19.1: The Free Particles
By definition, the particle does not feel any external force, therefore \(V(x)=0\), and the TISEq is written simply:
\[ - \dfrac{\hbar^2}{2m} \dfrac{d^2\psi}{dx^2} = E \psi(x). \label{20.1.1} \]
This equation can be rearranged to:
\[ \dfrac{d^2\psi}{dx^2} =- \dfrac{2mE}{\hbar^2} \psi(x), \label{20.1.2} \]
which corresponds to a mathematical problem where the second derivative of a function should be equal to a constant, \(- \dfrac{2mE}{\hbar^2}\) multiplied by the function itself. Such a problem is easily solved by the function:
\[ \psi(x) = A \exp(\pm ikx). \label{20.1.3} \]
The first and second derivatives of this function are:
\[\begin{equation} \begin{aligned} \dfrac{d \psi(x)}{dx} &= \pm ik A \exp(\pm ikx) = \pm ik \psi(x) \\ \dfrac{d^2 \psi(x)}{dx^2} &= \mp k^2 A \exp(\pm ikx) = -(\pm k^2) \psi(x). \end{aligned} \end{equation} \label{20.1.4} \]
Comparing the second derivative in Equation \ref{20.1.4} with Equation \ref{20.1.2}, we immediately see that if we set:
\[ k^2 = \dfrac{2mE}{\hbar^2}, \label{20.1.5} \]
we solve the original differential equation. Considering de Broglie’s equation, Equation 17.5.4 , we can replace \(k=\dfrac{p}{\hbar}\), to obtain:
\[ E = \dfrac{k^2 \hbar^2}{2m} = \dfrac{p^2}{2m}, \label{20.1.6} \]
which is exactly the classical value of the kinetic energy of a free particle moving in one direction of space. Since the function in Equation \ref{20.1.3} solves the Schrödinger equation for the free particle, it is called an eigenfunction (or eigenstate ) of the TISEq. The energy result of Equation \ref{20.1.6} is called eigenvalue of the TISEq. Notice that, since \(k\) is continuous in the eigenfunction, the energy eigenvalue is also continuous (i.e., all values of \(E\) are acceptable).