Inspired by the success of De Broglie’s matter wave ideas, Erwin Schrödinger sought a wave equation that would represent the dynamics of electrons as waves and would produce the atomic spectra naturally as a consequence of the wave equation without ad hoc rules. Schrödinger came up with a completely new wave equation that predicted the spectrum of the hydrogen atom correctly.
In Schrödinger’s equation, in place of the position and velocity of electron we describe an electron by its wave function \(\psi\) [read: “sigh”] which is a function of space and time, just as any other wave functions, such as the wave function for the sound or light waves. In one-dimensional situation, say the motion on a straight line along \(x\)-axis only, Schrödinger’s wave equation takes the following form.
\begin{equation}
i \hbar\ \dfrac{\partial \psi}{\partial t} = -\dfrac{\hbar^2}{2m}\ \dfrac{\partial^2 \psi}{\partial x^2} + U(x) \psi, \tag{51.2}
\end{equation}
where \(i = \sqrt{-1}\text{,}\) \(\hbar = h/2\pi\text{,}\) and \(U(x)\) is the potential energy. This equation is a complex equation, meaning that it involved the imaginary number \(i\text{.}\) The solution of this equation will the wave function \(\psi\text{,}\) which will be a complex quantity. All this will be mystery if you have not encountered complex numbers in your courses so far. You will find an introductory treatment of complex numbers in your algebra and calculus textbooks.
In the stationary state, e.g., one of the Bohr states, the electron has a definite energy \(E\text{.}\) In that case the solution takes the following form:
\begin{equation}
\psi(x,t) = \psi_E(x)\: e^{-iEt/\hbar},\tag{51.3}
\end{equation}
where
\(\psi_E(x)\) is time-independent stationary wave function corresponding to the stationary state of energy
\(E\text{.}\) Putting this form of the solution in Eq.
(51.2) we obtain the following equation, called the time-independent Schörodinger equation.
\begin{equation}
\dfrac{d^2 \psi_E}{d x^2} = -\dfrac{2m}{\hbar^2}\left[ E - U(x) \right] \: \psi_E. \tag{51.4}
\end{equation}
The solutions of this equation will give the time-independent part of the wave function when the electron is in one of the Bohr orbits. For a three-dimensional situation such as the electron in a hydrogen atom the equation becomes
\begin{equation}
\dfrac{\partial^2 \psi_E}{\partial x^2} + \dfrac{\partial^2 \psi_E}{\partial y^2} + \dfrac{\partial^2 \psi_E}{\partial z^2}= - \dfrac{2m}{\hbar^2}\left[ E - U(x,y,z) \right] \: \psi_E. \tag{51.5}
\end{equation}
In the case of Hydrogen atom the potential is the electric potential energy as we have seen above when we discussed the Bohr model.
\begin{equation}
U(x,y,z) = - \dfrac{1}{4\pi\epsilon_0}\:\dfrac{e^2}{\sqrt{x^2 + y^2 + z^2}}. \tag{51.6}
\end{equation}
It turns out that the solution of Eq.
(51.5) with
\(U\) for Hydrogen atom requires more advance math than is required for this book. Suffice to say that the solution of the Schrödinger equation with potential energy given by Eq.
(51.6) consists of allowed values of
\(E\) and the wave function that goes with that particular
\(E\text{.}\) The difference in energies between allowed states completely agree with the observed frequencies of light seen in the Hydrogen spectrum that mediate the transition between those states.
Since we will not solve the mathematically more demanding problem of the Hydrogen atom, we will solve illustrative ome-dimensional models that obey the one-dimensional equation Eq.
(51.4). The solutions of these problems give us valuable insights into problems of practical interest.
