Quantization of Arbitrary Hamiltonians

QUANTI ZATI ON OF ARBI TRARY HAMI LTONI ANS Jose Javier Garcia Moreta Graduate student of at the UPV/EHU (University of Basque country) In Solid State Physics Address: Practicantes Adan y Grijalba 2 5 G P.O 644 48920 Portugalete Vizcaya (Spain) Phone: (00) 34 685 77 16 53 E-mail: [email protected] PACS: 03.65.Ca , 03.65.Db, 02.60.Nm, 04.60.Ds ABSTRACT: I n this paper we discuss a method to apply the Quantization rules to arbitrary Hamiltonians t hat are not necessarily Polynomials in the momentum variable “p”, although for H ( x, p )   ( x)  F ( x, p ) , where F is some arbitrary analytic function of p and t he position of simplicity we apply our method to one dimensional Hamiltonias in the form the particle, to this purpose we apply results of Fractional Calculus and the Usual Quantization rules to formulate an Differential-integral linear equation equivalent to the Schröedinguer ˆ  Hˆ and all the equation that can be solved for big Energies En by means of a Neumann-series, through all the † paper we suppose that we can define our H so it s operator version satisfies H possible “Stat ionary states” are Orthonormal with real eigenvalues , to match with the usual  Keywords: Quantization, Fractional calculus, Taylor series. results in QM 1.Quantization rules under fractional Calculus: A generalization of the Taylor series using fractional Calculus [1] is: f (x   )     m r D m r f ( x)  0 r  ฀ (1) m   ( m  r  1) for any real ‘r’ ,here we have introduced the differintegral operator D q f , q  0 , and  D ab f  D a D b f   [ q ]1 c  d [ q ]1 x Dxq f ( x)  dtf (t )( x  t )   Dx q f ( x)  dtf (t )( x  t ) q 1 (2) (1  r ) dx 1  q 1 x ( q ) c ( Riemann-Liouville Differintegral) (Cauchy’s formula for repeated integration) d  r d  r Dx  r  D p  r  dx   r dp   r and (3) Using the Poisson sum formula, the series in (1) becomes:  (x)=  e2 imx    r  2 i c i 1  e s  d D   r f ( x) (  )  1 c i e sx (   r  1) ds (4)  m  So our Hamiltonian H ( x, p )   ( x)  F ( x, p ) (F is an analytic function of “p” and “x”) applying the fractional operator at p=0 and using (2) becomes:  ( x)   d  p  r Dp  r F ( x, 0) (  )  H (q, p) D p  r F ( x, 0)  D p  r F ( x, p )  (   r  1) p 0  (5) The Classical Hamilton-Jacobi equation takes the form of a differential-Integral equation:  S   r    ( x)   d    x S Dp  r F ( x, 0) (  )   0 (6)   (   r  1) t And Hamilton equations in this case are:  p 1   r H d Dp   r F ( x, 0) (  )  x    (   r ) p d  ( x)    d p  r F ( x, 0) H  Dp   r  (  )  p   (   r  1) x x (7) dx  F ( x, 0)  Where our F satisfies for the derivatives: D p   r  Dp   r F ( x, 0) , sub-index x x ‘p’ or ‘x’ means that the fractional derivative is taken over the variable x or p. And the Hamilton principle for Our Hamiltonian (variational version of Mechanics)   dt  xp    ( x)   d   p  r  Dp  r F ( x, 0) (  )   0  b  (   r  1)  (8)  a Now we must introduce the usual Quantization rules for our Hamiltonian so:   pˆ   r    i  Dx   r  ( x) Hˆ   i  r t (9) 1 2   And if we define the operator fˆ , gˆ  ( fˆ .gˆ  gˆ . fˆ ) then the Analogue of Schröedinguer equation is:    d   (  ) Dˆ p  r F ( xˆ , 0), (i)  r Dˆ x  r  ( x)   (   r  1) 1 t i  d  (10)  n i dt H  0 and En  n 1 t For a time-independent Hamiltonian we can define the Stationary states satisfying :  n ( x)  n  d     (  ) D p  r F ( xˆ , 0), (i)  r Dˆ x  r  n ( x)   ( x)n  n ( x)  (   r  1) 1  (11) So the Wave function can be put in the form  ( x)   Cn  n ( x)e and n  are the  it n n Eigenvalues of the linear differential-integral equation (5), for small values of “lambda” (Big Energies) we can find an approximate solution, if we can put (5) in the form:  n ( x) 1  n  d       (  ) D p  r F ( xˆ , 0), (i)  r Dˆ x  r  n ( x)   B( x)  (   r  1) 1    B ( x)  0 and H is the Hamiltonian defined before in (5)   d     (   r  1) 1     (  ) Dp  r F ( xˆ , 0), (i)  r Dˆ x  r   ( x)   Hˆ  (12) or  n ( x)  B( x) Then we could apply the “Neumman series” for | n | 1 1 1  n Hˆ  n ( x)   nk Hˆ k B ( x) and Hˆ k is the k-th iterated Kernel.  k 0 We also can define a conserved Probability current with    * in the form:  2    d t    ( ) (   r  1)   J  m  * ( x) D p   r F ( xˆ, 0), (i)  r Dˆ x  r  ( x)   x (13) x  The usual Schröedinguer equation is recovered setting  ( x)  V ( x) r=0 and F ( x, p )  p2 so the infinite series (1) has only a quadratic term in p, and applying the 2m usual quantization method. Another problem is to see if we can find a Functional I, of the form: I [ ,  * ]   dx  dtF [ ,  * , Dx  r  , Dx  r  * ,  t ,  t  * ] (14)     So taking the variations  and  * we get Our Schröedinguer equation (10) with the conserved current (12), The Euler-Lagrange equation associated with I and F are:     (1) n Dx n r   I F   F   F    Dxn  r        0   t     t    n    (15) variations respect to  , in order to get the conserved current (13) our Functional ‘F’ And a similar equation (with The complex conjugate of the wave function) holds for must be invariant under the changes    * and  ( x)   ( x)ei for any phase alpha. Although we have considered a one dimensional Hamiltonian, the method can also be applied to these separable Hamiltonians of the form: H TOTAL   H i ( xi , pi ) and H TOTAL   H i ( xi , pi ) N N (16) i 1 i 1 In these cases the Taylor series of HTOTAL is the sum or product of the Taylor series of every Hamiltonian Hi i=1,….,N , so (5) becomes a sum or a product of Integral equations , for the second case (product) we can put the solution in the form:  i ( xi ) (Wave function of the system)  E (n )  E N N (Energy levels) i 1 i 1 i i Finally another useful identity for the propagator (green function) is:  i   d        (  ) D p  r F ( xˆ , 0), (i)   r Dˆ x  r   ( x)  G (r , s )   (r  s)  (   r  1) 1  t  r  ( x, t ) The rest of expressions of QM remain unaltered, for example “Ehrenfrest (17) theorem” for an observable Aˆ is the same as with the usual SE equation:   [ Aˆ , Hˆ ] due to the linearity of H and the fact that we still have (time- d  i d  dt  n independent Hamiltonian) i dt An illustrative example comes from the Hamiltonian H  xp  g ( x) so the infinite series given in (1) becomes a finite one , here alpha is a real or complex number and g depends only on the position of the particle, then the Schröedinguer equation reads   x(i) Dx   (i) Dx ( x )  g ( x) t i (18) The semi-classical solution can be written as   eiS /  where ‘S’ satisfies the non- linear equation S ( x, t )   Et   du   S  S E  g (u )  x    g ( x)  0 x  x  t (19) x0 u due to invariance of the Hamiltonian under time translations H=Energy , another example is the Hamiltonian of a particle under a potential V(x) plus the term involving a fractional power of the momentum p 2  , in this case we can define an Energy functional E ( ,  * ) in a similar manner to the equation (14) E (,  * )    Dx    Dx *   V ( x) *  c 2   Dx   Dx *  2 (20) 2m E  0 with  * ‘c’ is a real number defined so the extremal of Euler-Lagrange equation the constraint    cons tan t gives us the usual Eigenvalue equation  2 d 2    V ( x)   i  2 n   2 d 2 2 (21) 2m dx dx However due to the complex number (i ) 2   a2   ib2  the Hamiltonian will not be self-adjoint (Hermitian) so H  H † and the Energies will no longer be real numbers which can yield to several unphysical conclusions. For the case of the Hamiltonian H 2  p 2   ( x) involving a fractional power of H , Eigenvalue problem    2 Dx2     n  En  n , the quantization rule for the Energy using the method of separation of variables and assuming we can find a solution to the becomes Hˆ    i  Dt2  , the time evolution then is obtained by solving the 2 fractional differential equation  i  Dt2 y (t )  En y (t ) and the Wave function can be 2 described as the linear combination 1     ( x, t )    e0 (t )  e En (t )   n ( x)    k 1    1   eu (t )   k 1 u Et  ,u  1       (22)  n ( i )2 k 0    ( k /  ) 0  With Et      1   u 1  1  1/    2  e k k E e t   , n d (23)   References: [1] Hulman R. “Fractional Calculus and the Taylor-Riemann Series,” Undergrad. J. Math. Vol.6(1) (2005). [2] Lloyd N. Trefethen and David Bau, III, “ Numerical Linear Algebra, Society for Industrial and Applied Mathematics” , 1997. ISBN 0-89871-361-7. [3] Kilbas A. , Srivastava, H.M and Trujillo J.J “Theory and Applications of Fractional Differential Equations”, Amsterdam, Netherlands, Elsevier, Febrary (2006). [4] Sakurai J.J “Modern Quantum Mechanics” Addison-Wesley (1994) [5] Omnes, Roland (1999). “Understanding Quantum Mechanics.” Princeton University Press. [6] Sobolev, S.L.: “Applications of Functional Analysis in Mathematical Physics”, AMS, 1963