Question

. A thin solid barrier in the x-y plane has a circutar hole, the size of the first Bohr orbit (of the H atom), whose radius is 0.0529 nm (and diameter = 0.1058 nm). A non-relativistic electron traveling in the z-direction with vx 0 m/s passes througlh the hole. What is the range within which vx is likely to be after it passes the hole. (for extra credit of 2 points, discuss briefly, how this speed compares with that of vi-2.19x106 m/s, the speed of the electron in the first Bohr orbit of the H atom)

Answer 1

Diameter of the hole is given to be

d = 0.1058 x 10^-9 m

so, uncertainty in the position of the electron will be atmost

x = 0.1058 x 10^-9 m

now use uncertainty relation for the conjugate pair x and p_x,

Diameter of the hole is given to be d = 0.1058 times 10^-9 m so, uncertainty in the position of the electron will be atmost x = 0.1058 x 10^-9 m now use uncertainty relation for the conjugate pair x and p_x, Delta p_x greaterthanorequalto planck/Delta x rightwardsdoublearrow Delta p_x = 9.9675 times 10^-25 kgm/s so, the momentum of the electron along the x direction will then be atleast p_x therefore, v_x = p_x/m = 1.095 times 10^6 m/s. this is the speed which the electron is likely to have after passing through the hole. This speed is half of the speed of the electron in the first Bohr orbit of the H atom.

=> $\Delta p_x \geq 9.9675\times 10^{-25}kgm/s$

so, the momentum of the electron along the x direction will then be atleast p_x = 9.9675 x 10^-25 kgm/s

therefore, $v_x=\frac{p_x}{m} = 1.095\times 10^6 m/s$ .

this is the speed which the electron is likely to have after passing through the hole. This speed is half of the speed of the electron in the first Bohr orbit of the H atom.