Answer
In the time independent Scrodinger wave equation (as given in the hints of the question), we see that if E < Vmin, then ψ'' and ψ always have the same sign. For example, If ψ is positive, then ψ'' is also positive . We can say that, ψ always curves away from the position axis . But, for it to be normalisable it has got to go to zero as x→−∞ . Now, for a non zero ψ, at some point the function has to deviate from zero, in the positive direction (let's say). Here the slope is positive and increasing, hence ψ increases with x . This curve can’t ever turn over and headback toward the axis, else that would require a negative second derivative. Similarly, if it starts out heading negative, it becomes more and more negative. Hence, we cannot come back to zero in either of the cases .
Work Step by Step
The above argument can be supported by the following illustration:
