Question

A free proton has a wave function Psi (x) = A sin (kx), where k = 1.2 times 10^10 m^-1 What is the proton's lambda? What is the proton's momentum? What is the proton's speed? Normalize Psi (x) if the wave only exists inside an infinite square well with width a = 2.1 m, (so that Psi (x) = A sin (kx) between 0 < x < a and Psi (x) = 0 otherwise).

Answer 1

a] The wavelength of the proton will be:

$\lambda = \frac{2\pi}{k} = \frac{2\pi}{1.2\times 10^{10}} = 5.236 \times 10^{-10}m$

b] Proton's momentum is given by:

$p = \frac{h}{\lambda}=\frac{6.626\times 10^{-34}}{5.236\times 10^{-10}} = 1.2655\times 10^{-24}kgm/s$

c] p = mv

m = 1.67 x 10^-27 kg

=> v = 757.768 m/s

d] The Normalization condition here will be:

$\int^{2.1}_0 \psi (x)\psi^*(x)dx = 1$

so, $A\int^{2.1}_0 sin^2(kx)dx = 1$

=> $A\int^{2.1}_0 \frac{1-cos2(kx)}{2}dx = 1$

=> $A[\frac{x}{2}-\frac{sin2(kx)}{2(2)}]^{2.1}_0 = 1$

=> $A[1.05-\frac{sin2(k2.1)}{4}] = 1$

=> $A = \frac{1}{[1.05-0.25sin(4.2k)]}$ .