Ekuasaun schrodinger(Juvinal dos Reis Soares)

Ekuasaun Schrodinger la depende ba tempo 𝐸𝑚𝑒𝑐𝑎𝑛𝑖𝑐𝑎 = 𝐸𝑐𝑖𝑛𝑒𝑡𝑖𝑐𝑎 + 𝐸𝑝𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 𝐸 = 𝐸𝑐 + 𝑉 → 𝑒𝑞𝑢𝑎çã𝑜 1 𝐸𝑐 =

1 𝑚𝑣 2 + 𝑉 → 𝑒𝑞𝑢𝑎çã𝑜 2 2

𝑃 = 𝑚𝑣 → 𝑒𝑞𝑢𝑎çã𝑜 3 𝑃2 = 𝑚2 𝑣 2 → 𝑒𝑞𝑢𝑎çã𝑜 4 𝑣2 =

𝑝2 → 𝑒𝑞𝑢𝑎çã𝑜 5 𝑚2

1 𝑝2 𝐸 = 𝑚 2 + 𝑉 → 𝑒𝑞𝑢𝑎çã𝑜 6 2 𝑚 𝐸=

1 𝑃2 𝑚 + 𝑉 𝑒𝑞𝑢𝑎çã𝑜 7 2 𝑚. 𝑚

𝑝2 𝐸= + 𝑉 → 𝑒𝑞𝑢𝑎çã𝑜 2𝑚 𝑝2

𝐸 = 2𝑚 + 𝑉 → ita ko’alia konaba funsaun da onda entau husi sorin2x ita multiplika ho 𝜓 sai hanesan tuir mai ne’e 𝑝2 𝐸𝜓 = 𝜓 + 𝑉𝜓 2𝑚 𝜓 = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) 𝜕𝜓 = 𝑖𝑘𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) 𝜕𝑥 𝜕 2𝜓 = 𝑖 2 𝑘 2 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) 𝜕𝑥 2 𝜕 2𝜓 = 𝑖 2𝑘2 𝜓 2 𝜕𝑥 𝑘=

2Π → 𝑒𝑞𝑢𝑎çã𝑜 9 𝜆

𝜆=

ℎ → 𝑒𝑞𝑢𝑎çã𝑜 10 𝑃

ℎ = 2Πℏ → 𝑒𝑞𝑢𝑎çã𝑜 11

𝑘=

2Π ℎ 𝑃

𝑘=

2ΠP → 𝑒𝑞𝑢𝑎çã𝑜 12 ℎ

𝑘=

2ΠP 2Πℏ

2ΠP = k2Πℏ 𝑃=

k2Πℏ → 2Π

𝑃 = ℏ𝑘 → 𝑒𝑞𝑢𝑎çã𝑜 13 𝑘=

𝑃 → 𝑒𝑞𝑢𝑎çã𝑜 14 ℏ

𝜕 2𝜓 = 𝑖 2𝑘2 𝜓 𝜕𝑥 2 𝑖 = √−1 𝜕2𝜓 𝜕𝑥 2 𝜕2𝜓 𝜕𝑥 2

= −𝑘 2 𝜓 sinal negative mosu tanba numeru imajinariu i nia valor -1 𝑃2

= − ℏ2 𝜓 → tanba k kuadradu entau P ho hbar ita halo mos kuadrado

−𝑃2 𝜓 = −ℏ2

𝜕 2𝜓 𝜕𝑥 2

ℏ2 𝜕 2 𝜓

𝐸𝜓 = − 2𝑚 𝜕𝑥 2 + 𝑉𝜓 ekuasaun schrodinger la deepened ba tempo ℏ2 𝜕 2 𝜓 𝐸𝜓 = − + 𝑉𝜓 2𝑚 𝜕𝑥 2

𝜓 = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) 𝜕𝜓 = −𝑖𝜔𝑡𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) 𝜕𝑡 𝜕𝜓 = −𝑖𝜔𝑡𝜓 𝜕𝑡 𝜔 = 2𝛱𝑓 𝑒𝑞𝑢𝑎çã𝑜 15 𝐸 = ℎf → 𝑒𝑞𝑢𝑎çã𝑜 16

𝑓=

𝐸 → 𝑒𝑞𝑢𝑎çã𝑜 17 ℎ

𝜔 = 2𝛱𝑓 𝐸 𝜔 = 2𝛱 ( ) ℎ 𝜔ℎ = 𝐸2𝛱 𝐸=

ℎ 𝜔 2𝛱

𝐸 = ℏ𝜔 𝑒𝑞𝑢𝑎çã𝑜 18 𝜔=

𝐸 → 𝑒𝑞𝑢𝑎çã𝑜 19 ℏ

𝜕𝜓 = −𝑖𝜔𝑡𝜓 𝜕𝑡

𝜕𝜓 𝐸 = −𝑖 𝜓 𝜕𝑡 ℏ 𝜕𝜓 𝐸 𝑖 = −𝑖 𝜓 𝜕𝑡 ℏ 𝑖 𝜕𝜓 𝐸 = −𝑖 2 𝜓 𝜕𝑡 𝑖ℏ 𝜕𝜓 𝑖2 = − 𝐸𝜓 𝜕𝑡 𝑖ℏ 𝐸𝜓 = 𝑖ℏ 𝑖ℏ

𝜕𝜓 𝜕𝑥

𝜕𝜓 ℏ2 𝜕 2 𝜓 =− + 𝑉𝜓 𝜕𝑡 2𝑚 𝜕𝑥 2

Tamba V= 0 Maka 𝑖ℏ

𝜕𝜓 ℏ2 𝜕 2 𝜓 =− → 𝑒𝑞𝑢𝑎çã𝑜 𝑠𝑐ℎ𝑟𝑜𝑑𝑖𝑛𝑔𝑒𝑟 𝑑𝑒𝑝𝑒𝑛𝑑𝑒 𝑏𝑎 𝑡𝑒𝑚𝑝𝑜 𝜕𝑡 2𝑚 𝜕𝑥 2 JUVINAL DOS REIS SOARES