Which of the following wave function is acceptable as the solution of the Schrodinger equation for all values of x?

Problem – Which of the following wave function is acceptable as the solution of the Schrodinger equation for all values of x?:

(a) $\psi(x) = A sec(x)$

(b) $\psi(x) = A tan(x)$

(c) $\psi(x) = A e^{x^{2}}$

(d) $\psi(x) = A e^{-x^{2}}$

Solution:

In, option (a) & (b)  -> $\psi(x)$ is not finite at x=$\frac{\pi}{2}$

In, option (c) -> $\psi(x)$ is not finite at x=$\pm\infty$

But, option (d) -> $\psi(x) = A e^{-x^{2}}$ is finite everywhere.

Hence, option (d) is the right answer.