Question

The wavefunction for a particle is defined by:
ψ(X)= [NCos(2πx/L, for –L/4≤x≤L/4
[ 0, otherwise
Determine
i) the normalization constant N, and
ii) the probability that the particle will be found between x = 0 and x = L / 8.

Answer 1

i) Normalization constant could be found using:

$$\int_{-L/4}^{L/4}{|\psi(x)|^2 dx}=1$$

This gives:

$$\int_{-L/4}^{L/4}{N^2*\cos^2{\frac{2\pi x}{L}} dx}=N^2*\frac{L}{4}=1$$

$$N=\frac{2}{\sqrt{L}}$$

So, normalized wavefunction is

$$\psi(x)=\frac{2}{\sqrt{L}}*\cos{\frac{2\pi x}{L}}$$

ii) The probability to find a particle between 0 and L/8 equals:

$$P=\int_{0}^{L/8}{|\psi(x)|^2 dx}=\int_{0}^{L/8}{\frac{4}{L}*\cos^2{\frac{2\pi x}{L}} dx}=\frac{4}{L}*\frac{L(2+\pi)}{16\pi}$$

$$P=\frac{(2+\pi)}{4\pi}\approx 0.41$$