each resonance corresponds to a degenerate but well defined structure. By averaging and de-coherence these structures are shown to form ensembles which are consistent with the usual quantum description of a spin. Keywords: Bell states, Bell’s Inequalities, spin theory, quantum theory, statistical interpretation, entanglement, non-hermitian states. PACS: Quantum statistical mechanics, 05.30. Quantum Mechanics, 03.65. Entanglement and quantum non-locality, 03.65.Ud 1. INTRODUCTION In spite of its tremendous success in describing the properties of microscopic systems, the debate over whether quantum mechanics is the most fundamental theory has been going on since its inception1. The basic property of quantum mechanics that belies it as the most fundamental theory is its statistical nature2. The history is well known: EPR3 showed that two non-commuting operators are simultaneously elements of physical reality, thereby concluding that quantum mechanics is incomplete, albeit they assumed locality. Bohr4 replied with complementarity, arguing for completeness; thirty years later Bell5, questioned the locality assumption6, and the conclusion drawn that any deeper theory than quantum mechanics must be non-local. In subsequent years the idea that entangled states can persist to space-like separations became well accepted7.
    [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]
  • Kinematic Relativity of Quantum Mechanics: Free Particle with Different Boundary Conditions
    Journal of Applied Mathematics and Physics, 2017, 5, 853-861 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 Kinematic Relativity of Quantum Mechanics: Free Particle with Different Boundary Conditions Gintautas P. Kamuntavičius 1, G. Kamuntavičius 2 1Department of Physics, Vytautas Magnus University, Kaunas, Lithuania 2Christ’s College, University of Cambridge, Cambridge, UK How to cite this paper: Kamuntavičius, Abstract G.P. and Kamuntavičius, G. (2017) Kinema- tic Relativity of Quantum Mechanics: Free An investigation of origins of the quantum mechanical momentum operator Particle with Different Boundary Condi- has shown that it corresponds to the nonrelativistic momentum of classical tions. Journal of Applied Mathematics and special relativity theory rather than the relativistic one, as has been uncondi- Physics, 5, 853-861. https://doi.org/10.4236/jamp.2017.54075 tionally believed in traditional relativistic quantum mechanics until now. Taking this correspondence into account, relativistic momentum and energy Received: March 16, 2017 operators are defined. Schrödinger equations with relativistic kinematics are Accepted: April 27, 2017 introduced and investigated for a free particle and a particle trapped in the Published: April 30, 2017 deep potential well. Copyright © 2017 by authors and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Commons Attribution International Special Relativity, Quantum Mechanics, Relativistic Wave Equations, License (CC BY 4.0). Solutions
    [Show full text]