PT-symmetric Solutions of Schrodinger Equation with position-dependent mass via Point Canonical Transformation
aa r X i v : . [ qu a n t - ph ] S e p PT-symmetric Solutions of Schr¨odinger Equation withposition-dependent mass via Point CanonicalTransformation
Cevdet TezcanFaculty of Engineering, Ba¸skent University, Baglıca Campus, Ankara, TurkeyRamazan Sever ∗ Department of Physics, Middle East Technical University, 06531 Ankara, TurkeyNovember 27, 2018
Abstract
PT-symmetric solutions of Schr¨odinger equation are obtained for the Scarf and gener-alized harmonic oscillator potentials with the position-dependent mass. A general pointcanonical transformation is applied by using a free parameter. Three different forms ofmass distributions are used. A set of the energy eigenvalues of the bound states andcorresponding wave functions for target potentials are obtained as a function of the freeparameter.PACS numbers: 03.65.-w; 03.65.Ge; 12.39.FdKeywords: Position-dependent mass, Point canonical transformation, Effective mass Schr¨odingerequation, generalized harmonic oscillator, Scarf potential ∗ Corresponding author: [email protected] Introduction
Exact solutions of the effective mass Schr¨odinger equation(SE) for some physical potentialshave received much attention. Important applications are obtained in the fields of materialscience and condensed matter physics such as semiconductors [1], quantum well and quantumdots [2], H clusters [3], quantum liquids [4], graded alloys and semiconductor heterostructures[5,6]. Recently, number of exact solutions on these topics increased [6-23]. Various solutionmethods are used in the calculations. The point canonical transformations (PCT) is one ofthese methods providing exact solution of energy eigenvalues and corresponding eigenfunctions[24-28]. It is also used for solving the Schr¨odinger equation with position-dependent effectivemass for some potentials [8-13]. In the present work, we solve three different potentials withthe three mass distributions. The point canonical transformation is taken in the more generalform introducing a free parameter. This general form of the transformation will provide us aset of solutions for different values of the free parameter.On the other hand, there has been considerable work on non-Hermitian Hamiltomians inrecent years. Much attentioan has also been focused on PT -symmetric Hamiltonians. Followingthe early studies of Bender et al . [29], the PT -symmetry formulation has been successfullyutilized by many authors [30-36]. The PT -symmetric but non-Hermitian Hamiltonians havereal spectra whether the Hamiltonians are Hermitian or not. Non-Hermitian Hamiltonianswith real or complex spectra have also been analyzed by using different methods [31-34,37].Non-Hermitian but PT -symmetric models have applications in different fields, such as optics[38], nuclear physics [39], condensed matter [40] and population biology [41]. There are somerecent works on these topics[42-44].The contents of the paper is as follows. In section 2, we present briefly the solution of theSchr¨odinger equation by using point canonical transformation. In section 3, we introduce someapplications for specific potentials. Results are discussed in section 4.2 Method
We write the most general form of the Hamiltonian as [45] H = 14 h m η pm λ pm µ + m µ pm λ pm η + V ( x ) i (1)where the parameters η, λ and µ are called the ambiguity parameters. They are constraintby the relation η + λ + µ = −
1. Different forms of the Hamiltonian are used in the literaturedepending on the choices of set of parameters. Here, we take the set such that η = µ = 0, λ = 1[46,47] introduced by Ben Danie-Duke. Thus, the Hamiltonian is invariant under instantaneousGalilean transformation [47]. The Schr¨odinger equation with ¯ h = 1 takes the form − ddx " m ( x ) d Ψ dx + V ( x )Ψ( x ) = E Ψ( x ) . (2)The wave function should be continuous at the mass discontinuity and verify the followingcondition 1 m ( x ) d Ψ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − = 1 m ( x ) d Ψ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + . (3)On the other hand a Hamiltonian is said to be PT-symmetric if[ P T, H ] = 0 (4)where P is the parity operator and T is the time reversal operator. They act on the positionand momentum states as P : x → − x, P → − P and T : x → x, P → − P, i → − i. (5)Thus, we get the following conditions to have a PT-symmetric Hamiltonian m ( x ) = m ( − x ) and V ∗ ( − x ) = V ( x ) . (6)3e shall use a general form of PCT with a free parameter to solve the Schr¨odinger equationfor any potential V ( x ). Defining the transformation with a free parameter β Ψ( x ) = m β ( x ) φ ( x ) (7)The SE takes − m φ ′′ + (2 β − m ′ m φ ′ + β ( β − m ′ m ! φ + β m ′′ m ! φ + V ( x ) φ = Eφ. (8)It is solved by Roy [42] for β = . Roy obtained this form by making the transformation y = Z x m ( t ) dt. (9)In the computations, three different position-dependent mass distributions[47] will be used.The reference potentials are PT-symmetric Scarf II[49,50] and generalized generalized harmonicoscillator[51] potentials. We will consider two different values of β . β = β = , Eq. (8) has the following compact form − d φdx + Ω( x ) φ ( x ) = Eφ ( x ) (10)where Ω( x ) = 38 m ′ m ! − m ′′ m ! + mV − ( m − E (11)Eq.(10) has the same form with the constant mass SE. m ( x ) = (cid:18) α + x x (cid:19) i) The Scarf II potential is 4( x ) = − λsech ( x ) − iµsech ( x ) tanh( x ) (12)The energy eigenvalues and corresponding wave functions are E n = − ( n − p − , n = 0 , , , ... < s + t −
12 (13)and φ n ( x ) = Γ( n − p + ) n ! Γ( − p ) z − p ( z ∗ ) − q P ( − p − , − q − ) n ( i sinh( y )) (14)where z = 1 − i sinh( x )2 , (15) p = − ± q + λ + µ ,= − ± t , (16) q = − ± q + λ − µ = − ± s . (17)For the Scarf II potential and the position-dependent mass case, we obtain the effective potentialas V ( x ) = x α + x ! ( Ω( x ) − − α ) x (1 + x ) )( α + x ) + (1 − α )(1 − x )(1 + x ) ( α + x ) + " ( α + x x ) − E ) (18)5here m ′ m = 4(1 − α ) x (1 + x )( α + x ) (19)and m ” m = 4(1 − α )(1 + x ) ( α + x ) h − αx ) + ( α + x )(1 − x ) i . (20)ii) PT-symmetric generalized harmonic oscillatorΩ( x ) = ( x − iε ) + g − ( x − iε ) . (21)The energy eigenvalues and corresponding wave functions are E n = 4 n − qg + 2 , n = 0 , , , ... (22)and φ n ( x ) = e − ( x − iε ) ( x − iε ) − pg + L − qgn (( x − iε ) ) (23)where q = ± V ( x ) = x α + x ! Ω( x ) − m ′ m ! + 14 m ′′ m ! + α + x x ! − E (24)where m ′ m and m ′′ m are given in Eqs. (20 , M ( x ) = (cid:18) α + x x (cid:19) = m i)Scarf II potentialThe form of the Scarf II potential is given in Eq. (13).where m ′ m , m ′′ m and Ω( x ) are given in Eqs. (20 , , = − ( n − p − n = 0 , , , ... < s + t −
12 (25)ii) PT-symmetric generalized oscillatorIt is given in
Eq. (22)The potential is given in
Eq. (13) V ( x ) = x α + x ! { Ω( x ) − ( m ′ m ) + 12 ( m ′′ m ) + [( α + x x ) − E } (26) V ( x ) = x α + x ! Ω( x ) − m ′ m ! + 12 m ′′ m ! + α + x x ! − E (27)where m ′ m , m ′′ m and Ω( x ) are given in Eqs. (20 , , φ n ( y ) = Γ( n − p + ) n ! Γ( − p ) z − p ( z ∗ ) − q P ( − p − , − q − ) n ( i sinh( y )) (28)where z, p and q are defined in Eqs. (16 , , β = − γ y = Z x m γ ( t ) dt. (29)Then, Eq. (3) takes the form − m m γ φ ′′ + (cid:18) γ β − m γ − m ′ (cid:19) φ ′ + β ( β − m ′ m ! φ + β m ′′ m ! φ + V φ = Eφ (30)To remove the term involving first derivative of the wave function, we impose γ + 2 β − β to get the exact solution.Thus, we get − φ ′′ + Ω( y ) φ = Eφ (31)where Ω( y ) = − β m − γ ( β − m ′ m ! + m ′′ m + ( V − E ) m − γ + E (32)and also V ( x ) = m γ − Ω( y ) + 2 − γγm γ − γ + 64 m ′ m ! + m ′′ m + E (cid:16) − m − γ (cid:17) . (33) m ( x ) = (cid:18) α + x x (cid:19) γ The new independent variable is y = h x + ( α −
1) tan − ( x ) i . (34)i) Scarf II potentialThe scarf II potential for the potential and position dependent mass has the form V ( x ) = m γ − Ω( y ) + 2 − γ m γ − γ + 64 m ′ m ! + m ′′ m + E (cid:16) − m − γ (cid:17) (35)where m ′ m and m ′′ m are given in Eqs. (35 , m ′ m = 4(1 − α ) γ x (1 + x )( α + x ) (36)and m ′′ m = 4(1 − αγ (1 + x ) ( α + x ) h k − − α ) x + ( α + x )(1 − x ) i (37)8here k = γ ii) PT-symmetric generalized oscillatorSolution of the SE for the PT-symmetric generalized oscillator, Eq. (22), gives us energyeigenvalues and corresponding wave functions as E n = 4 n − qρ + 2 , n = 0 , , , ... (38)and φ n ( y ) = e − ( y − iε ) ( y − iε ) − pρ + L − qρn (cid:16) ( y − iε ) (cid:17) (39)where q = ± We have applied the point canonical transformation in a general form by introducing a freeparameter to solve the Schr¨odinger equation for the Rosen-Morse and Scarf potentials withspatially dependent mass. We have obtained a set of exactly solvable target potentials byusing two position-dependent mass distributions. Energy eigenvalues and corresponding wavefunctions for the target potentials are written in the compact form.
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