.. DATA In your research on new solid-state devices, you are studying a solid-state structure that can be modelled accurately

as an electron in a one-dimensional infinite potential well (box) of width L. In one of your experiments, electromagnetic radiation is absorbed in transitions in which the initial state is the n= 1 ground state. You measure that light of frequency

is absorbed and that the next higher absorbed frequency is. (a) What is quantum number nfor the final state

in each of the transitions that leads to the absorption of photons of these frequencies? (b) What is the width Lof the potential well? (c) What is the longest wavelength in air of light that can be absorbed by an electron if it is initially in the n= 1 state?

The set of numbers used to describe the position and energy of the electron in an atom are called quantum numbers. There are four quantum numbers, namely, principal, azimuthal, magnetic and spin quantum numbers. The values of the conserved quantities of a quantum system are given by quantum numbers

(a) From equation (1), the ground-level energy corresponding to n = l is given by:

Let n is the number of the ?nal level in the ?rst transition; 50, form equation (1), its energy is given by:

And so the next higher level corresponds to 77, + 1 and so its energy is:

We know that the energy of the absorbed photon equals the energy siparation between the two levels that the transition

occurs between them, AE : E7-

Substituting for Eml from equation (2) and for En, we get:

So, for the second transition, We have:

Substituting for E73 from equation (2) and for En+11 we get:

Substituting equation (4) by equation (3), we get:

Solving this quadratic equation, we get n = 3;

Therefore, the quantum number n for the final state in each transitions are n = 3 and n = 4

(b) First, we plug our values for f1 and n = 3 into equation (3), and solve for E1, so We get the ground-level energy:

Now, we plug this value into equation (I) (where n : 1) and solve for L, so we get the width of the box

_\

therefore the length of the potential well is 0.900nm

(c) From equation (2), the longest wavelength of a photon corresponds to the smallest energy that photon can have-

And the photon with the smallest energy (longest wavelength) that can be absorbed by an electron in the ground-level is

that makes a transition to the ?rst excited state n = 2 occurs-

Thus, we have:

From equations (1) and (2), we get:

Finally, We plug our values for E1, so we get the longest wavelength:

Therefore the longest wavelength is 889nm