It is well-known that the wavefunction and its first derivative must continuous at boundaries between regions having different values for the static potential, and reasons for this have been given. However, it appears that for quantum tunneling through a square barrier in which the height of the barrier is changed sinusoidally with time it is not necessary for the first derivative of the wavefunction to be continuous. Please tell me if this is correct and provide a reason if that is possible.
Dear Prof. Hagmann
I have not calculated it yet...
But if the first derivative of the wave function is not continuous at a point x in the space, how can you determine the kinetic energy of this particle at this point?
How can you apply the Laplace operator at this point?
Thank you for your answer.
Eva
Please consider the one-dimensional time-dependent Schrodinger equation where the wavefunction and the potential are functions of both x and t. Then the Laplacian of the wavefunction would be equal to the sum of two terms where the first is a constant multiplied by the potential and the wavefunction, and the second is a constant multiplied by the derivative of the wavefunction in respect to time. Since we are interested in a point where the potential is discontinuous at the point of interest this suggests that the laplacian of the wavefunction would also be discontinuous, so this is my answer to your second question.
I have asked my question because I have an analytical solution for the problem of a particle interacting with a square barrier in which the height is modulated sinusoidally with time, and I find that requiring both the wavefunction and its derivative in respect to x be continuous at both boundaries provides an over-determined system of equations; two times those that are required. I have had two physicists who specialize in the analysis of field emission and other aspects of quantum analysis examine my solution and they have found no errors in my analysis. I would be happy to send this solution to you to obtain your comments, but I cannot publish my results until this problem is solved.
Would you be willing to examine my paper which I mentioned to see if there is an error in my derivation which suggests that one particular problem is over-determined by a factor of two when requiring continuity in both the wavefunction and its spatial derivative? If so I would like to contact you in a manner that would not make this manuscript public at the present time.
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