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What if there is a jump discontinuity on a wave function where the first derivative of which is still continuous? What is the problem with such wave function?

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What if there is a jump discontinuity on a wave function where the first derivative of which is still continuous? What is the problem with such wave function?

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strangerep

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This is best discussed (initially at least) by example. Do you have a particular specific example in mind?What if there is a jump discontinuity on a wave function where the first derivative of which is still continuous?

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bhobba

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All these issues are fixed in the Rigged Hilbert Space formalism.

Its based on distribution theory which really should be in the armory of any applied mathematician. It makes Fourier transform theory a snap for example, otherwise you become bogged down is difficult issues of convergence etc.

I recommend the following book:

https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20&tag=pfamazon01-20

Thanks

Bill

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Nugatory

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It would be helpful if you could provide a specific example of a problem involving a discontinuous wave function - I'm still not sure what you're thinking here.

Generally we require that the wave function be continuous across its domain, and expect to find discontinuities in the first derivative only where the potential becomes infinite (infinite square well, delta-function potentials, and the like). Discontinuities in the first derivative do mean that the second derivative is undefined so we can't solve Schrodinger's equation across the discontinuity, but that's not the same thing as saying that it is ill-defined. We can solve Schrodinger's equation on each side of the discontinuity and then take the requirement that the wave function be continuous as a boundary condition. But all of this is standard fare in first-year classes, so I presume that you're thinking of something more difficult.

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The significance of the wavefunction must be intimately related with the probability of finding the particle, and mathematical functions not satisfying this are just pure mathematical issues, but representing no physical situation.

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bhobba

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Hmmmm. Think about the Dirac Delta function.The wavefunction of a quantum mechanical partice must be continuous,

As I alluded to the answer lies in Rigged Hilbert Spaces.

The test functions are the physically realizable ones ie are continuously differentiable etc etc and have nice mathematical properties. The dual with all sorts of weird stuff like the Dirac Delta function is introduced for mathematical convenience. That includes non continuous functions etc as well as functions that do not fall off at infinity.

Thanks

Bill

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