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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. |
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Answer» Solution :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` |
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