1. (Kittel 7.1) Density of orbitals in one and two dimensions.(a) Show that the density of orbitals of a free electron in one dimension isL (2mπλheindependent of E.D₁ (ɛ) = -where L is the length of the line.(b). Show that in two dimensions, for a square of area A,Am7h²1/2D₂(E) =

To Derive : The density of oribitals of free electrons in one dimension is. Assumption : The density of state is, D∈= dnd∈ Here dn= No. of electronic quantum state of inergy interval ∈ and ∈td∈,D∈ = 2dnd∈ for each energy level, there is two spin direction Explanation : The total number of state can obtained as, n = k2πL2 n = kLπdndk=Lπ For one dimension the density can be obtained as, D∈ = 2dndk.dkd∈...........................1 For free electron gas is, ∈= h2k22m d∈dk=h2m×2m∈h2 = hm×2m∈ d∈dk= h2∈m From equation number 1 , D∈= 2×Lπ×1hm2∈ D∈=1π2mh2∈12 Thus the proved.To Derive : In the 2-D , for square of area A is independent of ε. Explanation : For The two dimension , n= πk22πL2= πk24π2L2 Here area of x space isπk2 ,area of square isA=L2 n= Ak24πdndk=Ak2π= A2π×2m∈h2D∈=2dndk.dkd∈ = 2×A2π×2m∈h2×1hm2∈ D(∈)= Amπh2 Thus, the independent of ε.

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1. (Kittel 7.1) Density of orbitals in one and two dimensions. (a) Show that the density of orbitals of a free electron in one dimension is where L is the length of the line. L (2m πħ²ε D₁(e) == 1/2

1. (Kittel 7.1) Density of orbitals in one and two dimensions. (a) Show that the density of orbitals of a free electron in one dimension is where L is the length of the line. L (2m πħ²ε D₁(e) == 1/2

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

1. (Kittel 7.1) Density of orbitals in one and two dimensions.
(a) Show that the density of orbitals of a free electron in one dimension is
L (2m
πλhe
independent of E.
D₁ (ɛ) = -
where L is the length of the line.
(b). Show that in two dimensions, for a square of area A,
Am
7h²
1/2
D₂(E) =

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