There Is No Solution to the Time-Independent Schrödinger Equation When the Energy Is Less Than the Potential
Theorem
In a region where the energy is less than the potential, no solution to the time-independent Schrödinger equation exists. That is, no wave function exists.
Explanation
Saying that no solution exists means that no physically meaningful solution exists.
In quantum mechanics, the wave function is interpreted probabilistically. That is, the magnitude of the wave function is treated as a probability density function, and for this to be possible the wave function must be square-integrable. In other words, the wave function $\psi$ must satisfy the following equation.
$$ \int_{-\infty}^{\infty} \left| \psi \right|^{2} dx \lt \infty $$
Therefore, the theorem above states that when the energy is less than the potential, no square-integrable solution exists.
Proof
In the step potential, barrier potential, well potential, and so on, the potential is a piecewise-constant function, so let us assume that the potential $U$ does not depend on $x$. The time-independent Schrödinger equation is as follows.
$$ \dfrac{-\hbar^2}{2m} \dfrac{\partial ^2 }{\partial x^2}\psi + U\psi=E\psi $$
$$ \implies \dfrac{\partial ^2}{\partial x^2} \psi = \dfrac{2m}{\hbar^2}(U-E)\psi $$
Here, since we are currently showing that there is no solution when the energy is less than the potential, on the right-hand side we have
$$ U>E \implies U-E>0 $$
Therefore $\dfrac{2m}{\hbar^2}(U-E)>0$, and this can be denoted as $\kappa ^2$ for some arbitrary constant $\kappa$.
$$ \dfrac{2m}{\hbar^2}(U-E) \equiv \kappa ^2 $$
$\kappa$ is read as kappa. Then the equation is as follows.
$$ \dfrac{\partial ^2}{\partial x^2} \psi = \kappa ^2 \psi $$
Since this is a second-order differential equation with a positive coefficient, it has the following solution.
$$ \psi (x) = Ae^{\kappa x}+Be^{-\kappa x} $$
Let us check whether the solution is square-integrable.
$$ \begin{align*} \int_{-\infty}^{\infty} |\psi|^2 dx &= \int_{-\infty}^{\infty} |Ae^{\kappa x}+Be^{-\kappa x}|^2 dx \\ &= \int_{-\infty}^{\infty} (A^2e^{2\kappa x}+2AB + B^2e^{-2\kappa x}) dx \\ &= \left[ \frac{A^2}{2\kappa} e^{2\kappa x}+2ABx + \frac{B^2}{-2\kappa} e^{-2\kappa x} \right]_{-\infty}^{\infty} \\ &= \infty \end{align*} $$
Since the solution is not square-integrable and diverges, this $\psi$ is not something we wish to deal with physically (quantum-mechanically). Therefore, in the region where $U>E$, no wave function exists.
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