Question

A electron is located in unidimensional square potential well with infinitely high walls. The width of the well equal to `l` is such that energy level is very dense. Find the density of the energy levels `dN//dE`,i.e., their number per unit energy interval,as a function of `E`. Calculate `dN//dE` for `E= 1.0 eV` if `l= 1.0cm`.

Answer 1

We have found that
`E_(n)=(n^(2)pi^(2) ħ^(2))/(2ml^(2))`
Let `N(E )=` number of states upto `E`. This number is `n`. The number of states upto `E+dE is N(E+dE)=N(E )+dN(E )=1` and `(dN(E ))/(dE)=(1)/(DeltaE)`
Where `DeltaE=` difference in engines between the `n^(th)` & `(n+1)^(th)` level
`=((n+1)^(2)-n^(2))/(2ml^(2))pi^(2) ħ^(2)=(2n+1)/(2ml^(2))pi^(2) ħ^(2)`
`~=(pi^(2) ħ^(2))/(2ml)xxsqrt((2ml^(2))/(pi^(2) ħ^(2)))sqrt(E)xx2`
`=(pi ħ)/(l)sqrt((2)/(m)sqrt(E ))`
Thus `(dN(E ))/(dE)=(l)/(pi ħ)sqrt((m)/(2E))`
For the given case this gives `(dN(E ))/(dE)= 0.816xx10^(7)` level per `eV`