exp(ax) is a acceptable wave function or not?
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Yes it is an acceptable wave function because it is single valued,having finite first order derivative and it is continuous everywhere; which are the conditions for a function to be a wave function.
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Is e^ikx an acceptable wave function? It depends on what you mean by “acceptable”. Mathematically, it is defined for all real values of and it is time independant. So it is a wavefunction.
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