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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

$$\psi (x,t)=A\cdot {\mathrm{e}}^{i(kx-\omega t)}=A\cdot {\mathrm{e}}^{\frac{i}{\hslash }(px-Et)},$$

whith $E=E_{\mathrm{kin}}=\hslash \cdot \omega$ and $p=\hslash \cdot k$. 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 $x$,

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

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 $\psi (x,t)$ as a product of a position-dependent factor ${\mathrm{e}}^{+ikx}$ and a time-dependent phase ${\mathrm{e}}^{-i\omega t}$, such like

$$\begin{array}{r}\begin{array}{rcl}\psi (x,t)& =& A\cdot {\mathrm{e}}^{ikx}\cdot {\mathrm{e}}^{-i\omega t}\\ & =& \psi \left(x\right)\cdot {\mathrm{e}}^{-i\omega t}.\end{array}\end{array}$$

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

$$\frac{{\partial }^2\psi }{\partial x^2}=-k^2\psi =-\frac{p^2}{{\hslash }^2}\psi =-\frac{2m}{{\hslash }^2}{E}_{\mathrm{kin}}\psi .$$

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

$$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+E_{\mathrm{pot}}\psi =E\psi$$

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

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

and an ansatz in three dimensions,

$$\psi (x,y,z,t)=\psi (x,y,z)\cdot {\mathrm{e}}^{-i\omega t},$$

in order to obtain the three-dimensional stationary Schrödinger equation

$$-\frac{{\hslash }^2}{2m}\mathrm{\Delta }\psi +E_{\mathrm{pot}}\psi =E\psi .$$

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

$$\frac{\partial \psi }{\partial t}=-\frac{i}{\hslash }{E}_{\mathrm{kin}}\psi .$$

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

$$-\frac{{\hslash }^2}{2m}\mathrm{\Delta }\psi =i\hslash \frac{\partial \psi }{\partial t}.$$

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 $E_{\mathrm{pot}}=E_{\mathrm{pot}}(r,t)$ the equation

$$-\frac{{\hslash }^2}{2m}\mathrm{\Delta }\psi (r,t)+E_{\mathrm{pot}}(r,t)\psi (r,t)=i\hslash \frac{\partial }{\partial t}\psi (r,t),$$

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.