Discussion
Which of the following functions can be an acceptable solution for Schrödinger equation for all values of x ?
Answer: A
The acceptable function is \(\psi(x) = Ae^{-\alpha x^{2}}\) (with \(\alpha \gt 0\)).
An acceptable wave function must be finite, single-valued and continuous everywhere, and square integrable. The Gaussian is all of these: it is smooth and falls to zero at large |x|, so \(\int|\psi|^{2}dx\) is finite. It is the ground state of the harmonic oscillator.
\(Ae^{\alpha x^{2}}\) blows up as \(x \to \pm\infty\) and cannot be normalised. A tan x and A sec x become infinite at \(x = \pi/2\) and its repeats, so they fail for some values of x.
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