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The Schrödinger Equation

Introducing the equation

First we discuss the simple case of a free particle with mass \(m\) at uniform motion with velocity \(u\) along the \(x\) direction. We set the motion at a constant potential (\(E_{\mathrm{pot}} = 0\)) and get a matter wave like

\[\psi \left(x,t\right) = A \cdot \mathrm{e}^{i\left( k x - \omega t\right)} = A \cdot \mathrm{e}^{\frac{i}{\hbar}\left(p x - E t\right)} \mathrm{,}\]

whith \(E = E_{\mathrm{kin}} = \hbar \cdot \omega\) and \(p = \hbar \cdot k\). Because the formula is analogue with an electromagnetic wave, we use the same formula as for an electromagnetic wave propagating with a phase velocity of \(u\) along \(x\),

\[\frac{\partial^2 \psi}{\partial x^2} - \frac{1}{u^2} \frac{\partial^2 \psi}{\partial t^2} = 0 \mathrm{.}\]

In order to describe stationary quantum states, namly the momentum \(p\) and enegery \(E\) do not depent on time, we are seeking for a stationary solution of the above equation. Since they are indepent of time, we can write the genernal solution for the wave equation \(\psi \left(x,t\right)\) as a product of a position-dependent factor \(\mathrm{e}^{+ikx}\) and a time-dependent phase \(\mathrm{e}^{-i \omega t}\), such like

\[\begin{split}\begin{eqnarray} \psi \left(x,t\right) & = & A \cdot \mathrm{e}^{i k x} \cdot \mathrm{e}^{-i \omega t}\\ {} & = & \psi \left(x\right) \cdot \mathrm{e}^{-i \omega t}\mathrm{.} \end{eqnarray}\end{split}\]

If we now use this ansatz in the wave equation, we obtain for the derivative in space

\[\frac{\partial^2 \psi}{\partial x^2} = -k^2 \psi = - \frac{p^2}{\hbar^2} \psi = - \frac{2 m}{\hbar^2} E_{\mathrm{kin}} \psi \mathrm{.}\]

In the general case our particle might move within a force field. If it is a conservative field, we can assign a potential energy \(E_{\mathrm{pot}}\) to every point in space keeping the total energy constant \(E = E_{\mathrm{kin}} + E_{\mathrm{pot}}\). Then, we directly obtain the one-dimension stationary Schrödinger equation

\[- \frac{\hbar^2}{2 m} \, \frac{\partial^2 \psi}{\partial x^2} + E_{\mathrm{pot}} \psi = E \psi\]

In the case of a three-dimensional motion of our particle we can analogously use the three-dimensional wave equation,

\[\Delta \psi- \frac{1}{u^2} \frac{\partial^2 \psi}{\partial t^2} = 0 \mathrm{,}\]

and an ansatz in three dimensions,

\[\psi \left(x,y,z,t\right) = \psi \left(x,y,z\right) \cdot \mathrm{e}^{-i \omega t} \mathrm{,}\]

in order to obtain the three-dimensional stationary Schrödinger equation

\[- \frac{\hbar^2}{2 m} \, \Delta \psi + E_{\mathrm{pot}} \psi = E \psi \mathrm{.}\]

If we, in addition, calculate the first order derivative of our matter wave with respect to time, we obtain (for \(E_{\mathrm{pot}} = 0\))

\[\frac{\partial \psi}{\partial t} = - \frac{i}{\hbar} E_{\mathrm{kin}} \, \psi \mathrm{.}\]

In the case of a free particle, the condition \(E_{\mathrm{pot}} = 0\) implies \(E_{\mathrm{kin}} = \; \mathrm{const.}\) and we can combine the derivative in space and in time in order to derive this three-dimensionsial time-dependent equation

\[- \frac{\hbar^2}{2 m} \, \Delta \psi = i \hbar \frac{\partial \psi}{\partial t} \mathrm{.}\]

Please note, in the case of non-stationary problems, namely where \(E = E \left( t \right)\) and \(p = p \left( t \right)\), the simple derivative \(\partial^2 \psi / \partial t^2 = - \omega^2 \psi\) does not hold true anymore and more time-dependent parameters have to be considered. However, Schrödinger proposed for a time-dependent potential energy \(E_{\mathrm{pot}} = E_{\mathrm{pot}} \left(r, t \right)\) the equation

\[- \frac{\hbar^2}{2 m} \, \Delta \psi\left(r, t \right) + E_{\mathrm{pot}} \left(r, t \right) \psi\left(r, t \right) = i \hbar \frac{\partial}{\partial t} \psi\left(r, t \right) \mathrm{,}\]

which was confirmed by numerous experiments. For stationary problems, one can again separate the wave function into a position- and a time-dependent factor and get the stationary Schrödinger equation as stated above.