Question

The wave function of a particle is 𝜓(𝑥) = { 𝐴𝑐𝑜𝑠( 2𝜋𝑥 𝐿 ) 𝑓𝑜𝑟 − 𝐿 4 ≤ 𝑥 ≤ 𝐿 4 0 elsewhere i) Determine the normalization constant A. ii) What is the probability that the particle will be found between x= 0 and x = L/6 if we measured its position? iii) Find the expectation values for the operators x, p, and p 2 .

Answer 1

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

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