Can the solution of a spinless particle for a Schrodinger equation/Klein Gordon equation be a square wave? If yes, what are those energy potential conditions?
Hello!
The sinc wave packet (e.g, see International Journal of Quantum Chemistry, Vol. 92 (2003) pp. 205-211) has almost a square shape, which seems to be useful for precise, time-dependent reactive scattering calculations.
Cheers,
Alexis
This is a very straightforward problem; all that one needs to do is to convert the Schrödinger equation into an equation for the external potential. I attach a Mathematica notebook in which I solve the problem exactly (note: I have not been as rigorous as I should have been in treating the boundary conditions).
Dear Shalemder, if you are addressing me, I fail to comprehend your message. The main question on this page is whether there exists a potential for which the corresponding wave function has the shape of a square (more generally, a rectangle), and I believe that I explicitly showed that yes such potential exists, albeit that for the ideal case of a sharp square/rectangle this potential is not an ordinary function, but a distribution. If you want to add things up, by the linearity of the problem you have the superposition principle at your disposal.
Hi Behnam> Sorry I could not be put in clearly. The difference between square wave like 1, -1, 1, -1 and 1,0,1,0 is that in the former the probability of finding the particle is uniform in the whole universe. On the other hand 1, 0, 1, 0, ... is that it means the particle does not occur at repeated intervals. I agree that potential calculations are simple. My intuition is that it should result in positive and negative delta functions at specific intervals. The question is: does having a 1, 0, 1, 0... kind of square wave makes sense for a particle?
Dear Shalender, the wave function is strictly required to be normalizable. If you wish to have a wave function that has the shape of a train of pulses, this train cannot be infinitely long; it must be finite, or failing to be identically vanishing, it will necessarily not have a finite norm. Other possibility to be adopted is that of imposing a periodic (or anti-periodic or mixed) boundary condition on the problem, the sequence of zeroes and ones repeating themselves after a given period (here the single-particle states are Bloch functions and the relevant function corresponds to the one at k=0, the centre of the Brillouin zone). If you relax the condition of normalizability, you have more freedom to do things, but then your problem is no longer a quantum-mechanical one -- it reduced to a game of playing with differential equations.
Incidentally, whether one has 1, -1, 1, -1,... or 1,0,1,0,... makes little difference, since norm is defined in terms of the absolute value of the underlying wave function.
Really, guys? The answer is, "Mathematically, YES -- it only requires an infinite potential," AND, "Physically, NO -- there are no infinite potentials in nature." This is covered in every Introductory Quantum Mechanics course. (Of course, you can get arbitrarily close to a square wave by making a big enough potential at the edge; I suppose for an Engineer that would suffice. :-)
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