In my notes, I have the Time Independent Schrodinger equation for a free particle ∂2ψ∂x2+p2ℏ2ψ=0
The solution to this is given, in my notes, as ψ(x)=Ce(ipxℏ)
Now, since (1) is a second order homogeneous equation with constant coefficients, given the coefficients we have, we get a pair of complex roots:r1,2=±ipℏ
Thus, the most general solution looks something like:ψ(x)=c1cos(pxℏ)+c2sin(pxℏ)
However, instead of writing the solution as a cosine plus a sin, the professor seems to have taken a special case of the general solution (with c1=1 and c2=i) and converted the resulting ψ(x)=cos(pxℏ)+isin(pxℏ) into exponential form, using eiθ=cosθ+isinθ to get (2).
The main question I have concerning this is: shouldn't we be going after real solutions, and ignoring the complex ones for this particular situation? According to my understanding Ψ(x,t) is complex but ψ(x) should be real. Thanks in advance.
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