European Journal of Physics       PAPER Revisiting the quantum harmonic oscillator via unilateral Fourier transforms To cite this article: Pedro H F Nogueira and Antonio S de Castro 2016 Eur. J. Phys. 37 015402   View the article online for updates and enhancements. 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Revisiting the quantum harmonic oscillator via unilateral Fourier transforms Pedro H F Nogueira and Antonio S de Castro1 UNESP, Campus de Guaratinguetá, Departamento de Física e Química, 12516–410 Guaratinguetá, SP, Brazil E-mail: pedrofusconogueira@gmail.com and castro@pq.cnpq.br Received 21 September 2015, revised 10 November 2015 Accepted for publication 20 November 2015 Published 9 December 2015 Abstract The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integr- able eigenfunctions. Keywords: Fourier transform, harmonic oscillator, Fourier sine and cosine transform 1. Introduction The integral transform methods are useful and powerful methods of solving ordinary linear differential equations because they can convert the original equation into a simpler differential equation or into an algebraic equation. Nevertheless, the inversion of the transform for reconstructing the original function may be a rather complicated calculation. If this is the case and one does not find the integral in tables, the method will be worthless. The harmonic oscillator is ubiquitous in the literature on quantum mechanics because it can be solved in closed form with a variety of methods and its solution can be useful for approximations or exact solutions of various problems. The quantum harmonic oscillator is usually solved with the help of the power series method [1], by the algebraic method based on the algebra of operators [2], or by the path integral approach [3] . In recent times, the one- dimensional harmonic oscillator has also been approached by the exponential Fourier transform [4–7] and Laplace transform [8] methods. However, all of the accounts of the quantum harmonic oscillator via the exponential Fourier transform present calculations which are based on false basic assertions. European Journal of Physics Eur. J. Phys. 37 (2016) 015402 (8pp) doi:10.1088/0143-0807/37/1/015402 1 Author to whom any correspondence should be addressed. 0143-0807/16/015402+08$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1 mailto:pedrofusconogueira@gmail.com mailto:castro@pq.cnpq.br http://dx.doi.org/10.1088/0143-0807/37/1/015402 http://crossmark.crossref.org/dialog/?doi=10.1088/0143-0807/37/1/015402&domain=pdf&date_stamp=2015-12-09 http://crossmark.crossref.org/dialog/?doi=10.1088/0143-0807/37/1/015402&domain=pdf&date_stamp=2015-12-09 The observation that the eigenfunction and its Fourier transform satisfy formally identical differential equations and identical boundary conditions at infinity led Muñoz [4] to conclude that the eigenfunction and its corresponding Fourier-transformed function differ at most by a proportionality constant. His critical flaw was not to consider that the eigenfunction and its corresponding Fourier transform are functions of different but interrelated variables and that there is a definite scaling property involving the pair of Fourier transforms: f cx F k c c .{ ( )} ( ) = Ponomarenko [5] stated that the necessary and sufficient condition for the eigenfunction to have a definite parity can be expressed in terms of the solution of 1 1z( )- =  , but he missed the fact that this equation has many more solutions than those which have z expressed by integer numbers. Engel [6] came to the conclusion that Pono- marenko’s method may be used ‘if the requirement of definite parity of the eigenstates is replaced by that of normalization’. In addition, he argued that Ponomarenko failed to consider a Fourier-transformed function valid on the whole axis because the origin is a singular point of the corresponding Fourier-transformed equation. However, he himself apparently failed to observe that the transformed equation allows Fourier-transformed solutions with definite parities despite the mentioned singularity, and that the changing k to k- makes k a even (odd) when a is an even (odd) integer. It should be mentioned, though, that Engel perceived that the relation between the nth moment of a function and the nth derivative of its transform at the origin plays an indispensable role for determining the bound-state solutions. Palma and Raff [7] developed a strategy for approaching the stationary states of the time-independent Schrödinger equation with a large class of potentials, but they erroneously returned to single valuedness of the eigenfunction in order to eliminate irrelevant overall phase factors. In view of the fact that there are significant confusions regarding the quantum harmonic oscillator via the exponential Fourier transform method, we will revise the problem with the closely related unilateral Fourier (sine and cosine) transform method. Except for the one- dimensional double δ-function potential [9], this method does not seem to have been directly applied to the Schrödinger equation. We will show that the unilateral Fourier transform is a straightforward and efficient manner in which bound-state solutions in nonrelativistic quantum mechanics can be treated by applying it to the harmonic oscillator. We will show that the relation between the convergence of the nth moment of the eigenfunction (in the sense of a conveniently weighted integral) and the derivatives of the corresponding Fourier-trans- formed function at the origin, already perceived by Engel in connection with the exponential Fourier transform [6], is inept at finding the unique solution of the problem. We will also show that both definite parity and square integrability of the eigenfunctions are requisites just sufficient to determinate the proper solution. To prepare the ground, we will first give a short review of a few relevant properties of the Fourier sine and cosine transforms. 2. Fourier transforms and their main properties The exponential Fourier transform pair is defined by [10–17] x x x x 1 2 d e 1 2 d e . 1 x x i 1 i { ( )} ( ) ( ) { ( )} ( ) ( ) ( )   ò ò f k p f k f p k k = F = F = = F k k -¥ +¥ + - -¥ +¥ - For odd ( x xs s( ) ( )f f- = - + ) and even ( x xc c( ) ( )f f- = + + ) functions there are two modifications of the exponential Fourier transform. Defining xz = and k ,k= the Fourier sine and cosine transforms of the functions s ( )f z and c ( )f z are expressed by Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 2 k k k k 2 d sin 2 d cos , 2 s s s 0 s c c c 0 c { ( )} ( ) ( ) { ( )} ( ) ( ) ( )   ò ò f z p z f z z f z p z f z z = F = = F = ¥ ¥ and the inversions are accomplished by means of k k k k k k k k 2 d sin 2 d cos . 3 s 1 s s 0 s c 1 c c 0 c { ( )} ( ) ( ) { ( )} ( ) ( ) ( )   ò ò f z p z f z p z F = = F F = = F - ¥ - ¥ Given sf and ,cf a sufficient condition for the existence of the unilateral Fourier transforms (and their inverses) is ensured if sf and cf ( sF and cF ) are absolutely integrable on 0, .[ )¥ In particular, sf and cf ( sF and cF ) must vanish as z  ¥ (k  ¥). Furthermore, the unilateral Fourier transform pairs satisfy Parseval’s formulas k k k k d d d d . 40 s 2 0 s 2 0 c 2 0 c 2 ∣ ( )∣ ∣ ( )∣ ∣ ( )∣ ∣ ( )∣ ( ) ò ò ò ò z f z z f z = F = F ¥ ¥ ¥ ¥ It is instructive to note that the inversion of the unilateral Fourier transforms requires that sf and cf satisfy different boundary conditions at the origin, namely lim 00 sf =z and lim d d 0.0 cf z =z The convenience of using the sine or cosine transform is dictated by these boundary conditions. Note also that k k k k k k lim d d 1 2 d lim d d 1 2 d lim d d lim d d 0, 5 n n n n n n n n n n n n 0 2 1 s 2 1 0 2 1 s 0 2 c 2 0 2 c 0 2 s 2 0 2 1 c 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ò ò f z z p f z z p f z z f z z = - F = - F = = z z z z  + + ¥ +  ¥   + + and k k k k k k k k lim d d 1 2 d lim d d 1 2 d lim d d lim d d 0. 6 k n n n n k n n n n k n n k n n 0 2 1 s 2 1 0 2 1 s 0 2 c 2 0 2 c 0 2 s 2 0 2 1 c 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ò ò p z z f z p z z f z F = - F = - F = F =  + + ¥ +  ¥   + + In applications to differential equations, it is necessary to know how to express the transforms of functions involving the derivatives of ( )f z in terms of k .( )F The unilateral Fourier transforms have the derivative properties Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 3 k k k k k k d d d d d d , 7 2 2 2 ( ) ( ) ( ) ( ) ( ) ( )   z f z z f z z = - F - F = - F ⎧⎨⎩ ⎫⎬⎭ ⎧⎨⎩ ⎫⎬⎭ where the operator  and the function k( )F refer either to the Fourier sine or to the Fourier cosine transform of the functions satisfying the proper boundary conditions for ensuring the existence of inverse transforms. 3. The quantum harmonic oscillator 3.1. The eigenvalue problem We are now prepared to address the quantum harmonic oscillator problem. The one- dimensional time-independent Schrödinger equation for the harmonic oscillator in dimen- sionless variables can be written as x x x x x d d 2 0, , . 8 2 2 2( )( ) ( ) ( ) ( )y e y+ - = Î -¥ +¥ Equation (8) is an eigenvalue equation for the characteristic pair ,( )e y with e Î and x xd . 92∣ ( )∣ ( )ò y < ¥ -¥ +¥ Because x = 0 is a regular point of equation (8), ψ is analytic at the origin, i.e. xd dn n x 0∣ ∣ ∣y < ¥= for all n .Î Because x( )y - is also a solution of equation (8), the linear combinations x x( ) ( )y y - are also solutions, so two different eigenfunctions with well defined parities can be built. Thus, it suffices to concentrate attention on the positive half- line and use boundary conditions on ψ at the origin and at infinity. Eigenfunctions and their first derivatives continuous on the whole line with well defined parities can be constructed by taking symmetric and antisymmetric linear combinations of ψ defined on the positive side of the x-axis. Incidentally, the combinations x x( ) ( )y y - share the same eigenvalue so that at first glance one would expect a twofold degeneracy, but we will show that the requirement of continuity of the eigenfunctions and their first derivatives invalidates one of the two combinations for an given eigenvalue, in accordance with the nondegeneracy theorem (a general result valid for bound states in one-dimensional nonsingular potentials) [18]. As x 0, the solution with definite parity varies as x ,d where δ is 0 or 1 regardless the magnitude of ε. The homogeneous Neumann condition ( xd d 0x 0∣y == ) develops for 0d = but not for 1d = , whereas the homogeneous Dirichlet boundary condition ( 0x 0y == ) develops for 1d = but not for 0.d = The continuity of ψ at the origin excludes the possibility of an odd-parity eigenfunction for 0,d = and the continuity of xd dy at the origin excludes the possibility of an even-parity eigenfunction for 1.d = Thus, 0d = for even solutions, and 1d = for odd solutions. On the other hand, the square-integrable asymptotic form of the solution as x  ¥ is given by x x e x 22( )y ~ a - with arbitrary α. We write x x ex 22( ) ( )y f= in such a way that f is a solution of the equation x x x x x x d d 2 d d 2 1 0. 10 2 2 ( ) ( ) ( ) ( ) ( )f f e f+ + + = Notice that the parity of f is the same as that of ψ and that x = 0 is a regular point of (10). As a matter of fact, f varies in the neighbourhood of the origin as x .d Notice also that one has to Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 4 find a particular solution of (10) in such a way that f behaves like x e x2a - for sufficiently large x . This condition, added by the regularity at x = 0 ensures the existence of the Fourier sine (cosine) transform for f odd (even). 3.2. Fourier sine and cosine transforms Restricting our attention on the positive half-line ( xz = ), the unilateral Fourier transform establishes a mapping of the second-order equation for f into an integrable first-order equation for cF or :sF k k k k k d d 2 1 2 2 0, 11 ( ) ( ) ( )eF + + - F =⎜ ⎟⎛ ⎝ ⎞ ⎠ where Φ denotes cF or .sF The solution of (11) is expressed as k A k e , 12a a k 42( ) ( )( )F = - where A a( ) is an arbitrary constant of integration and a 1 2 e= - Î is as yet undetermined. Due to the fact that k ea a klog= and klog is multivalued, this solution can be cast into the form k A k me e , . 13a a k ma4 i22( ) ( )( ) F = Îp- This form explicitly shows that k( )F has infinite branches if a is an irrational number, and q branches if a p q,= where p and q are integers with q 0.¹ It is true that when a is not an integer number k( )F is a multivalued function but their dissimilar branches differ by k-independent phase factors (e mai2p ) without physical con- sequences thanks to the normalization condition expressed by (9) and to Parseval’s formulas. In plain terms, the overall phase factors can be absorbed into the constant of integration. Notice that k = 0 is itself a singular point of equation (11) and so the neighbourhood of k = 0 needs careful handling because Φ may exhibit a singularity at the origin. The point of danger lies in the exponent of k. Parseval’s formulas expressed by (4), related to square-integrable eigenfunctions, demand a 1 2> - to guarantee convergence. The derivatives of cf and sf tend to finite limits as ζ approaches the origin. Thus, the two first lines of equation (5) demand that the n2 th moment of cF and the n2 1( )+ th moment of sF are finite numbers so that a 1.> - Note that this condition is weaker than that originating from Parseval’s formulas. It is a pity that the last line of equation (5) proves clumsy to impose restrictions on a. The derivatives of Φ near the origin impose more restrictions on the allowed values for a. Using the property of the gamma function z z z1( ) ( )G + = G (see, e.g., [19], [20]), one finds k k A a a n klim d d 1 1 lim 14 k n n a k a n 0 0 ( ) ( ) ( ) ( )( )F = G + G + -  - for any branch of k .( )F Because z( )G has no zeros but has simple poles at z n ,= - with n ,Î equation (14) is equal to zero when a n n1= - - , so a n 1 - with a .Î Taking into account the restriction resulting from Parseval’s formulas, one can say that Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 5 k k a n a n k k a n a lim d d 0 if 2 1, with and 0 lim d d 0 if 2 , with . 15k n n k n n 0 2 2 0 2 1 2 1 ( ) ( ) ( )     F = - Î ¹ F = Î   + + At large ζ the exponentially decreasing factors in cf and sf always predominate over any power increasing factor and so the n2 th moment of cf and the ( n2 1+ )th moment of sf are finite numbers. In this case, the properties of the unilateral Fourier transforms expressed by the first two lines of (6) imply that k k k k lim d d , lim d d . 16 k n n k n n0 2 c 2 0 2 1 s 2 1 ( ) ( ) ( )F < ¥ F < ¥   + + Due to the exponent of k, equation (16) is satisfied if a n2 for cF and a n2 1 + for sF with a .Î Thus, with the aid of (15) one finds a = n for both cF and .sF In principle, the spectrum has been determined: n 1 2.ne = + 3.3. The inversion of the Fourier sine and cosine transforms In order to reconstruct cf and sf on the half-line one needs to calculate the integrals related to the inverse Fourier transforms. It follows that A k k k A k k k 2 d e cos 2 d e sin . 17 n n n k n n n k c 0 4 s 0 4 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ò ò f z p z f z p z = = ¥ - ¥ - From (3.952.7) and (3.952.8) of [16], one finds A F n A F n e 2 , 1 2 , e 2 1 2 , 3 2 , . 18 n n n n c c 1 1 2 s s 1 1 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) f z z f z z z = - = - + z z - - ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ The confluent hypergeometric function or Kummer’s function, F a b z, , ,1 1 1 1( ) also denoted M a b z, , ,1 1( ) is defined by the series (see, e.g. [19, 20]) F a b z b a a j b j z j b, , , 0, 1, 2, . 19 j j 1 1 1 1 1 1 0 1 1 1( ) ( ) ( ) ( )( ) ! ( )å= G G G + G + ¹ - - ¼ = ¥ This series converges for all z Î and has asymptotic behaviour prescribed by F a b z b z b a z a z , , e e , 2 arg 3 2. 20 z a a z a b1 1 1 1 1 i 1 1 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ∣ ∣ p p G  G - + G - < < p ¥ + - - The presence of e z in (20) ruins the asymptotic behaviour of n c ( )f and n s ( )f decided before beyond doubt ( e 2za z- ). This unfavourable situation can be remedied by considering the poles of a1( )G and demanding a n .1 = - In this case, F n b z, ,1 1 1( )- behaves asymptotically as zn and the series (19) is truncated at j n=  in such a way that the confluent hypergeometric function results in polynomial in z of degree not exceeding n . Therefore, n is even for cf and n is odd for .sf As a matter of fact, F n b z, ,1 1 1( )- is proportional to the generalized Laguerre polynomial L zn b 11 ( )( )-  with z 0, ,[ )Î ¥ and L zn 1 2 ( )( )-  and L zn 1 2 ( )( )+  are proportional to H zn2 1 2( ) and z H zn 1 2 2 1 1 2( )- + respectively, where H zn 1 2( ) is the Hermite polynomial. Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 6 Therefore, F n 2, 1 2,1 1 2( )z- with n even and F n 2 1 2, 3 2,1 1 2( )z- + with n odd are proportional to H ,n ( )z with the desired properties Hd d 0n2 0∣z =z and H 0.n2 1 0 =z+  Hence, cf and sf take the form A H A H e e . 21 n n n n n n c 2 2 2 s 2 1 2 1 2 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) f z z f z z = = z z - + + - + It is worthwhile to note that the last line of (6) imposes severe restrictions on the allowed values for a without postulating the convergence of the moments of f. Now the exponent of k makes equation (14) vanish for real values of a subject to the conditions a n2 1> + for ,cF and a n2> for .sF Additional restrictions arising from (15) and (16) make a = n for both cF and .sF The formulas (3.952.9) and (3.952.10) of [16] allow us to obtain the integrals for cf and sf expressed by (17) at once in terms of confluent hypergeometric functions. Eventually, the good asymptotic behaviour of cf and sf prescribed by the normalization condition establishes a n2= for cf and a n2 1= + for ,sf with cf and sf realized in terms of Hermite polynomials. The only remaining question is how to write the eigenfunctions. 3.4. Complete solution of the problem Following up our earlier comments about eigenfunctions of definite parity, one could think about antisymmetric and symmetric extensions of n c 2 ( )( )f z and .n s 2 1 ( )( )f z+ Nevertheless, antisymmetric (symmetric) extensions of n c 2( )f ( n s 2 1( )f + ) are not allowed because the solution of (10) is infinitely differentiable at the origin. This further constraint makes ny even (odd) parity for n even (odd), namely x x e e 22 n n n n n n 2 2 c 2 c 2 2 1 2 s 2 1 s 2 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y f z f z y f z f z = + - = - - z z + + + ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ so that the spectrum is nondegenerate, in agreement with the nondegeneracy theorem [18]. Finally, using the property H x H x1 ,n n n( ) ( ) ( )- = - the solution of the original problem is expressed as n x N H x 1 2 e , 23 n n n x n 22( ) ( ) ( ) e y = + = - where Nn are normalization constants. 4. Conclusion In conclusion, we have shown that the complete solution of the one-dimensional quantum harmonic oscillator can be approached via the unilateral Fourier transform method. Single- valuedness of the eigenfunction is not a fair request. The convergence of the moments of the unilateral Fourier transform is not enough to do the job and the convergence of the moments of e 22 ( )y zz- is difficult to handle, especially because one has to appeal to the properties of the confluent hypergeometric function. Fortunately, the inversion of the Fourier sine and cosine transforms results in tabulated integrals and the proper bound-state solutions can be straightforwardly determined, just requiring definite parity and square-integrable eigenfunctions. Eur. J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 7 Acknowledgments This work was supported in part by means of funds provided by FAPESP and CNPq. References [1] Sommerfeld A 1929 Wave Mechanics (New York: Academic) [2] Sakurai J J 1994 Modern Quantum Mechanics (Reading, MA: Addison-Wesley) [3] Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw-Hill) [4] Muñoz G 1998 Integral equations and the simple harmonic oscillator Am. J. Phys. 66 254–6 [5] Ponomarenko S A 2004 Quantum harmonic oscillator revisited: a Fourier transform approach Am. J. Phys. 72 1259–60 [6] Engel A 2004 Comment on Quantum harmonic oscillator revisited: a Fourier transform approach by A. A. Ponomarenko Am. J. Phys. 72 1259–60 Engel A 2006 Am. J. Phys 74 837 [7] Palma G and Raff U 2011 A novel application of a Fourier integral representation of bound states in quantum mechanics Am. J. 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J. Phys. 37 (2016) 015402 P H F Nogueira and A S de Castro 8 http://dx.doi.org/10.1119/1.18855 http://dx.doi.org/10.1119/1.18855 http://dx.doi.org/10.1119/1.18855 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.1677395 http://dx.doi.org/10.1119/1.2221343 http://dx.doi.org/10.1119/1.3531975 http://dx.doi.org/10.1119/1.3531975 http://dx.doi.org/10.1119/1.3531975 http://dx.doi.org/10.1088/0143-0807/34/1/199 http://dx.doi.org/10.1088/0143-0807/34/1/199 http://dx.doi.org/10.1088/0143-0807/34/1/199 1. Introduction 2. Fourier transforms and their main properties 3. The quantum harmonic oscillator 3.1. The eigenvalue problem 3.2. Fourier sine and cosine transforms 3.3. The inversion of the Fourier sine and cosine transforms 3.4. Complete solution of the problem 4. Conclusion Acknowledgments References