Further information 9.1 A justification of the Schrodinger equation

The form of the Schrodinger equation can be justified to a certain extent by showing that it implies the de Broglie relation for a freely moving particle. Free motion means motion in a region where the potential energy is zero (V=0 everywhere). Then 

The function sin kx is a wave of wavelength A=2jc/fc, as we can see by comparing sin kx with sin(27w/A), the standard form of a harmonic wave with wavelength A. To verify that sin kx is indeed a solution, we insert = sin foe into both sides of the differential equation and use 

According to the Schrodinger equation, the final term of this expression is equal to E jf, so it follows that = kW/2m and k= 2mEy /h. 

we note that the energy of the particle is entirely kinetic (because y = 0 everywhere), so the total energy of the particle is just its kinetic energy  

Because E is related to kby E = kW/2m, it follows from a comparison of the two equations tbit p = kh. Therefore, the linear momentmn is related to the wavelength of the wavefimction hy 

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