6-5 ① We have ψ(x) = a sin(kx) + b cos(kx) and the boundary condition is that ψ = 0 when x = 0.
Since cos(0) = 1, this can only be true if b =0. In other words, the wavefunction simplifies to:
ψ(x) = a sin(kx) in the region x=0 to x = L, and ψ(x) = 0 outside this region.
② Thinking about a sine curve, we know that sin(kx) = 0 when kx = nπ, where n is any integer. Therefore it must be true that kL = nπ.
Hence k = nπ/L and we can write the wavefunction (in the region between x = 0 and L) as:
