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Advanced Physics Homework Help
Charge Density in Quantum well Sub-bands.
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[QUOTE="kpsr, post: 5404496, member: 585959"] [h2]Homework Statement [/h2] I am trying to calculate the charge density in the first subband (n=1) of the quantum well of length L as shown in the below figure. [ATTACH=full]185731[/ATTACH] here [ATTACH=full]185732[/ATTACH] is the electron wave function for the first sub-band and its value (from relevant equation 1) is given as [B]Ψ[SUB]1[/SUB][SUP]2[/SUP](x) = 2/L*sin[SUP]2[/SUP](πx/L) [/B] from Schrodinger equation total electron charge density is given by [B]n(x) = N[SUB]1[/SUB]×Ψ[SUB]1[/SUB][SUP]2[/SUP](x) [/B] where [B]N[SUB]1[/SUB][/B] is the electron charge density in the first subband(n=1). hence for charge density at x = L/2 the above equation becomes [B]n(L/2) = N[SUB]1[/SUB]×Ψ[SUB]1[/SUB][SUP]2[/SUP](L/2) = [B]N[SUB]1[/SUB][B]×2/L*sin[SUP]2[/SUP](π/2) [COLOR=#ff0000]--result 1[/COLOR][/B][/B][/B] But from wave function normalization,that is the probability of finding electron between 0 and L is one(from relevant equation 2) [B]Ψ[SUB]1[/SUB][SUP]2[/SUP](L/2) ≈1[/B], then the total electron charge density at x=L/2 is [B][B]n(L/2)[/B] = N[SUB]1[/SUB]×1 [COLOR=#ff0000]--result 2[/COLOR] [B][COLOR=#ff0000][/COLOR][/B] 2. Homework Equations [/B] wave-function for the electrons in the quantum well sub bands [ATTACH=full]185733[/ATTACH] wave function normalization(that is the probability of finding electron between 0 and L is one) [ATTACH=full]185734[/ATTACH] [h2]The Attempt at a Solution[/h2] When i made attempts for the solutions i got two results [B][B][COLOR=#ff0000]result 1[/COLOR] [/B][/B]and[B] [B][COLOR=#ff0000]result 2[/COLOR] [/B][/B]as above. Now my question is which is the correct result from these two..? [B][B][B][B][B][/B][/B][/B][/B][/B] [/QUOTE]
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Charge Density in Quantum well Sub-bands.
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