Question

Consider a particle of mass m trapped in an infinite one-dimensional box of width 

0.2 nm. It is found that when the energy of the particle is 230 eV, its eigen function has 

five antinodes. Calculate the mass of the particle. Show that the particle can never have 

energy equal to 1 keV

Answer 1

For a particle in a box potential we know, that the eigenfunction of n-th stationnary state has exactly n antinodes. Thus we deduce that $E_5 = \frac{\pi^2 \hbar^2}{2mL^2} \cdot 25 = 230eV$. From this we find that $m = \frac{25\cdot \pi^2 \cdot 1.1\cdot 10^{-68}}{2 \cdot 4\cdot 10^{-20} \cdot 2.3 \cdot 1.6 \cdot 10^{-19}} \approx 9.2 \cdot 10^{-29} kg$ which is approximately 100 electron masses.

The energy can take values of a form $E_n = \frac{\pi^2 \hbar^2}{2mL^2}\cdot n^2$ and thus $\frac{E_n}{n^2} = const$. We see that for $E= 1keV$ we should have $\frac{E}{n^2} = \frac{E_5}{5^2}$ and thus $n^2 = 25\cdot \frac{100}{23}$ which is not an integer and thus the value 1keV can't be attained.