A Fundamental Form of the Schrodinger Equation

Abstract

We propose a first order equation from which the Schrodinger equation can be derived. Matrices that obey certain properties are introduced for this purpose. We start by constructing the solutions of this equation in one dimension and solve the problem of electron scattering from a step potential. We show that the sum of the spin up and down, reflection and transmission coefficients, is equal to the quantum mechanical results for this problem. Furthermore, we present a three dimensional (3D) version of the equation which can be used to derive the Schrodinger equation in 3D.

Appendix: Transmission and Reflection Coefficients for \(E>V_0\)

$$\begin{aligned} T_1= & {} \frac{(E+m)(E-{V_0})^{1/2}}{(E-{V_0}+m)E^{1/2}}\left| \frac{C}{A} \right| ^2 \nonumber \\= & {} \frac{4 \sqrt{1-\frac{{V_0}}{E}} \left( E^2+m \sqrt{E (E-{V_0})}-E \left( m+\sqrt{E (E-{V_0})}+{V_0}\right) \right) ^2}{(E+m) (E+m-{V_0}) {V_0}^2}\end{aligned}$$

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$$\begin{aligned} T_2= & {} \frac{(E+m)(E-{V_0})^{1/2}}{(E-{V_0}+m)E^{1/2}}\left| \frac{C'}{A} \right| ^2 \nonumber \\= & {} \frac{4 m \sqrt{E (E-{V_0})} {V_0}^2}{(E+m) (E+m-{V_0}) \left( -2 E-2 \sqrt{E (E-{V_0})}+{V_0}\right) ^2} \end{aligned}$$

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$$\begin{aligned} R_1= & {} \left| \frac{B}{A} \right| ^2 \nonumber \\= & {} \frac{(E-m)^2 {V_0}^2}{(E+m)^2 \left( -2 E-2 \sqrt{E (E-{V_0})}+{V_0}\right) ^2} \end{aligned}$$

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$$\begin{aligned} R_2= & {} \left| \frac{B'}{A} \right| ^2 \nonumber \\= & {} \frac{4 E m {V_0}^2}{(E+m)^2 \left( -2 E-2 \sqrt{E (E-{V_0})}+{V_0}\right) ^2} \end{aligned}$$

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1.1 Coefficients for $E<V_0$

$$\begin{aligned} \frac{B}{A}= & {} \frac{(-E+m) {V_0}}{(E+m) \left( 2 E-{V_0}+2 i \sqrt{E (-E+{V_0})}\right) } \end{aligned}$$

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$$\begin{aligned} \frac{B'}{A}= & {} \frac{2 \sqrt{E m} {V_0}}{(E+m) \left( -2 i E+i {V_0}+2 \sqrt{E (-E+{V_0})}\right) } \end{aligned}$$

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$$\begin{aligned} \frac{D}{A}= & {} \frac{2 \left( E^2+E (m-{V_0})+i \left( m \sqrt{E (-E+{V_0})}+\sqrt{E^3 (-E+{V_0})}\right) \right) }{(E+m) \left( 2 E-{V_0}+2 i \sqrt{E (-E+{V_0})}\right) } \end{aligned}$$

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$$\begin{aligned} \frac{D'}{A}= & {} \frac{2 \sqrt{E m} {V_0}}{(E+m) \left( -2 i E+i {V_0}+2 \sqrt{E (-E+{V_0})}\right) } \end{aligned}$$

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