Question

What is the maximum precision with which the momentum of an electron can be known if the uncertainty in the position of electron is $\pm 0.001\mathring{A}?$ Will there be any problem in describing the momentum if it has a value of $\frac{h}{2\pi a_0}$ where $a_0$ is Bohr's radius of first orbit, i.e., $0.529\mathring{A}?$

Answer 1

According to Heisenberg's uncertainty principle,

$△x.△p=\frac{h}{4r}$

uncertainty in momentum can be calculated as,

$△p=\frac{h}{4r}△x=6.63\times 10^{-34}/(4\times 3.14\times 0.001\times {10}^{-10})=5.27\times {10}^{-22}Ns$

If momentum $=\frac{h}{2\pi r_o}=\frac{6.63\times {10}^{-34}}{2\times 3.14\times 0.529\times {10}^{-10}}=2\times {10}^{-24}Ns$

Uncertainity In momentum is $=\frac{5.27\times {10}^{-22}}{2\times {10}^{-24}}$

$=263.5$ times as large as the momentum itself.