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## Homework Statement

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.

here

**Ψ**

_{1}^{2}(x) = 2/L*sin^{2}(πx/L)from Schrodinger equation total electron charge density is given by

**n(x) = N**

_{1}×Ψ_{1}^{2}(x)where

**N**is the electron charge density in the first subband(n=1).

_{1}hence for charge density at x = L/2 the above equation becomes

**n(L/2) = N**

_{1}×Ψ_{1}^{2}(L/2) =**N**_{1}**×2/L*sin**^{2}(π/2) --result 1But from wave function normalization,that is the probability of finding electron between 0 and L is one(from relevant equation 2)

**Ψ**, then the total electron charge density at x=L/2 is

_{1}^{2}(L/2) ≈1

2. Homework Equations

**n(L/2)**= N_{1}×1 --result 22. Homework Equations

wave-function for the electrons in the quantum well sub bands

wave function normalization(that is the probability of finding electron between 0 and L is one)

## The Attempt at a Solution

When i made attempts for the solutions i got two results

**and**

**result 1****as above.**

**result 2**Now my question is which is the correct result from these two..?