You want to modify the finite potential well of Fig. 39-7 to allow its trapped electron to exist in more than four quantum states. Could you do so by making the well (a) wider or narrower, (b) deeper or shallower?

The energy E of an electron in a finite potential well similar to an infinite well varies with the width of the well L as

$\mathit{E}\mathbf{\infty }\frac{\mathbf{1}}{{\mathbf{L}}^{\mathbf{2}}}$ ..... (I)

For a fixed depth, more than four bound states can be allowed if the energy level of each bound state is reduced from its initial value.

When the width becomes wider energy at the $n^{th}$state, it decreases that it can be less than $U_0$. Therefore it can allow its trapped electron to exist in more than four quantum states.

From equation (I) this is achievable if L is increased, that is the well is made wider.

Bound states are possible in a particular quantum state as long as the corresponding energy is less than the height of the potential well. Thus more energy levels and hence more states can be accommodated by increasing the height of the potential that is making the well deeper.