How to compute the levy path integral with zero potential?

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In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as



$$K(x_b,t_b|x_a,t_a)=int_x_t_a=x_a,x_t_b=x_bDx(t)expleft-fracihint_t_a^t_bdtV(x(t))right$$



where h is the Planck constant. It is known that in Feynman functional measure (generated by the process of the Brownian motion) and with zero potential ($V(x)=0$), the amplitude can be computed exactly, but what is the case with the non-Gaussian case? In the paper of Prof.Nikolai Laskin it said it can be computed with the measure generated by the $alpha$ stable Levy motion ($1<alpha<2$), but in this case the probability density function is so different from the Brownian case, so how to compute the amplitude?










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    In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as



    $$K(x_b,t_b|x_a,t_a)=int_x_t_a=x_a,x_t_b=x_bDx(t)expleft-fracihint_t_a^t_bdtV(x(t))right$$



    where h is the Planck constant. It is known that in Feynman functional measure (generated by the process of the Brownian motion) and with zero potential ($V(x)=0$), the amplitude can be computed exactly, but what is the case with the non-Gaussian case? In the paper of Prof.Nikolai Laskin it said it can be computed with the measure generated by the $alpha$ stable Levy motion ($1<alpha<2$), but in this case the probability density function is so different from the Brownian case, so how to compute the amplitude?










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      In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as



      $$K(x_b,t_b|x_a,t_a)=int_x_t_a=x_a,x_t_b=x_bDx(t)expleft-fracihint_t_a^t_bdtV(x(t))right$$



      where h is the Planck constant. It is known that in Feynman functional measure (generated by the process of the Brownian motion) and with zero potential ($V(x)=0$), the amplitude can be computed exactly, but what is the case with the non-Gaussian case? In the paper of Prof.Nikolai Laskin it said it can be computed with the measure generated by the $alpha$ stable Levy motion ($1<alpha<2$), but in this case the probability density function is so different from the Brownian case, so how to compute the amplitude?










      share|cite|improve this question















      In quantum mechanics, if we have the quantum particle moving in the potential $V$ then the quantum-mechanical amplitude $K(x_b,t_b| x_a,t_a)$ can be written as



      $$K(x_b,t_b|x_a,t_a)=int_x_t_a=x_a,x_t_b=x_bDx(t)expleft-fracihint_t_a^t_bdtV(x(t))right$$



      where h is the Planck constant. It is known that in Feynman functional measure (generated by the process of the Brownian motion) and with zero potential ($V(x)=0$), the amplitude can be computed exactly, but what is the case with the non-Gaussian case? In the paper of Prof.Nikolai Laskin it said it can be computed with the measure generated by the $alpha$ stable Levy motion ($1<alpha<2$), but in this case the probability density function is so different from the Brownian case, so how to compute the amplitude?







      stochastic-processes mathematical-physics stochastic-analysis quantum-mechanics levy-processes






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      edited Aug 30 at 14:11

























      asked Aug 30 at 5:17









      yuanfei huang

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          The detailed calculations of a free particle propagator for the Levy case are presented:
          in the Sec 3.1 “Free particle” (pages 405-409) in the book:
          Fractional Dynamics
          Recent Advances
          https://doi.org/10.1142/8087 | October 2011, Pages: 532
          Edited By: Joseph Klafter (Tel Aviv University, Israel), S C Lim (Multimedia University, Malaysia) and Ralf Metzler (Technische Universität Munchen, Germany)
          https://www.worldscientific.com/worldscibooks/10.1142/8087



          or



          in the Chapter 7 “A Free Particle Quantum Kernel” (pages 101 – 117) in the book:
          Fractional Quantum Mechanics
          https://doi.org/10.1142/10541 | July 2018, Pages: 360
          By (author): Nick Laskin (TopQuark Inc., Canada)
          https://www.worldscientific.com/worldscibooks/10.1142/10541






          share|cite|improve this answer








          New contributor




          N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
          Check out our Code of Conduct.
























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            It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.






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              2 Answers
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              2 Answers
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              active

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              up vote
              0
              down vote



              accepted










              The detailed calculations of a free particle propagator for the Levy case are presented:
              in the Sec 3.1 “Free particle” (pages 405-409) in the book:
              Fractional Dynamics
              Recent Advances
              https://doi.org/10.1142/8087 | October 2011, Pages: 532
              Edited By: Joseph Klafter (Tel Aviv University, Israel), S C Lim (Multimedia University, Malaysia) and Ralf Metzler (Technische Universität Munchen, Germany)
              https://www.worldscientific.com/worldscibooks/10.1142/8087



              or



              in the Chapter 7 “A Free Particle Quantum Kernel” (pages 101 – 117) in the book:
              Fractional Quantum Mechanics
              https://doi.org/10.1142/10541 | July 2018, Pages: 360
              By (author): Nick Laskin (TopQuark Inc., Canada)
              https://www.worldscientific.com/worldscibooks/10.1142/10541






              share|cite|improve this answer








              New contributor




              N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
              Check out our Code of Conduct.





















                up vote
                0
                down vote



                accepted










                The detailed calculations of a free particle propagator for the Levy case are presented:
                in the Sec 3.1 “Free particle” (pages 405-409) in the book:
                Fractional Dynamics
                Recent Advances
                https://doi.org/10.1142/8087 | October 2011, Pages: 532
                Edited By: Joseph Klafter (Tel Aviv University, Israel), S C Lim (Multimedia University, Malaysia) and Ralf Metzler (Technische Universität Munchen, Germany)
                https://www.worldscientific.com/worldscibooks/10.1142/8087



                or



                in the Chapter 7 “A Free Particle Quantum Kernel” (pages 101 – 117) in the book:
                Fractional Quantum Mechanics
                https://doi.org/10.1142/10541 | July 2018, Pages: 360
                By (author): Nick Laskin (TopQuark Inc., Canada)
                https://www.worldscientific.com/worldscibooks/10.1142/10541






                share|cite|improve this answer








                New contributor




                N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.



















                  up vote
                  0
                  down vote



                  accepted







                  up vote
                  0
                  down vote



                  accepted






                  The detailed calculations of a free particle propagator for the Levy case are presented:
                  in the Sec 3.1 “Free particle” (pages 405-409) in the book:
                  Fractional Dynamics
                  Recent Advances
                  https://doi.org/10.1142/8087 | October 2011, Pages: 532
                  Edited By: Joseph Klafter (Tel Aviv University, Israel), S C Lim (Multimedia University, Malaysia) and Ralf Metzler (Technische Universität Munchen, Germany)
                  https://www.worldscientific.com/worldscibooks/10.1142/8087



                  or



                  in the Chapter 7 “A Free Particle Quantum Kernel” (pages 101 – 117) in the book:
                  Fractional Quantum Mechanics
                  https://doi.org/10.1142/10541 | July 2018, Pages: 360
                  By (author): Nick Laskin (TopQuark Inc., Canada)
                  https://www.worldscientific.com/worldscibooks/10.1142/10541






                  share|cite|improve this answer








                  New contributor




                  N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  The detailed calculations of a free particle propagator for the Levy case are presented:
                  in the Sec 3.1 “Free particle” (pages 405-409) in the book:
                  Fractional Dynamics
                  Recent Advances
                  https://doi.org/10.1142/8087 | October 2011, Pages: 532
                  Edited By: Joseph Klafter (Tel Aviv University, Israel), S C Lim (Multimedia University, Malaysia) and Ralf Metzler (Technische Universität Munchen, Germany)
                  https://www.worldscientific.com/worldscibooks/10.1142/8087



                  or



                  in the Chapter 7 “A Free Particle Quantum Kernel” (pages 101 – 117) in the book:
                  Fractional Quantum Mechanics
                  https://doi.org/10.1142/10541 | July 2018, Pages: 360
                  By (author): Nick Laskin (TopQuark Inc., Canada)
                  https://www.worldscientific.com/worldscibooks/10.1142/10541







                  share|cite|improve this answer








                  New contributor




                  N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  share|cite|improve this answer



                  share|cite|improve this answer






                  New contributor




                  N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.









                  answered Sep 7 at 1:08









                  N Laskin

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                  Check out our Code of Conduct.





                  New contributor





                  N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.






                  N Laskin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.




















                      up vote
                      0
                      down vote













                      It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.






                      share|cite|improve this answer
























                        up vote
                        0
                        down vote













                        It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.






                        share|cite|improve this answer






















                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.






                          share|cite|improve this answer












                          It turns out that for the Levy path integral, the calculation of the amplitude of a quantum particle uses the Fourier translation of the probability density function, since this representation is an integral of an exponential function which is easy to compute.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Sep 3 at 16:18









                          yuanfei huang

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