Question

To how small a region must an electron be confined for borderline relativistic speeds say$\mathbf{\text{0.05}}$to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near${\mathbf{\text{10}}}^{\mathbf{\text{-10}}}\mathbf{\text{m}}$? For a lead atom "inner-shell" electron of orbit radius${\mathbf{\text{10}}}^{\mathbf{\text{-12}}}\mathbf{\text{m}}$?

Short answer

a) The position measurement constraint $\text{(Δx)}$is ${\text{3.86×10}}^{\text{-12}}\text{m}$.

b) For atoms having inner shell radiuses close to ${\text{10}}^{\text{-12}}\text{m}$, such as the lead atom, the relativistic treatment may be required. The relativistic effects on the hydrogen atom, with an orbital radius of ${\text{10}}^{\text{-10}}\text{m}$, will, nevertheless, be insignificant.

Step-by-step solution

Step 1:Derivation for position measurement constraint.

Given the uncertainty in the velocity$\text{v}$, we need to figure out the position measurement$\text{(Δx)}$constraint.

$\because \text{ΔpΔx}\ge \frac{\hslash }{2}\Rightarrow \text{mΔvΔx}\ge \frac{\hslash }{2}$

$\therefore \text{mΔvΔx}\ge \frac{\hslash }{2}$

$\therefore 9.1\times {10}^{-31}kg\times 0.05\times 3\times {10}^{8}m/s\times \Delta x³\frac{1.054\times {10}^{-34}\frac{k{g}^{-3}{m}^2}{s}}{2}$

$\therefore \text{Δx}\ge \frac{{\text{1.054×10}}^{\text{-34}}\text{m}}{{\text{2×1.37×10}}^{\text{-23}}}$

$\therefore \text{Δx}\ge {\text{3.86×10}}^{\text{-12}}\text{m}$.

Relativistic effect for hydrogen atom.

According to our findings, an electron confined to a box, such as an atom, with dimensions in the range ofwill begin to display relativistic effects. In other words, for atoms with inner shell radius near to, such as the lead atom, the relativistic treatment may be required. The relativistic effects on the hydrogen atom, which has an orbital radius of, will, nevertheless, be insignificant.