A new approach to the schrodinger equation with position dependent mass and its implications in quantum dots and semiconductors rami ahmad elnabulsi 1 may 2020 journal of physics and chemistry of solids, vol. Obtaining the schrodinger wave equation let us now construct our wave equation by reverse engineering, i. Pdf on the solution of the schrodinger equation with position. The schrodinger equation for the position dependent mass pdm has found its way into standard textbooks of quantum mechanics fluegge, schiff. Schrodinger equation for a charged spin12 particle in an electromagnetic. Scattering of positiondependent mass schrodinger equation with. The point canonical transformation pct approach is used to solve the schr.
Some physical examples are given for bound and scattering systems. We study the temporal evolution of solutions of 1d schrodinger equation with positiondependent mass inside an infinite well. The positiondependent effective mass schrodinger equation. Lecture 6 schrodinger equation and relationship to electron. A new approach to the schrodinger equation with positiondependent mass and its implications in quantum dots and semiconductors rami ahmad elnabulsi 1 may 2020 journal of physics and chemistry of solids, vol. Wave functions and energy spectra have been obtained analytically. The effective hamiltonian contains the kinetic energy with position dependent mass, which is inverse to the hopping integral, and effective potential energy. To provide a better physical insight into the solutions, some figures are included. Exact solution of the schrodinger equation for a particle with positiondependent mass pdm in a halfharmonic potential is considered and the corresponding exact eigenvalues and eigenfunctions are derived. Recently, the study of the nonrelativistic equation with the position dependent effective mass has attracted a lot of attention to many authors. A singular oscillator hamiltonian with a positiondependent effective mass has been constructed. Panahi department of physics, university of guilan, rasht. Et any linear combination of stationary states each with a different allowed energy of the system is also a valid solution of the schrodinger equation stationary states in fact all possible solutions to the schrodinger equation can be written in this way.
The method includes a co ordinate transformation ct followed by functional transformation ft of wave function. The problem of ddimensional schrodinger equations with a positiondependent mass is analyzed in the framework of firstorder intertwining operators. Here we have studied first and secondorder intertwining approach to generate isospectral partner potentials of positiondependent effective mass schroedinger equation. Pdf analytical solution to position dependent mass schrodinger. No closed algebraic solution is possible or required for this part just explain which equation needs to be solved. Probability densities and uncertainties for the pdm system are obtained and a comparison with those of a halfharmonic potential. The wavefunction is a complex variable and one cant attribute any distinct physical meaning to it. Position dependent mass schrodinger equation and isospectral. We derive the effective lowenergy hamiltonian for the tightbinding model with the hopping integral slowly varying along the chain. To explain our findings with the help of an illustration, we have used point canonical transformation pct to obtain the general solution of the position dependent mass schrodinger equation corresponding to a potential and mass function.
The prime stands for derivative of the function with respect to its argument. Mar 16, 2011 analytical solution to position dependent mass schrodinger equation article pdf available in journal of modern optics 58. This equation is typically described as having only one space dimension x, because the only other independent variable is the time t. Calculate the fisher entropy for positiondependent mass schrodinger equation. The second two boundary conditions require asink 1s bsinks28 ak 1 cosk 1s bkcosks29 which is satis ed when k 1 cotk 1s kcotks30 we can write k 1 p k2 2k2. Chapter11 the schrodinger equation in 3d description of the motion of two interacting particles 11. Nevertheless, the dependent variable u may represent a second space dimension, if, for example, the displacement u takes place in ydirection, as in the case of a string that. Properties of quasioscillator in positiondependent mass. Quantum physics ii, lecture notes 1 mit opencourseware. Time dependent equation this is the equation of motion for the quantum state. The main difference is that matter particles have mass, and photons don t.
Moreover, we have studied the ramsauertownsend effect with the positiondependent mass in quantum mechanics. Thus one dimensional positiondependent mass 1d pdm schrodinger equation is written as d1d. Derivation of schrodinger equation for a system with position. Recently, the study of the nonrelativistic equation with the positiondependent effective mass has attracted a lot of attention to many authors. B 39, 434441 1989 positiondependent effective mass for inhomogeneous semiconductors. The interest in physical systems with positiondependent mass pdm is. We will consider solving the 1d time dependent schrodinger equation using the finite difference time development method fdtd. The secondorder intertwiner is constructed directly by taking it as second order linear differential operator with position depndent coefficients and the system of equations arising from the intertwining.
A new approach to the schrodinger equation with position. Analytical solution to position dependent mass schrodinger. The scattering of the positiondependent mass schrodinger equation due to both single dirac delta and double dirac delta potentials has been. Schrodinger equation is considered within positiondependent mass formalism. In general, explicit derivation of the shannon information entropy is quite difficult. The sc hr o ding er w av e equati on macquarie university. Time dependent schrodinger equation the time dependent schrodinger equation for one spatial dimension is of the form for a free particle where ux 0 the wavefunction solution can be put in the form of a plane wave for other problems, the potential ux serves to set boundary conditions on the spatial part of the wavefunction and it is helpful to separate the equation into the time. Solutions of onedimensional effective mass schrodinger equation. Point canonical transformation versus deformed shape. Abstracta systematic approach is adopted to extract an effective lowenergy hamiltonian for crystals with a slowly varying inhomogeneity, resolving several controversies. Aug 25, 2008 quantum chemical systems with a position. Firstorder intertwining operators and positiondependent.
We just remember the time dependent schr odinger equation. The morse potential and the pdm,0 mr me r are expanded in the series about the origin 40. Request pdf schrodinger equation with positiondependent mass generally speaking, the effective mass is taken as a constant in the traditional wave equations. Exactly solvable models play an extremely important role in many fields of quantum mechanics. Remarks on the solution of the position dependent mass schrodinger equation. Exact solution of the schrodinger equation for a particle. Analytic results in the positiondependent mass schrodinger problem by m.
Schrodinger equation with positiondependent mass request pdf. Properties of quasioscillator in positiondependent mass formalism. Schrodingers famed equation, for a single particle in the 3dimensional case in the presence of a potential v. Dirac equation with positiondependent effective mass and.
Indeed, many of the important questions in the theory of solids concern. The most general form is the timedependent schrodinger equation tdse, which gives a description of a system evolving with time 143. Effective schrodinger equation with general ordering. The wave equation is a partial differential equation that may constrain some scalar function u u x1, x2, xn.
Since the positiondependent mass operator and the momentum operator no longer commute, the generalization of the standard hamiltonian is not trivial. Displacement operator for quantum systems with position. For nonrelativistic quantum physics the basic equation to be solved is the schr odinger equation. We analyze the behavior of the wavefunctions for scattered states in light of the parameters involved.
Equivalence between the position dependent effective mass schrodinger equation and quasiexactly solvable hamiltonians. Schrodinger equation, position dependent mass, heun equation 1 introduction the study of the schrodinger equation with a position dependent mass pdm has been a matter ofinterest since the early days of solid state physics. The schrodinger equation with positiondependent mass. In classical mechanics the motion of a particle is usually described using the timedependent position ixt as the dynamical variable. Analytic results in the positiondependent mass non. Wavepacket revival for the schrodinger equation with position. Pdf using a recently developed technique to solve the schroinger equation for constant mass, we have studied the regime in which the mass varies with. Present the solutions of this system with hyperbolic potential. Analytical solution to position dependent mass schrodinger equation article pdf available in journal of modern optics 58. Analytical solution to position dependent mass for 3d. Comparison theorems for the positiondependent mass. Bardapurkar 32 introduction quantum mechanics is an essential part of undergraduate syllabus in physics as well as in chemistry. The positiondependent effective mass schrodinger equation exhibiting a similar position dependence for both the potential and mass is exactly solved.
C timedependent states electron wavepacket and group velocity effective mass case of low energy electrons 9. The method includes a coordinate transformation ct followed by functional transformation ft of wave function. From the hamiltonian equations to the schrodinger equation. Exactly solvable potentials esps of position dependent mass pdm schrodinger equation are generated from hulthen potential parent system by using extended transformation et method. Physics 216 spring 2012 quantum mechanics of a charged. Analytical solution to position dependent mass schr\ odinger equation. Exact solution of position dependent mass schrodinger equation by supersymmetric quantum mechanics. The one dimensional time dependent schrodinger equation for a particle of mass m is. In classical mechanics the motion of a particle is usually described using the time dependent position ixt as the dynamical variable. We show that the parameters of the potential play a crucial role. Using a recently developed technique to solve schrodinger equation for constant mass, we stud ied the regime in which mass varies with. Dirac equation with position dependent effective mass and we study the onedimensional non hermitian imaginary potential with a real energy spectrum in the frame work of the. The above equation finally takes the form 4 we have considered a real quantum mechanical potential v a x, which is termed as aquantum system aqs. The quantity u may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating.
Generation of exactly solvable potentials of position. Equivalence between the positiondependent effective mass schrodinger equation and quasiexactly solvable hamiltonians. The positiondependent effective mass schrodinger equation exhibiting a similar position dependence for both the potential and mass is. Pdf comparison theorems for the positiondependent mass. Presuming that the wavefunction represents a state of definite energy e, the equation can be separated by the requirement. Analytical solution to position dependent mass for 3dschrodinger equation. By inserting a continous cons of position eigenstates eq. Solutions of the nonrelativistic wave equation with position. This wavefunction depends on position and on time and it is a complex number.
In this study, we introduce a new approach to construct the schrodinger equation with positiondependent mass characterized by an emerging particles effective mass, in perfect analogy with the problems involving a positiondependent mass particle in semiconductor heterostructures. For a free particle the time dependent schrodinger equation takes the form. Mass function of parent system gets transformed to that of generated system. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass schrodinger equation corresponding to a potential and mass function. In the present paper we consider the quantum differential equations resulting from position dependent mass pdm particles. Dirac equation with positiondependent effective mass and solvable potentials in schrodinger equation by. B the dependence of amplitude probability on position.
The problem of ddimensional schrodinger equations with a position dependent mass is analyzed in the framework of firstorder intertwining operators. Schrodinger equation with positiondependent mass springerlink. For derivation of the pdm schrodinger equation see k. Generally speaking, the effective mass is taken as a constant in the traditional wave equations. We have obtained the reflection as well as transmission coefficients and current density for both potentials. Versions there are several equations that go by schrodingers name. Our first theorem states that the corresponding eigenvalues of a constantmass hamiltonian and of a bendanielduke pdm hamiltonian with the same potential are ordered if the constant and positiondependent masses are ordered everywhere.
Analytical solution to position dependent mass schrodinger equation. Pdf analytical solution to position dependent mass. Fisher information for the positiondependent mass schrodinger. In quantum mechanics, the pdm schrodinger equation has been already. A novel approach for constructing kinetic energy operators with. Schrodinger equation, positiondependent mass, heun equation 1 introduction the study of the schrodinger equation with a positiondependent mass pdm has been a matter ofinterest since the early days of solid state physics. The secondorder intertwiner is constructed directly by taking it as secondorder linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship.
Thus one dimensional positiondependent mass 1dpdm schrodinger equation is written as. Changing of ordering in the kinetic energy leads to change of the effective potential energy and leaves. We shall first postulate the wave function for the simplest conceivable system. Schrodinger equation for a charged spin12 particle. Comparison theorems for the positiondependent mass schrodinger equation. Alan doolittle lecture 6 schrodinger equation and relationship to electron motion in crystals reading. Pdf exact solutions of the positiondependenteffective. Kulikov theoretical physics department, ffeks, dniepropetrovsk national university, 72 gagarin avenue, dniepropetrovsk 49010, ukraine correspondence should be addressed to d. Since the position dependent mass operator and the momentum operator no longer commute, the generalization of the standard hamiltonian is not trivial. The wave equation in one space dimension can be written as follows. Exactly solvable potentials esps of positiondependent mass pdm schrodinger equation are generated from hulthen potential parent system by using extended transformation et method. Effective hamiltonian with positiondependent mass and. In this work, the point canonical transformation method applied to schrodinger equations with a position.
The form of the schrodinger equation depends on the physical situation see below for special cases. C time dependent states electron wavepacket and group velocity effective mass case of low energy electrons 9. The schrodinger equation with a positiondependent mass sepdm is employed in many areas of quantum physics. Shape invariance was also addressed in this setting and the energy spectra were. The traditional approach is to expand the wavefunction in a set of traveling waves, at least in the asymptotic region. Exact solutions of the positiondependenteffective mass. Thermodynamic properties, information entropy, and uncertainty in coordinate and momentum spaces are calculated. However the derivation of the excitedstate wavefunctions in. In wave mechanics the dynamical variable is a wavefunction.
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