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Find the particular solution of the time dependet Schrodinger equation for a freely moving particle of mass m. |
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Answer» Solution :THEL Schrodinger equation in one dimension for a free particle is `i ħ(del Psi)/(delt)=-( ħ^(2))/(2m)(del^(2)Psi)/(delx^(2))` we WRITE `Psi(x,t)= varphi(x)chi(t)`. Then `((i)/ ħ(SQRT(2mEx-Et)) ħ)/(chi)(dchi)/(dt)=-( ħ^(2))/(2m)(1)/(varphi)(d^(2)varphi)/(dx^(2))=E`,say Then `chi(t)~exp(-(IET)/(h))` `varphi(x)~exp(isqrt(2mE)/( ħ)x)` `E` must be real and positive if `varphi(x)` is to be bounded everywhere. Then `Psi(x,t)=Const exp((i)/ (ħ)(sqrt(2mE)x-Et))` This particular solution DESCRIBES plane waves. |
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