Phase-Space Matrix Representation of Differential Equations for Obtaining the Energy Spectrum of Model Quantum Systems

Home , Integrating factor

arXiv:2108.11487v1 [quant-ph] 25 Aug 2021 a) systems quantum model of equatio spectrum differential of representation matrix Phase-space lcrncmi:[email protected] mail: Electronic ewrs hs-pc,mdlqatmsses nryspec energy systems, quantum model phase-space, Keywords: ato Hydrogen the rotor, oscillator. rigid the oscillator, harmonic the as Schrödingermodels e independent time the differe and ordinary physics the mathematical between the relationship the of prese method understanding the the addition, In relations. recurrence textboo with methods in Schrödinger equat time-independent shown the approaches of analyses some asymptotic simplifies m presented eigenfunctio quantum method and for eigenvalues The Schrödinger equation the independent find time to dimensional method a developed we ordin tions order second of representation phase-space the Employing 2021) August 27 (Dated: Cali, Icesi, Universidad Colombia Sciences, Chemical of Department Arango A. Carlos Morales, C. Juan 1, a) 1 sfrotiigteenergy the obtaining for ns trum o,adpwrseries power and ion, r ieeta equa- differential ary tdhr facilitates here nted uto fphysical of quation ta qain of equations ntial ,adteMorse the and m, dlsystems. odel s ae on based ks, so h 1- the of ns I. INTRODUCTION

The 1-dimensional time independent Schrödinger equation (TISE) can be solved analyt- ically for few physical models. The harmonic oscillator, the rigid rotor, the Hydrogen atom, and the Morse oscillator are examples of physical models with known analytical solution of the TISE (1). The analytical solution of the TISE for a physical model is usually obtained by using the ansatz of a wavefunction as a product of two functions, one of these functions acts as an integrating factor (2), the other function produces a differential equation solv- able either by Frobenius series method or by directly comparing with a template ordinary differential equation (ODE) with known solution (3). Examples of template ODEs of physical interest are the Hermite, Laguerre, Legendre, and confluent hypergeometric equations. Before using the wavefunction product ansatz is necessary to write the TISE in terms of a non-dimensional coordinate, and make a transformation of coordinates that depends on the potential energy function. It is important to mention that not always straightforward to find a product wavefunction with an integrating factor that produces a TISE easy to solve by Frobenuis method or comparable with a template ODE with known solution.

The analysis in phase-space of second order ODEs uses to geometrical and mathematical tools that allow to understand qualitatively the behaviour of the solutions of the ODE without having an exact solution (4, 5, 6). The solutions to a 1-dimensional second order ODE in phase-space are trajectories in the 2-dimensional phase-space instead of the 1- dimensional function obtained by solving the ODE in space representation. A stability analysis of the tangent phase-space vector field of an ODE allows to find and classify fixed points, and bound phase-space trajectories (7). The phase-space analysis of the TISE gives the bound states of a quantum systems in terms of the fixed points and the bound trajectories (8).

A 1-dimensional second order ODE can be described in phase-space as a couple of first order ODEs. This set of ODEs in phase-space can be written in matrix fashion by defining a phase-space vector as the concatenation of the function and its space derivative. The resulting matrix equation relates the tangent to the phase-space vector with the product of a2 2 matrix times the phase-space vector (5). Since the TISE for a physical system depends × on a variable which is a physical quantity, with dimensions of length, it is necessary to define a dimensionless variable before using integrating factors and transformation of coordinates

2 to obtain an equation comparable with the template ODEs (9). As a second order ODE, the TISE can be represented in terms of the phase-space wavefunction vector. The matrix representation of the TISE in phase-space allows to use integrating factors that make the matrix form of the TISE comparable with the matrix form of the template ODE. In this work we introduce a method based on the phase-space representation of both, the TISE, and a template second order ODE. The method allows to obtain directly nondimensionalization constants, integrating factors, and the solution of the TISE for model quantum systems. Instead of using a comparison term by term between the TISE and the template ODE, the method of this work is based in an algebraic equation obtained by comparing the phase- space matrix representation of the TISE and a template ODE with known solution in the form of orthonormal polynomials. Section III of this article displays the use of the equations deduced in section II to calculate the energy spectrum and the eigenfunctions of model quantum systems: the harmonic oscillator, the rigid rotor, the hydrogen atom, and the Morse oscillator.

The method developed and presented in this article is limited to finding analytical solutions of 1-dimensional quantum systems, of academic interest, in terms of orthogonal polynomials. A generalization of this method has been used to obtain numerically bound and resonance states of the rotationally excited Hydrogen molecule(8). There are different approaches to find eigenvalues of the TISE using phase-space representation. Phase-space integration based in the Wentzel–Kramers–Brillouin (WKB) method has been used to calculate accurately the eigenvalues of anharmonic oscillators, and square and logartihmic poten- tials (10, 11). Primitive semi-classical methods that use phase-space integration based in the Wilson-Sommerfeld quantization rules provide excellent results for anharmonic oscillators (12).

II. THEORY AND METHODS

A. Phase-space representation of the time independent Schr¨odinger equation

A homogeneous second order linear diﬀerential equation, P (x)y′′(x) + Q(x)y′(x) + y(x) R(x)y(x) = 0, can be written in terms of the phase-space vector y(x)= y′(x) , obtaining 3 a matrix equation y′(x)= D(x)y(x) with matrix D = D(x) given by

0 1 D = . (1)  R(x) Q(x)  − P (x) − P (x)  The 1-dimensional time independent Schr¨odinger equation (TISE) is given by ~2 ψ′′(q)+ B(q)ψ′(q)+ V (q)ψ = Eψ(q), (2) − 2m

The physical coordinate q can be written as q = xcx, with x as a dimensionless coordinate, and xc a constant carrying out the length dimension. The dimensionless TISE is given by

ϕ′′(x)+ b(x)ϕ′(x)+ v(x)ϕ(x)= εϕ(x), (3) −

B(q) V (q) E ~2 with ϕ(x)= ψ(q), b(x) = , v(x)= , ε = , and a = 2 . The dimensionless TISE axc a a 2mxc ϕ can be written in matrix fashion by using the phase-space wavefunction vector ϕ = ϕ′ , ϕ′ = Aϕ, (4)

with A = A(x) is given by 0 1 A =   , (5) k2(x) b(x) −  and k2(x)=(ε v(x)). − It is convenient to write the wave function ϕ as a product ϕ = fg of two functions

f = f(x) and g = g(x), with derivative ϕ′ =(f ′g + fg′). The product function ϕ = fg, and f its derivative give a matrix equation for ϕ in terms of the vector f = f ′ , ϕ = Bf, (6)

with matrix B = B(x) given by g 0 B =   . (7) g′ g   The x-derivative of ϕ = Bf is given by

ϕ′ = B′f + Bf ′, (8)

with

g′ 0 B′ =   . (9) g′′ g′   4 The use of ϕ = Bf in equation (4) gives ϕ′ = ABf. This result for ϕ′ can be used in the left-hand-side of equation (8) to obtain ABf = B′f + Bf ′, which can be solved for f ′ giving

1 f ′ = B− (AB B′) f − (10) = Cf,

1 with the matrix C = B− (AB B′) given explicitly by − 0 1 C =  2 ′ ′′ ′  . (11) gk bg +g 2g − b − g − g  Matrices C and D, equation (11) and (1) respectively, can be equaled to obtain a set of ODEs for the integrating factor g 2 gk bg′ + g′′ R − = , (12) − g −P 2g Q b ′ = . (13) − g −P Integration of equation (13) gives the function g in terms of the known functions P , Q and b, Q + bP g = exp dx . (14) Z 2P Equations (12) and (13) are combined to obtain the algebraic equation b b2 k2 + ′ = G(x), (15) 2 − 4 2 ′ ′ Q 2QP +2P (Q 2R) 2 with G(x)= − 2 − . Since k is a function of the energy E, equation (15) can − 4P be used to obtain the energy eigenvalues of the TISE. The left hand side (LHS) of equation (15) depends on the coefficients of the TISE, meanwhile the right hand side (RHS) depends on the coefficients of a template equation. Examples of template ODEs employed in physics with the respective G functions are shown in the second line of the first column and the second column, respectively, of table I.

The template ODEs P (x)y′′(x) + Q(x)y′(x) + R(x)y(x) = 0 usually have polynomial solutions y(x)= pλ,ν(x) of degree λ and order ν. Examples polynomial solutions for template ODEs, and the mathematical conditions for the polynomial solutions, are given in the third line of the ﬁrst column of table I. The integrating factor g(x) of equation (14), and the polynomial solutions pλ,ν(x) of the template ODE give the solution of the TISE

ψ(q(x)) = Nλ,νg(x)pλ,ν(x), (16) with Nλ,ν as a normalization constant.

5 Template ODE G function

Hermite y 2xy + 2λy = 0 G(x)=1+2λ x2 ′′ − ′ − x ( , ), y = H (x); λ Z, λ 0 ∈ −∞ ∞ λ ∈ ≥ Associated Legendre

2 2 2 2 m m 1+(x 1)(l+1)l (1 x )y′′ 2xy′ + l(l + 1) 1 x2 y = 0 G(x)= − (x2 −1)2 − − − − − − x [ 1, 1], y = P m(x); l,m Z, l m l ∈ − l ∈ − ≤ ≤ Polar Associated Legendre 1 2 m2 1 ( /4 m ) y θ θy θ l l 2 y θ G θ l l −2 ′′( ) + cot ′( )+ ( + 1) sin θ ( ) = 0 ( )= 4 + ( +1)+ sin θ − y = P m(cos θ); l,m Z, l m l l ∈ − ≤ ≤ Associated Laguerre

1 1+ν+2λ 1 ν2 xy + (ν + 1 x)y + λy = 0 G(x)= + + − 2 ′′ − ′ − 4 2x 4x x [0, ), y = Lν (x); λ, ν Z, λ 0, ν > 0 ∈ ∞ λ ∈ ≥ Conﬂuent Hypergeometric

1 c 2a c(2 c) xy + (c x)y ay = 0 G(x)= + − + −2 ′′ − ′ − − 4 2x 4x F (a, c, x); a Z, a 0, c> 0 1 1 ∈ ≤ TABLE I. Examples of template ODEs used in physics with their respective polynomial solutions, and G functions.

III. APPLICATIONS

A. The harmonic oscillator

In terms of the dimensionless coordinate x R, the TISE for the harmonic oscillator ∈ ~2 1 2 2 2 ′′ 2 ϕ (x)+ 2 mω xc x ϕ(x)= Eϕ(x), (17) − 2mxc has b = 0, and 2mx2 k2 = c E 1 mω2x2x2 . (18) ~2 − 2 c The value of k2 of equation (18) can be used on the LHS of equation (15) producing a quadratic term in x that can be cancelled only by using the Hermite G(x) of table I , 2mx2 c E 1 mω2x2x2 =1+2λ x2. (19) ~2 − 2 c − 6 The equality in (19) is hold if

2mx2 c E =1+2λ, (20) ~2 m2ω2x4 c x2 = x2. (21) − ~2 −

~ 1/2 Equation (21) gives the constant xc = mω . Substitution of this value of xc in equation (20) gives the energy E = ~ω (λ + 1/2). The integrating factor g of equation (14) can be obtained for the harmonic oscillator and x2/2 the Hermite equation, g(x) = e− . Bound solutions of the Hermite diﬀerential equation are given by the Hermite polynomials y = H (x) with λ Z and λ 0. The bound solution λ ∈ ≥ of the Harmonic oscillator is given by

q2 q ψλ(q)= Nλ exp 2x2 Hλ x , (22) − c c with Nλ as a normalization constant.

B. The Rigid Rotor

In spherical coordinates, the Schr¨odinger equation for a rigid rotor is given by

~2 1 ∂ 1 ∂2 sin θ + ψ(θ,φ)= Eψ(θ,φ), (23) − 2µR2 sin θ ∂θ sin2 θ ∂φ2 with µ and R as the reduced mass and bond length of the rotor, respectively. The use of a wave function as a product ψ(θ,φ)=Θ(θ)Φ(φ) gives

1 d dΘ sin θ sin θ + β sin2 θ = C, (24) Θ dθ dθ 1 d2Φ = C, (25) −Φ dφ2 with C as a constant and the dimensionless β = 2µR2E/~2. Normalized 2π-periodic solutions of (25) are given by Φ(φ) = 1 eimφ with m Z. The use of Φ(φ) = 1 eimφ in √2π ∈ √2π equation (25) produces C = m2; the use of this C in equation (24) produces

d2Θ dΘ β sin2 θ m2 + cot θ + − Θ=0. (26) dθ2 dθ sin2 θ

The use of the vector Θ = Θ, Θ′ allows to write this equation in matrix fashion, { } Θ′ = A(θ)Θ. The matrix A has exactly the same form of equation (5) with b = cot θ and − 7 2 2 2 β sin θ m 2 2− k = sin θ . The use of these values of b and k on the RHS of equation (15) gives

2 1 1/4 m + β + − = G(θ). (27) 4 sin2 θ

The LHS of this equation matches with the G(θ) of the polar form of the Associated Legendre 2IE equation with β = l(l + 1). Recalling that β = ~2 , the energy of the rigid rotor is given by ~2 El = 2I l(l + 1). Finally, the use of equation (14) gives

g(θ)=1. (28)

m m The solution of equation (26) is given by Θl,m(θ)= Nl,mPl (cos θ) with Pl as an associated

Legendre polynomial, and Nl,m a normalization constant. The full solution for the rigid rotor, equation (23), is given by

m imφ ψ(θ,φ)= Nl,mPl (cos θ)e (29) m = Yl (θ,φ),

m with Yl (θ,φ) as the spherical harmonic functions.

C. The radial equation for the Hydrogen atom

In terms of the dimensionless coordinate ρ, related to the physical coordinate r by r = rcρ, the dimensionless radial Schr¨odinger equation for Hydrogen is given by

2 2 R′′(ρ)+ R′(ρ)+ k (ρ)R(ρ)=0, (30) ρ l having 2 2 rc E 2rc l(l + 1) kl = 2 + 2 , (31) −a0Eg a0ρ − ρ

2 2 4πǫ0~ ~ with l a non-negative integer, a0 = 2 , and Eg = 2 , as the Bohr radius and mee − 2mea0 ground state energy, respectively. The coeﬃcient of the ﬁrst derivative of equation (30)

gives b = 2/ρ. The use of b = 2/ρ and the Associated Laguerre G(ρ) of table I in equation − − (15) gives 2 2 rc E 2rc l(l + 1) 1 1 ν 1+ ν +2λ 2 + 2 = + − 2 + . (32) − a0Eg a0ρ − ρ −4 4ρ 2ρ

8 The equality of (32) leads to

2 rc E 1 2 = , (33) −a0Eg −4 2r 1+ ν +2λ c = , (34) a0ρ 2ρ l(l + 1) 1 ν2 = − . (35) − ρ2 4ρ2

1 2 /2 a0Eg Equation (33) gives rc = 4E , equation (35) gives ν =2l + 1. Finally, using the results Eg of rc and ν in equation (34) produces E = n2 with n =1+ l + λ. Using the quantized value Eg a0n of the energy, E = n2 , the constant rc changes to rc = 2 . Polynomial solutions of the associated Laguerre ODE are obtained for λ and ν non negative integers, since l is a non-negative integer, n =1+ l + λ must be a positive integer, and l can take any value 0 l n 1. ≤ ≤ − l ρ/2 Finally, by using equation (14) the integrating factor g results in g(ρ) = ρ e− , which gives for the Hydrogen atom wave functions

l r 2r 2l+1 2r − na0 2r Rn,l( na0 )= Nn,l e Ln l 1( na0 ), (36) na0 − −

2l+1 with Ln l 1 as an associated Laguerre polynomial, and Nn,l as a normalization constant. − −

D. The Morse Oscillator

The dimensionless TISE for the Morse potential is given by

2 ~ 2αxcx αxcx 2 ϕ′′(x)+ De e− 2e− ϕ(x)= Eϕ(x), (37) − 2mxc −

2√2mDe αxcx with x R, α> 0, and D > 0. The transformation of coordinates y = e− gives ∈ e α~ 1 ε δ 1 χ′′(y)+ χ′(y)+ + χ(y)=0, (38) y y2 y − 4

2mE √2mDe with y > 0, χ(y) = ϕ(x), ε = α2~2 , and δ = α~ . The dimensionless TISE (38) leads 2 ε δ 1 2 to k = y2 + y 4 , and b(y) = 1/y. The use of this values of b and k in the LHS of − − equation (15), and the conﬂuent hypergeometric G, in the RHS of equation (15) produces

1+4ε δ 1 1 c 2a c(2 c) + = + − + − . (39) 4y2 y − 4 −4 2y 4y2

9 The equality (39) holds if

1+4ε = c(2 c), (40) − c 2a δ = − . (41) 2

Equations (40) and (41) give a = 1 δ √ ε and c = 1 2√ ε. Polynomial solutions 2 − ± − ± − of the confluent hypergeometric differential equation are obtained for non positive integer a and nonegative c. The number of bound solutions must increase as the value of δ increases. These requirements are fulfilled by a = 1 δ + √ ε and c = 1+2√ ε, which give 2 − − − 2 ε = δ + a 1 , and the energy eigenvalues are − − 2 2 n + 1/2 E = D 1 , (42) n − e − δ

with n = a. Bound states of the Morse oscillator have nonegative integer values of n such − that E > D . n − e y/2 √ ε The integrating factor g = g(y) of equation (14) is given by g = e− y − , and the full Morse wavefunction is

y/2 √ ε ψ(y)= Nn,ce− y − 1F1(n,c,y), (43)

2√2mDe αxcx with y = e− , c =1+2√ ε, N a normalization constant, and F (n,c,y) the α~ − n,c 1 1 conﬂuent hypergeometric function of the ﬁrst kind.

IV. CONCLUSIONS

The results obtained in section III have shown that the equations developed in section II work exactly for model quantum systems. Equations (14) and (15) establish a connection between the TISE of a model system and a template ODE, these equations only require the coeﬃcient functions of the TISE and the template ODE. The method presented here avoid the ﬁnding of nondimensionalization parameters and integrating factors based on asymptotic analyses of the TISE. The integrating factors are obtained directly from the equations of this method by using a simple factorization ansatz of the wavefunction. In general terms, the use of the method presented in this work facilitates the understanding of the relationship between the ODEs of the mathematical physics and the TISE for model quantum systems. The method presented in this work avoids the use of Frobenius or algebraic ladder operator

10 methods to obtain the energy spectrum of the TISE, instead, it resorts to elements from algebra, linear algebra, and integral and diﬀerential calculus, that are easier to use and reach by undergraduate students in courses of Quantum Mechanics or Physical Chemistry.

V. ACKNOWLEDGMENTS

This project has been fully ﬁnanced by the internal research grants of University Icesi.

REFERENCES

1P. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 5th ed. (Oxford, New York, NY, USA, 2011). 2W. Derrick and S. I. Grossman, Elementary Differential Equations, 4th ed. (Pearson, 1997). 3M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill book company, 1953). 4C. M. Falco, American Journal of Physics 44, 733 (1976). 5V. Arnol’d, Ordinary Differential Equations (Springer-Verlag, Heidelberg, Germany, 1992). 6J. H. Hubbard, J. M. McDill, A. Noonburg, and B. H. West, The College Mathematics Journal 25, 419 (1994). 7S. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA, 1994). 8J. S. Molano and C. A. Arango, Chemical Physics Letters 762, 138171 (2021). 9H. Langtangen and G. Pedersen, Scaling of Differential Equations, 1st ed. (Springer Inter- national Publishing, 2016). 10S. Nagabhushana, B. A. Kagali, and S. Vijay, American Journal of Physics 65, 563 (1997). 11W. N. Mei, American Journal of Physics 66, 541 (1998). 12S. Mukhopadhyay, K. Bhattacharyya, and R. K. Pathak, International Journal of Quantum Chemistry 82, 113 (2001).