DOCSLIB.ORG

Dirac -Function Potential in Quasiposition Representation of a Minimal-Length Scenario

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dirac -Function Potential in Quasiposition Representation of a Minimal-Length Scenario Eur. Phys. J. C (2018) 78:179 https://doi.org/10.1140/epjc/s10052-018-5659-6 Regular Article - Theoretical Physics Dirac δ-function potential in quasiposition representation of a minimal-length scenario M. F. Gusson, A. Oakes O. Gonçalves, R. O. Francisco, R. G. Furtado, J. C. Fabrisa, J. A. Nogueirab Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES 29075-910, Brazil Received: 26 April 2017 / Accepted: 22 February 2018 / Published online: 3 March 2018 © The Author(s) 2018. This article is an open access publication Abstract A minimal-length scenario can be considered as Although the first proposals for the existence of a mini- an effective description of quantum gravity effects. In quan- mal length were done by the beginning of 1930s [1–3], they tum mechanics the introduction of a minimal length can were not connected with quantum gravity, but instead with be accomplished through a generalization of Heisenberg’s a cut-off in nature that would remedy cumbersome diver- uncertainty principle. In this scenario, state eigenvectors of gences arising from quantization of systems with an infinite the position operator are no longer physical states and the number of degrees of freedom. The relevant role that grav- representation in momentum space or a representation in a ity plays in trying to probe a smaller and smaller region of quasiposition space must be used. In this work, we solve the the space-time was recognized by Bronstein [4] already in Schroedinger equation with a Dirac δ-function potential in 1936; however, his work did not attract a lot of attention. quasiposition space. We calculate the bound state energy and It was only in 1964 that Mead [5,6] once again proposed a the coefficients of reflection and transmission for the scatter- possible connection between gravitation and minimal length. ing states. We show that√ leading corrections are of order of Hence, we can assume that gravity may lead to an effec- the minimal length (O( β))and the coefficients of reflection tive cut-off in the ultraviolet. Furthermore, if we are con- and transmission are no longer the same for the Dirac delta vinced that gravitational effects are in order when a minimal well and barrier as in ordinary quantum mechanics. Further- length is introduced then a minimal-length scenario could more, assuming that the equivalence of the 1s state energy of be thought of as an effective description of quantum gravity the hydrogen atom and the bound state energy of the Dirac effects [7]. δ-function potential in the one-dimensional case is kept in a As far as we know, the introduction of a minimal-length minimal-length scenario, we also find that the leading cor- scenario can be carried out in three different ways [7– rection term for the ground state energy of the hydrogen atom 9]: a generalization of the Heisenberg uncertainty principle −25 1 is of the order of the minimal length and Δxmin ≤ 10 m. (GUP), a deformation of special relativity (DSR) and a mod- ification of the dispersion relation (MDR). Various problems connected with the minimal length have been studied in the context of non-relativistic quantum 1 Introduction mechanics. Among them are the harmonic oscillator [10–14], the hydrogen atom [15–21], step and barrier potentials [22– Gravity quantization has become a huge challenge for theo- 24], finite and infinite square wells [25,26]. In the relativistic retical physicists. Despite enormous efforts made, so far, it context, the Dirac equation has been studied in [27–33]. The was not possible to obtain a theory which can be considered Casimir effect has also been studied in a minimal-length sce- suitable and not even a consensus approach. Nevertheless, nario in [34–38]. most of the candidate theories for gravity quantization seem An interesting problem in quantum mechanics is the Dirac to have one common point: the prediction of the existence of δ-function potential. In general, the Dirac δ-function poten- a minimal length, that is, a limit for the precision of a length tial is used as a pedagogical exercise. Nevertheless, it has also measurement. been used to model physical quantum systems [39]. Maybe because the attractive Dirac δ-function potential is one of a e-mail: [email protected] 1 It is named doubly special relativity because of the existence of two b e-mail: [email protected] universal constants: the speed of light and the minimal length. 123 179 Page 2 of 7 Eur. Phys. J. C (2018) 78 :179 the simplest quantum systems which displays both a bound tainty principle (GUP). Since state and a set of continuous states, it has been used to model |[x ˆ, pˆ]| atomic and molecular systems [40–45]. In addition, the short- ΔxΔp ≥ , (1) range interactions in condensed matter with a large scattering 2 length can actually be modeled as a Dirac δ-function poten- a generalization of the uncertainty principle corresponds to a tial [46–50]. In quantum field theory, in order to treat the modification in the algebra of the operators. There are differ- Casimir effect more realistically, the boundary conditions ent suggestions of the modification of the commutation rela- 1 σ( )φ2( ) tion between the position and momentum operators which are replaced by the interaction potential 2 x x , where σ(x) represents the field of the material of the borders (back- implement a minimal-length scenario. Weare concerned with ground field). Hence, for sharply localized borders the back- the most usual of them, proposed by Kempf [10,11], which ground field can be modeled by a Dirac δ-function [51,52]. in a one-dimensional space is given by The Dirac δ-function potential by its very nature is a chal- [ˆ, ˆ]:= ¯ + β ˆ2 , lenging problem in a minimal-length scenario. x p ih 1 p (2) Ferkous [53] and Samar and Tkachuck [54] have inde- where β is a parameter related to the minimal length. The pendently calculated the bound state energy in “momen- commutation relation (2) corresponds to the GUP tum space”. In the two papers cited, the authors√ have found β β h¯ a√ correction for the expression of energy in and ΔxΔp ≥ 1 + β(Δp)2 + βˆp2 , (3) ( β ∼ Δxmin), but with different coefficients, therefore hav- 2 ing disagreeing outcomes. Samar and Tkachuck claim that which implies the existence of a non-zero minimal uncer- (−∞, ∞) √ this is because Ferkous considers p to belong to , tainty in the position Δxmin = h¯ β. π π whereas they consider p to belong to − √ , √ .Inthis 2 β 2 β Unfortunately, in this scenario the eigenstates of the posi- work, we propose to solve the problem of a non-relativistic tion operator are not physical sates2 and, consequently, the particle of mass m in the presence of the Dirac δ-function representation in position space can no longer be used, that is, potential in quasiposition space. Since the quasiposition an arbitrary state vector |ψ cannot be expanded in the basis space representation is used we can consider the cases of of state eigenvectors of the position operator {|x}. Hence the bound states and scattering states as well. We find the same obvious way to go ahead is to make use of the representation expression for the energy of the bound state as obtained by in momentum space: Ferkous. In addition, assuming that the equality between ∂ψ(˜ p) the 1s state energy of the hydrogen atom and the bound p|ˆx|ψ=ih¯ + βp2 , 1 ∂ (4) state energy of the Dirac δ-function potential in the one- p ˜ dimensional case when the coefficient of the δ-potential is p|ˆp|ψ=pψ(p). (5) replaced by the fine structure constant [41] is kept in a However, the representation in momentum space is not minimal-length scenario, we find that the leading correc- suitable in some cases, such as, for example, when the wave tion for the ground state energy of hydrogen atom is of the √ function has to satisfy a boundary condition at specifics order of the minimal length (O( β)), differently from the points. So, the representation in quasiposition space [56], values commonly found in the literature using perturbative methods [15–18,55], but according to the results obtained x ML|ˆx|ψ(t)=xψqp(x, t), (6) by Fityo et al. [20] and Bouaziz and Ferkous [21]usinga ∂2 ∂ψqp( , ) ML 2 x t non-perturbative approach. x |ˆp|ψ(t)=−ih¯ 1 − βh¯ , (7) ∂x2 ∂x The rest of this paper is organized as follows. In Sect. 2 we show how to introduce a minimal-length scenario and find to first order in the β parameter, is more appropriate.3 |x ML the time-independent Schroedinger equation in a quasiposi- are state vectors of maximal localization which satisfy [10] tion space representation. In Sect. 3 we solve the modified ML|ˆ| ML= , ∈ , Schroedinger equation and find the bound state energy and x x x x with x (8) (Δ ) = Δ = β, the coefficients of reflection and transmission for the scatter- x |x ML xmin h¯ (9) ing states. We present our conclusions in Sect. 4. 2 That is because the uncertainty ΔA of an operator Aˆ in any of its state eigenvectors |ψA must be zero, which is not the case for the position 2 Minimal-length scenario operator, since Δxmin > 0. 3 ˆ =ˆ Pedram√ [56] has proposed a representation in which x xo and ( β ˆ ) In quantum theory, a minimal-length scenario can be accom- ˆ = tan √ po ˆ ˆ p β ,wherexo and po are ordinary operators of position and plished by imposing a non-zero minimal uncertainty in the momentum, which obey the canonical commutation relation [ˆxo, pˆo]= measurement of position which leads to a generalized uncer- ih¯ .
Load more

Recommended publications
  • The Scattering and Shrinking of a Gaussian Wave Packet by Delta Function Potentials Fei
    The Scattering and Shrinking of a Gaussian Wave Packet by Delta Function Potentials by Fei Sun Submitted to the Department of Physics ARCHIVF in partial fulfillment of the requirements for the degree of Bachelor of Science r U at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 @ Massachusetts Institute of Technology 2012. All rights reserved. A u th or ........................ ..................... Department of Physics May 9, 2012 C ertified by .................. ...................... ................ Professor Edmund Bertschinger Thesis Supervisor, Department of Physics Thesis Supervisor Accepted by.................. / V Professor Nergis Mavalvala Senior Thesis Coordinator, Department of Physics The Scattering and Shrinking of a Gaussian Wave Packet by Delta Function Potentials by Fei Sun Submitted to the Department of Physics on May 9, 2012, in partial fulfillment of the requirements for the degree of Bachelor of Science Abstract In this thesis, we wish to test the hypothesis that scattering by a random potential causes localization of wave functions, and that this localization is governed by the Born postulate of quantum mechanics. We begin with a simple model system: a one-dimensional Gaussian wave packet incident from the left onto a delta function potential with a single scattering center. Then we proceed to study the more compli- cated models with double and triple scattering centers. Chapter 1 briefly describes the motivations behind this thesis and the phenomenon related to this research. Chapter 2 to Chapter 4 give the detailed calculations involved in the single, double and triple scattering cases; for each case, we work out the exact expressions of wave functions, write computer programs to numerically calculate the behavior of the wave packets, and use graphs to illustrate the results of the calculations.
    [Show full text]
  • Scattering from the Delta Potential
    Phys 341 Quantum Mechanics Day 10 Wed. 9/24 2.5 Scattering from the Delta Potential (Q7.1, Q11) Computational: Time-Dependent Discrete Schro Daily 4.W 4 Science Poster Session: Hedco7pm~9pm Fri., 9/26 2.6 The Finite Square Well (Q 11.1-.4) beginning Daily 4.F Mon. 9/29 2.6 The Finite Square Well (Q 11.1-.4) continuing Daily 5.M Tues 9/30 Weekly 5 5 Wed. 10/1 Review Ch 1 & 2 Daily 5.W Fri. 10/3 Exam 1 (Ch 1 & 2) Equipment Load our full Python package on computer Comp 5: discrete Time-Dependent Schro Griffith’s text Moore’s text Printout of roster with what pictures I have Check dailies Announcements: Daily 4.W Wednesday 9/24 Griffiths 2.5 Scattering from the Delta Potential (Q7.1, Q11) 1. Conceptual: State the rules from Q11.4 in terms of mathematical equations. Can you match the rules to equations in Griffiths? If you can, give equation numbers. 7. Starting Weekly, Computational: Follow the instruction in the handout “Discrete Time- Dependent Schrodinger” to simulate a Gaussian packet’s interacting with a delta-well. 2.5 The Delta-Function Potential 2.5.1 Bound States and Scattering States 1. Conceptual: Compare Griffith’s definition of a bound state with Q7.1. 2. Conceptual: Compare Griffith’s definition of tunneling with Q11.3. 3. Conceptual: Possible energy levels are quantized for what kind of states (bound, and/or unbound)? Why / why not? Griffiths seems to bring up scattering states out of nowhere. By scattering does he just mean transmission and reflection?" Spencer 2.5.2 The Delta-Function Recall from a few days ago that we’d encountered sink k a 0 for k ko 2 o k ko 2a for k ko 1 Phys 341 Quantum Mechanics Day 10 When we were dealing with the free particle, and we were planning on eventually sending the width of our finite well to infinity to arrive at the solution for the infinite well.
    [Show full text]
  • Revisiting Double Dirac Delta Potential
    Revisiting double Dirac delta potential Zafar Ahmed1, Sachin Kumar2, Mayank Sharma3, Vibhu Sharma3;4 1Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 2Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 3;4Amity Institute of Applied Sciences, Amity University, Noida, UP, 201313, India∗ (Dated: June 14, 2016) Abstract We study a general double Dirac delta potential to show that this is the simplest yet versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission (T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti- symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission [or zero reflectivity, R(E)] at energy E = 0 is non-intuitive. However, earlier this has been found and called \threshold anomaly". Here we show that it is a critical phenomenon and we can have 0 ≤ R(0) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke zero-energy and zero curvature eigenstate ( (x) = Ax + B) of delta well between two symmetric rigid walls for R(0) = 0. We resolve that the resonant energies and the perfect transmission energies are different and they arise differently. arXiv:1603.07726v4 [quant-ph] 13 Jun 2016 ∗Electronic address: 1:[email protected], 2: [email protected], 3: [email protected], 4:[email protected] 1 I. INTRODUCTION The general one-dimensional Double Dirac Delta Potential (DDDP) is written as [see Fig.
    [Show full text]
  • Paradoxical Quantum Scattering Off a Time Dependent Potential?
    Paradoxical Quantum Scattering off a Time Dependent Potential? Ori Reinhardt and Moshe Schwartz Department of Physics, Raymond and Beverly Sackler Faculty Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel We consider the quantum scattering off a time dependent barrier in one dimension. Our initial state is a right going eigenstate of the Hamiltonian at time t=0. It consists of a plane wave incoming from the left, a reflected plane wave on the left of the barrier and a transmitted wave on its right. We find that at later times, the evolving wave function has a finite overlap with left going eigenstates of the Hamiltonian at time t=0. For simplicity we present an exact result for a time dependent delta function potential. Then we show that our result is not an artifact of that specific choice of the potential. This surprising result does not agree with our interpretation of the eigenstates of the Hamiltonian at time t=0. A numerical study of evolving wave packets, does not find any corresponding real effect. Namely, we do not see on the right hand side of the barrier any evidence for a left going packet. Our conclusion is thus that the intriguing result mentioned above is intriguing only due to the semantics of the interpretation. PACS numbers: 03.65.-w, 03.65.Nk, 03.65.Xp, 66.35.+a The quantum problem of an incoming plain wave The eigenstates of the Hamiltonian (Eq. 1) can be encountering a static potential barrier in one dimension is classified according to the absolute value of incoming an old canonical text book example in which the reflection momentum and to whether they are right or left going.
    [Show full text]
  • Casimir Effect for Impurity in Periodic Background in One Dimension
    Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 53 (2020) 325401 (24pp) https://doi.org/10.1088/1751-8121/ab9463 Casimir effect for impurity in periodic background in one dimension M Bordag Institute for Theoretical Physics, Universität Leipzig Brüderstraße 16 04103 Leipzig, Germany E-mail: [email protected] Received 2 April 2020, revised 7 May 2020 Accepted for publication 19 May 2020 Published 27 July 2020 Abstract We consider a Bose gas in a one-dimensional periodic background formed by generalized delta function potentials with one and two impurities. We inves- tigate the scattering off these impurities and their bound state levels. Besides expected features, we observe a kind of long-range correlation between the bound state levels of two impurities. Further, we define and calculate the vac- uum energy of the impurity. It causes a force acting on the impurity relative to the background. We define the vacuum energy as a mode sum. In order toget a discrete spectrum we start from a finite lattice and use Chebychev polynomi- als to get a general expression. These allow also for quite easy investigation of impurities in finite lattices. Keywords: quantum vacuum, Casimir effect (theory), periodic potential (Some figures may appear in colour only in the online journal) 1. Introduction One-dimensional systems provide a large number of examples to study quantum effects. These allow frequently for quite explicit and instructive formulas, but at the same time there are many one-dimensional and quasi-one-dimensional systems in physics which are frequently also of practical interest like nanowires and carbon nanotubes.
    [Show full text]
  • Tunneling in Fractional Quantum Mechanics
    Tunneling in Fractional Quantum Mechanics E. Capelas de Oliveira1 and Jayme Vaz Jr.2 Departamento de Matem´atica Aplicada - IMECC Universidade Estadual de Campinas 13083-859 Campinas, SP, Brazil Abstract We study the tunneling through delta and double delta po- tentials in fractional quantum mechanics. After solving the fractional Schr¨odinger equation for these potentials, we cal- culate the corresponding reflection and transmission coeffi- cients. These coefficients have a very interesting behaviour. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmis- sion coefficient is given by = cos2 (π/α), where α is the T0 order of the derivative (1 < α 2). ≤ 1. Introduction In recent years the study of fractional integrodifferential equations applied to physics and other areas has grown. Some examples are [1, 2, 3], among many others. More recently, the fractional generalized Langevin equation is proposed to discuss the anoma- lous diffusive behavior of a harmonic oscillator driven by a two-parameter Mittag-Leffler noise [4]. Fractional Quantum Mechanics (FQM) is the theory of quantum mechanics based on the fractional Schr¨odinger equation (FSE). In this paper we consider the FSE as introduced by Laskin in [5, 6]. It was obtained in the context of the path integral approach to quantum mechanics. In this approach, path integrals are defined over L´evy flight paths, which is a natural generalization of the Brownian motion [7]. arXiv:1011.1948v2 [math-ph] 15 Mar 2011 There are some papers in the literature studying solutions of FSE.
    [Show full text]
  • Some Exact Results for the Schršdinger Wave Equation with a Time Dependent Potential Joel Campbell NASA Langley Research Center
    Some Exact Results for the Schršdinger Wave Equation with a Time Dependent Potential Joel Campbell NASA Langley Research Center, MS 488 Hampton, VA 23681 [email protected] Abstract The time dependent Schršdinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density. 02.30Em, 02.60-x, 03.65-w, 61.50-f Introduction Very few cases of the time dependent Schršdinger wave equation with a time dependent potential can be solved exactly. The cases that are known include the time dependent harmonic oscillator,[ 1-3] an example of an infinite potential well with a moving boundary,[4-6] and various other special cases. There remain, however, a large number of time dependent problems that, at least in principal, can be solved exactly. One case that has been looked at over the years is the delta function potential. The author first investigated this problem in the mid 1980s [7] as an unpublished work. This was later included as part of a PhD thesis dissertation [8] but not published in a journal.
    [Show full text]
  • The Attractive Nonlinear Delta-Function Potential
    The Attractive Nonlinear Delta-Function Potential M. I. Molina and C. A. Bustamante Facultad de Ciencias, Departamento de F´ısica, Universidad de Chile Casilla 653, Las Palmeras 3425, Santiago, Chile. arXiv:physics/0102053v1 [physics.ed-ph] 16 Feb 2001 1 Abstract We solve the continuous one-dimensional Schr¨odinger equation for the case of an inverted nonlinear delta–function potential located at the origin, ob- taining the bound state in closed form as a function of the nonlinear expo- nent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the non- linear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta–like profile. Further increase of the expo- nent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity. 2 The delta-function potential δ(x x ) has become a familiar sight in the − 0 landscape of most elementary courses on quantum mechanics, where it serves to illustrate the basic techniques in simple form. As a physical model, it has been used to represent a localized potential whose energy scale is greater than any other in the problem at hand and whose spatial extension is smaller than other relevant length scales of the problem. Arrays of delta-function potentials have been used to illustrate Bloch’s theorem in solid state physics and also in optics, where in the scalar approximation, wave propagation in a periodic medium resembles the dynamics of an electron in a crystal lattice.
    [Show full text]
  • Green Function for Delta Potential and One Loop Correction to Quantum Electrodynamic Scattering Process
    GREEN FUNCTION FOR DELTA POTENTIAL AND ONE LOOP CORRECTION TO QUANTUM ELECTRODYNAMIC SCATTERING PROCESS A project report for partial fulfillment of requirements for degree of Integrated Master of Science in Physics Submitted by Raj Kishore 410PH5084 Supervised by Dr. Biplab Ganguli Department of Physics and Astronomy National Institute of Technology Rourkela Abstract Greens function a technique used to solve in general non homogeneous dif- ferential equations. It is basically a correlation function. Its application in high energy physics in finding the propagators has discussed here. In this paper, its application in quantum physics to find Greens functions for quan- tum operators and its solutions has the main focus. By knowing the Greens function we can calculate density of states. In this paper, contains a detail calculations to find the Greens function for single and double delta function potential and then analysis of the bound state. In particle physics one of the main observable concern for the experiment is the scattering cross-section. Physicists used the perturbation theory to approximate results of the scattering process, as we do not have the ex- act knowledge of scattering process. In the higher order corrections, due to loops, the amplitude expression for the Feynman diagram diverges. We apply a theory of renormalization to get a physically possible results. This paper start with the basic introduction to renormalization theory then its application to quantum electrodynamics scattering process. I have briefly discussed all three types of loops found in QED and focused mainly on the vacuum polarization (photon self energy) corrections which give the charge renormalization.
    [Show full text]
  • Solution of the Quantum Harmonic Oscillator Plus a Delta-Function Potential at the Origin: the Oddness of Its Even-Parity Solutions
    Home Search Collections Journals About Contact us My IOPscience Solution of the quantum harmonic oscillator plus a delta-function potential at the origin: the oddness of its even-parity solutions This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Eur. J. Phys. 32 1377 (http://iopscience.iop.org/0143-0807/32/5/025) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 186.105.181.147 The article was downloaded on 09/09/2011 at 03:47 Please note that terms and conditions apply. IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 32 (2011) 1377–1384 doi:10.1088/0143-0807/32/5/025 Solution of the quantum harmonic oscillator plus a delta-function potential at the origin: the oddness of its even-parity solutions J Viana-Gomes and N M R Peres Physics Department, University of Minho, CFUM, P-4710-057 Braga, Portugal E-mail: zgomes@fisica.uminho.pt Received 3 May 2011, in final form 11 July 2011 Published 12 August 2011 Online at stacks.iop.org/EJP/32/1377 Abstract We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.
    [Show full text]
  • 3. Some One-Dimensional Potentials These Notes Are a Supplement to Sections 3.1, 3.3 and 3.5 in Hemmer’S Book
    TFY4215/FY1006 | Lecture notes 3 1 Lecture notes 3 3. Some one-dimensional potentials These notes are a supplement to sections 3.1, 3.3 and 3.5 in Hemmer's book. 3.1 General properties of energy eigenfunctions (Hemmer 3.1, B&J 3.6) For a particle moving in a one-dimensional potential V (x), the energy eigenfunctions are the acceptable solutions of the time-independent Schr¨odingerequation Hc = E : h¯2 @2 ! d2 2m − + V (x) = E ; or = [V (x) − E] : (T3.1) 2m @x2 dx2 h¯2 3.1.a Energy eigenfunctions can be chosen real Locally, this second-order differential equation has two independent solutions. Since V (x) and E both are real, we can notice that if a solution (x) (with energy E) of this equation is complex, then both the real and the imaginary parts of this solution, 1 1 <e[ (x)] = [ (x) + ∗(x)] and =m[ (x)] = [ (x) − ∗(x)]; 2 2i will satisfy (T3.1), for the energy E. This means that we can choose to work with two independent real solutions, if we wish (instead of the complex solutions (x) and ∗(x)). An example: For a free particle (V (x) = 0), 1 p (x) = eikx; with k = 2mE h¯ is a solution with energy E. But then also ∗(x) = exp(−ikx) is a solution with the same energy. If we wish, we can therefore choose to work with the real and imaginary parts of (x), which are respectively cos kx and sin kx, cf particle in a box. Working with real solutions is an advantage e.g.
    [Show full text]
  • Self-Adjoint Extensions of Operators and the Teaching of Quantum
    Self-adjoint extensions of operators and the teaching of quantum mechanics Guy BONNEAU ∗ Jacques FARAUT † Galliano VALENT ∗ Abstract For the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self- adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different physical situations. Some consequences are worked out, which could lead to experimental checks. arXiv:quant-ph/0103153v1 28 Mar 2001 PAR/LPTHE/99-43 December 1999 ∗Laboratoire de Physique Th´eorique et des Hautes Energies, Unit´eassoci´ee au CNRS UMR 7589, Universit´e Paris 7-Denis Diderot, 2 Place Jussieu, 75251 Paris Cedex 05. †Laboratoire d’Analyse Alg´ebrique, Universit´eParis 6-Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05. 2 Bonneau, Faraut, Valent 1 Introduction In most French universities, quantum mechanics is usually taught in the third year courses, separately from its applications to atomic, molecular and subnuclear physics, which are dealt with during the fourth year. In such “first contact” lectures, many mathematical subtleties are necessarily left aside. However, even in such commonly used examples as infinitely deep potential wells, overlooking the mathematical problems leads to contradictions which may be detected by a careful student and which have to do with a precise definition of the “observables” i.e. the self-adjoint operators. Of course, experts in the mathematical theory of unbounded operators in Hilbert spaces know the correct answer to these questions, but we think it could be useful to popularize these concepts among the teaching community and the more mature students of fourth year courses.
    [Show full text]