Answer to Question #220414 in Physics for Ethan

Question

An electron is confined within a region of width 1.0x10^-10 m.
(a) Estimate the minimum uncertainty in x - component of electron's momentum
(b) if the electron has momentum with magnitude equal to the uncertainty found in part
(a) what is its K.E.?
Mass of electron = 9.1x10^-31kg.

Answer 1

(a) The Heisenberg's uncertainty principle says

$$\Delta p\cdot \Delta x\geq \hbar$$

Hence,

$$p\sim\Delta p=\frac{\hbar}{\Delta x}\\ =\frac{1.05\times 10^{-34}\:\rm J\cdot s}{1.0\times 10^{-10} \:\rm m}=1.05\times 10^{-24}\:\rm kg\cdot m/s$$

(b)

$$E_k=\frac{p^2}{2m}=\frac{(1.05\times 10^{-24}\:\rm kg\cdot m/s)^2}{2\times 9.1\times 10^{-31}\:\rm kg}=6.1\times 10^{-19}\:\rm J$$