18.1: The Time-Independent Schrödinger Equation
We can start the derivation of the single-particle time-independent Schrödinger equation (TISEq) from the equation that describes the motion of a wave in classical mechanics:
\[ \psi(x,t)=\exp[i(kx-\omega t)], \label{19.1.1} \]
where \(x\) is the position, \(t\) is time, \(k=\dfrac{2\pi}{\lambda}\) is the wave vector, and \(\omega=2\pi\nu\) is the angular frequency of the wave. If we are not concerned with the time evolution, we can consider uniquely the derivatives of Equation \ref{19.1.1} with respect to the location, which are:
\[ \begin{aligned} \dfrac{\partial \psi}{\partial x} &=ik\exp[i(kx-\omega t)] = ik\psi, \\ \dfrac{\partial^2 \psi}{\partial x^2} &=i^2k^2\exp[i(kx-\omega t)] = -k^2\psi, \end{aligned} \label{19.1.2} \]
where we have used the fact that \(i^2=-1\).
Assuming that particles behaves as wave—as proven by de Broglie’s we can now use the first of de Broglie’s equation, Equation 17.5.4 , we can replace \(k=\dfrac{p}{\hbar}\) to obtain:
\[ \dfrac{\partial^2 \psi}{\partial x^2} = -\dfrac{p^2\psi}{\hbar^2}, \label{19.1.3} \]
which can be rearranged to:
\[ p^2 \psi = -\hbar^2 \dfrac{\partial^2 \psi}{\partial x^2}. \label{19.1.4} \]
The total energy associated with a wave moving in space is simply the sum of its kinetic and potential energies:
\[ E = \dfrac {p^{2}}{2m} + V(x), \label{19.1.5} \]
from which we can obtain:
\[ p^2 = 2m[E - V(x)], \label{19.1.6} \]
which we can then replace into Equation \ref{19.1.4} to obtain:
\[ 2m[E-V(x)]\psi = - \hbar^2 \dfrac{\partial^2 \psi}{\partial x^2}, \label{19.1.7} \]
which can then be rearranged to the famous time-independent Schrödinger equation (TISEq) :
\[ - \dfrac{\hbar^2}{2m} \dfrac{\partial^2 \psi}{\partial x^2} + V(x) \psi = E\psi, \label{19.1.8} \]
A two-body problem can also be treated by this equation if the mass \(m\) is replaced with a reduced mass \(\mu = \dfrac{m_1 m_2}{m_1+m_2}\).