Outline how atomic emission spectra provide evidence for the quantization of energy in atoms.

Consider an electron confined in a one-dimensional “box” of length L. The de Broglie waves associated with the electron are standing waves with wavelengths given by $\frac{{2L}}{n}$, where n=1, 2, 3, …

Show that the energy E_n of the electron is given by

$$En = \frac{{{n^2}{h^2}}}{{8{m_e}{L^2}}}$$

where h is Planck’s constant and m_e is the mass of the electron.

An electron is confined in a “box” of length L=1.0×10^–10m in the n=1 energy level. Its position as measured from one end of the box is (0.5±0.5)×10^–10m. Determine

(i) the momentum of the electron.

(ii) the uncertainty in the momentum.

all particles have an associated wavelength / OWTTE;

wavelength is given by $\lambda  = \frac{h}{p}$, where h is Planck’s constant and p is momentum;

from de Broglie hypothesis, ${p_n} = \frac{h}{{{\lambda _n}}} = \frac{{nh}}{{2L}}$;
kinetic energy given by ${E_K} = \frac{{{p^2}}}{{2{m_e}}}$;
combined and manipulated to obtain result;

(i) $\lambda  = \frac{{2L}}{n} = \frac{{2 \times 1.0 \times {{10}^{ - 10}}}}{1} = 2.0 \times {10^{ - 10}}$;
$p = \frac{h}{\lambda } = \frac{{6.6 \times {{10}^{ - 34}}}}{{2.0 \times {{10}^{ - 10}}}} = 3.3 \times {10^{ - 24}}{\rm{kgm}}{{\rm{s}}^{ - 1}}$;
Award [2] for alternative methods, e.g. calculating energy then momentum.

(ii) use of $\Delta x\Delta p \ge \frac{h}{{4\pi }}$;
to get $\Delta p \ge \frac{{6.6 \times {{10}^{ - 34}}}}{{4\pi  \times 0.5 \times {{10}^{ - 10}}}} = 1.1 \times {10^{ - 24}}{\rm{kgm}}{{\rm{s}}^{ - 1}}$;