Question

A system is defined by the wavefunction ψ(x) = Acos(2πx/L) for –L/4 ≤ x ≤ L/4.

Determine the normalization constant A and the probability that the particle will be

Found between x = 0 and x = L/8?

Answer 1

The normalization condition is the following (see http://farside.ph.utexas.edu/teaching/qmech/Quantum/node34.html):

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

Substituting the wavefunction, obtain:

$$\int_{-L/4}^{L/4}A^2\cos^2(2\pi x/L)dx = 1\\$$

Taking the integral, find:

$$A^2\left( \dfrac{L}{2\pi}\right)\dfrac{\pi }{2} = 1\\ A = \dfrac{2}{\sqrt{L}}$$

The probability of finding the particle between 0 and L/8 is:

$$\int_{0}^{L/8}\psi^2(x)dx =\int_{0}^{L/8}A^2\cos^2(2\pi x/L)dx = \dfrac{1}{4} + \dfrac{1}{2\pi} \approx 0.409$$

Answer. $A = \dfrac{2}{\sqrt{L}},p = 0.409$.