(Solved): 3. Determine the values of the Einstein coefficients A and B for electric dipole transitions betwee ...

3. Determine the values of the Einstein coefficients $A$ and $B$ for electric dipole transitions between the states $n$ and $n?1$ of a particle in an infinite 1-dimensional square well potential of the form $V(x)=0$ for $0<x<L$, and infinity elsewhere. In particular, confirm that, for large $n,B?L^2$ and $A?n^3{L}^{?4}$. Hint: for $n=1,2,3,\dots ,{?}_0^L\sin \left(\frac{(n?1)?}{L}x\right)x\sin \left(\frac{n?}{L}x\right)dx=?\frac{4L^2}{{?}^2}\frac{n(n?1)}{(2n?1{)}^2}$