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.
Ψ12(x) = 2/L*sin2(πx/L)
from Schrodinger equation total electron charge density is given by
n(x) = N1×Ψ12(x)
where N1 is the electron charge density in the first subband(n=1).
hence for charge density at x = L/2 the above equation becomes
n(L/2) = N1×Ψ12(L/2) = N1×2/L*sin2(π/2) --result 1
But from wave function normalization,that is the probability of finding electron between 0 and L is one(from relevant equation 2)
Ψ12(L/2) ≈1, then the total electron charge density at x=L/2 is n(L/2) = N1×1 --result 2
2. 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 result 1 and result 2 as above.
Now my question is which is the correct result from these two..?