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Density of states in 1D
Volume of one unit cell π‘‰π‘˜ = 2πœ‹
𝐿
𝑉𝑙𝑖𝑛𝑒 = 𝐾
𝑁 =2×
𝑉𝑙𝑖𝑛𝑒
𝐾
𝐾𝐿
=2×
=
2πœ‹
π‘‰π‘˜
πœ‹
𝐿
2π‘šπΈ
𝐾=√ 2
ℏ
√2π‘šπΈ
𝐿 √2π‘šπΈπΏ
ℏ2
𝑁=
=
πœ‹
β„πœ‹
1
1
1
βˆ’
𝑑𝑁 𝑑𝑁 π‘‘π‘˜ 2 (2π‘šπΈ) 2 2π‘šπΏ (2π‘šπΈ)βˆ’2 π‘šπΏ
=
=
=
𝑑𝐸 π‘‘π‘˜ 𝑑𝐸
β„πœ‹
β„πœ‹
1
𝐷(𝐸)1𝐷
1
(2π‘šπΈ)βˆ’2 π‘šπΏ
βˆ’
2 π‘š
(2π‘šπΈ)
β„πœ‹
=
=
=
𝐿
β„πœ‹
π‘š
1
β„πœ‹(2π‘šπΈ)2
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