Why these particular ansatz functions?

In principle, each of the four functions $\sin(kx)$ , $\cos(kx)$ , $\exp(kx)$ , $\exp(-kx)$ is a solution of the Schrödinger equation for a constant potential. However, only two of them (any two) are independent of one another. Using complex numbers, two of these functions can always be expressed in terms of the other two, for example:

$e^{ikx} = \cos(kx) + i\sin(kx)$

$\sin(kx) = \frac{e^{ikx}-e^{-ikx}}{2i}$

Since two functions are enough to represent every solution, a general ansatz consists of a superposition of two suitably chosen trial functions. If the trial functions are chosen cleverly, the $k$ -factors (as well as the weighting parameters) remain real.

In a region I (classically allowed region), one such ansatz is, for example,

$\psi_{\text{I}}(x) = A\sin(k_{\text{I}} x) + B\cos(k_{\text{I}} x),$

while in an adjacent region II (classically forbidden region) the general ansatz

$\psi_{\text{II}}(x) = C \exp(-k_{\text{II}} x) + D \exp(k_{\text{II}} x)$

leads to the goal.

The trial function $\exp(k_{\text{II}} x)$ with a positive exponent is not physically meaningful in region II: this exponential function grows without bound. This would mean an infinitely high probability of finding the particle as $x \to \infty$ . The corresponding term must therefore vanish ( $D = 0$ ).

In general, physicists prefer to work here with complex exponential functions $e^{ikx}$ and $e^{-ikx}$ rather than with sine and cosine functions. There are two main reasons for this:

Mathematically, the exponential function is easier to handle.
Physically, the complex exponential functions are simultaneously momentum eigenfunctions for a constant potential. They correspond exactly to a wave travelling to the right or to the left with momentum $\pm\hbar k$ .

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