Electron energy levels

An electron is in a certain energy state in a one-dimensional, infinite potential well from $x = 0$ to $x = L = 175 pm$. The electron’s probability density is zero at $x =0.250L$, and $x =0.333L$; it is not zero at intermediate values of x. The electron then jumps to the next lower energy level by emitting light. What is the change in the electron’s energy?

P.S please tell me how to get n (quantum number)

Answer

The energy levels of electron in infinite dimensional well are

$E(n)= n^2*h^2/(8*m*L^2)$

(see http://en.wikipedia.org/wiki/Particle_in_a_box#Energy_levels)

on the lowest energy level $n=1$, $L= \lambda/2$

for $n=2$ , $L = 2*\lambda/2$

and in general $L = n*\lambda/2$

(see http://en.wikipedia.org/wiki/Particle_in_a_box#Wavefunctions)

for the electron in problem $\lambda/2 = (0.333-0.25)*L =0.083*L$

thus $n=1/0.083 =12$

the transition is between energy levels $n =12$ and $n=11$

$E(12-11) =(12^2-11^2)*h^2/(8*m*L^2) =$

$=23* (6.62*10^{-34}/175*10^{-12})^2 *1/8/9.1*10^{-31}==4.529*10^-17 J =283 eV$