Aula 11 - Potenciais Unidimenssionais.pdf

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Cap. 1

08-10-2018

Prof. Dr. Elias Oliveira Serqueira 63 [email protected]

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ℏ2 𝑑 2 ψ π‘₯ βˆ’ + 𝑉 π‘₯ ψ π‘₯ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2

ℏ2 𝑑 2 ψ π‘₯ βˆ’ + 𝑉0 ψ π‘₯ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2 ℏ2 𝑑 2 ψ π‘₯ βˆ’ = 𝐸ψ π‘₯ 2 2π‘š 𝑑π‘₯

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ℏ2 𝑑 2 ψ π‘₯ βˆ’ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2

ℏ2 𝑑 2 ψ π‘₯ βˆ’ + 𝑉0 ψ π‘₯ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2

𝐸

𝑑2 ψ π‘₯ 2π‘šπΈ =βˆ’ 2 ψ π‘₯ 2 𝑑π‘₯ ℏ 𝑑2 ψ π‘₯ 2ψ π‘₯ = βˆ’π‘˜ 𝑑π‘₯ 2

𝑑2 ψ π‘₯ 2π‘š = βˆ’ 2 𝐸 βˆ’ 𝑉0 ψ π‘₯ 𝑑π‘₯ 2 ℏ 𝑑2 ψ π‘₯ 2π‘š = 2 𝑉0 βˆ’ 𝐸 ψ π‘₯ 2 𝑑π‘₯ ℏ 2π‘š 2 π‘ž = 2 𝑉0 βˆ’ 𝐸 ℏ 𝑑2 ψ π‘₯ 2ψ π‘₯ = π‘ž 𝑑π‘₯ 2

A solução:

ψ π‘₯ = 𝐴 cos π‘˜π‘₯ + 𝐡 sin π‘˜π‘₯ ψ π‘₯ = 𝐴 𝑒 π‘–π‘˜π‘₯ + 𝐡𝑒 βˆ’π‘–π‘˜π‘₯

ψ π‘₯ = 𝐢𝑒 π‘žπ‘₯ + 𝐷𝑒 βˆ’π‘žπ‘₯

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ℏ2 𝑑 2 ψ π‘₯ βˆ’ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2

ℏ2 𝑑 2 ψ π‘₯ βˆ’ + 𝑉0 ψ π‘₯ = 𝐸ψ π‘₯ 2π‘š 𝑑π‘₯ 2

𝐸

𝑑2 ψ π‘₯ 2π‘š = βˆ’ 2 𝐸 βˆ’ 𝑉0 ψ π‘₯ 𝑑π‘₯ 2 ℏ 𝑑2 ψ π‘₯ 2π‘šπΈ =βˆ’ 2 ψ π‘₯ 2 𝑑π‘₯ ℏ 𝑑2 ψ π‘₯ 2ψ π‘₯ = βˆ’π‘˜ 𝑑π‘₯ 2

π‘ž2

2π‘š = 2 𝐸 βˆ’ 𝑉0 ℏ

𝑑2 ψ π‘₯ 2ψ π‘₯ = βˆ’π‘ž 𝑑π‘₯ 2

A solução:

ψ π‘₯ =𝐴

𝑒 π‘–π‘˜π‘₯

+

𝐡𝑒 βˆ’π‘–π‘˜π‘₯

ψ π‘₯ = 𝐢𝑒 π‘–π‘žπ‘₯ + 𝐷𝑒 βˆ’π‘–π‘žπ‘₯

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Resulta

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Ou ainda

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Note que 78

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