6∫R√2m(E-V(ds=0←→6∫n(ds=0 Presently unknown ☐凸2+学=0 Presumed to be of the form 720+An20=0 Here A is an unknown constant,and the expression v2mT plays the role of"index of refraction"for the propagation of matter waves.The unknown equation now can be written as 72地+A2m(E-V(川中=0 Substitute the known free particle solution 沙=err B= V(⊙=0 into the above equation.We find the unknown constant A equals to 1/h2,therefore we have 2+01B-v(=0 for the general case.It is often written in a form -en+vnen and bears the name"Stationary Schrodinger Equation' C.Conclusion By a way of comparison,we obtained the equation of motion for a particle with definite energy moving in an external potential field V()-the Stationary Schrodinger Equation. From the procedure we stated above,here we stressed on the"wave propagation"side of the motion of micro-particle.v2mT=V2m(E-V()is treated as"refraction index"of matter waves It may happen for many cases that in some spatial districts E is less than V(),i.e.. E<V ()What will happen in such cases? δ R B A p 2m(E − V (~r))ds = 0 ⇐⇒ δ R B A n(~r)ds = 0 ⇓ ⇓ Presently unknown ? ⇐⇒ ∇2ψ + n 2ω 2 c 2 ψ = 0 ⇓ Presumed to be of the form ∇2ψ + An2ψ = 0 Here A is an unknown constant, and the expression √ 2mT plays the role of ”index of refraction” for the propagation of matter waves. The unknown equation now can be written as ∇2ψ + A [2m(E − V (~r))] ψ = 0 Substitute the known free particle solution ψ = e i ~ ~p·~r E = 1 2m ~p 2 V (~r) = 0 into the above equation. We find the unknown constant A equals to 1/~ 2 , therefore we have ∇2ψ + 2m ~ 2 [E − V (~r)] ψ = 0 for the general case. It is often written in a form − ~ 2 2m ∇2ψ (~r) + V (~r) ψ (~r) = Eψ (~r) and bears the name ”Stationary Schr¨odinger Equation” C. Conclusion By a way of comparison, we obtained the equation of motion for a particle with definite energy moving in an external potential field V (~r) - the Stationary Schr¨odinger Equation. From the procedure we stated above, here we stressed on the ”wave propagation” side of the motion of micro-particle. √ 2mT = p 2m(E − V (~r)) is treated as ”refraction index” of matter waves. It may happen for many cases that in some spatial districts E is less than V (~r), i.e., E < V (~r). What will happen in such cases? 6