Chapter 39, Problem 1Q

To determine

To rank:

The electrons according to their ground state energies

Answer to Problem 1Q

Solution:

The ranking of electrons according to their ground state energies is $E_a>E_c>E_b.$

Explanation of Solution

1) Concept:

Using the formula for ground state energy of an electron in one dimensional infinite potential well, we can find the relation between energy and width of the well, and then from the given values of widths, we can rank the electrons according to their ground state energies.

2) Formulae:

$E_n=\frac{n^2{h}^2}{8m{L}^2}$

3) Given:

i) $L_a=50 pm$

ii) $L_b=200 pm$

iii) $L_c=100 pm$

4)  Calculations:

The ground state energy of an electron in one dimensional infinite potential well is given by

$E=\frac{1^2{h}^2}{8m{L}^2}$

From this, we can interpret that $E\propto \frac{1}{L^2}$

Since,

$L_b>L_c>L_a$

Then,

$E_a>E_c>E_b$

Conclusion:

The energy of an electron in one dimensional infinite potential well increases with the decreasing length of the potential well.