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- W2034837517 abstract "The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a “free particle” obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order. The equations of motion contain quantities which have “fractional” dimensions, chosen such that the “energy” has the correct dimension <svg style=vertical-align:-2.22495pt;width:69.912498px; id=M1 height=18.612499 version=1.1 viewBox=0 0 69.912498 18.612499 width=69.912498 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,15.775)><path id=x5B d=M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z /></g><g transform=matrix(.017,-0,0,-.017,5.927,15.775)><path id=x1D440 d=M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
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t37 -4h417q23 0 33.5 5t25.5 21h23z /></g> <g transform=matrix(.012,-0,0,-.012,57.65,7.613)><use xlink:href=#x32/></g> <g transform=matrix(.017,-0,0,-.017,63.988,15.775)><path id=x5D d=M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z /></g> </svg>. The action <svg style=vertical-align:-0.23206pt;width:8.2875004px; id=M2 height=11.75 version=1.1 viewBox=0 0 8.2875004 11.75 width=8.2875004 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,11.4)><path id=x1D446 d=M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68
l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z /></g> </svg> is defined as a fractional time integral of the Lagrangian, and a “fractional Planck constant” is introduced. The TFSE corresponds to a “subdiffusion” equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order. The present work corrects a number of errors in Naber’s work. The correct continuity equation for the probability density is derived and a Green function solution for the case of a “free particle” is obtained. The total probability for a “free” particle is shown to go to zero in the limit of infinite time, in contrast with Naber’s result of a total probability greater than unity. A generalization to the case of a particle moving in a potential is also given." @default.
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- W2034837517 date "2013-01-01" @default.
- W2034837517 modified "2023-09-27" @default.
- W2034837517 title "Time Fractional Schrodinger Equation Revisited" @default.
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- W2034837517 doi "https://doi.org/10.1155/2013/290216" @default.
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