Ecuaciones Escalon Potencial

Escalón de Potencial


Se inicia con la ecuación de Schrödinger

〖-ℏ〗^2/2m (d^2 ψ_((x)))/(dx^2 )+Vψ_((x))=Eψ_((x))
En R_1

〖-ℏ〗^2/2m (d^2 ψ_(I(x)))/(dx^2 )=Eψ_(I(x))

(d^2ψ_(I(x)))/(dx^2 )+2mE/ℏ^2 ψ_I(x) =0

K^2=2mE/ℏ^2

ψ_((x) )=Ae^ikx+Be^(-ikx)

En R_2
〖-ℏ〗^2/2m (d^2 ψ_((x)))/(dx^2 )+V_0 ψ_((x))=Eψ_((x))

〖-ℏ〗^2/2m (d^2 ψ_((x)))/(dx^2 )=(E-V_0 ) ψ_((x))

(d^2ψ_((x)))/(dx^2 )=-2m/ℏ^2 (E-V_0 ) ψ_((x))

(d^2 ψ_((x)))/(dx^2 )+2m/ℏ^2 (E-V_0 ) ψ_((x) )=0

(d^2 ψ_((x)))/(dx^2 )-2m/ℏ^2 (V_0-E) ψ_((x) )=0

K^2=2m/ℏ^2 (V_0-E)

(d^2 ψ_((x)))/(dx^2 )-K^2ψ_((x) )=0

ψ_((x) )=Ce^kx+De^(-kx)

Igualar
ψ_I(0) =ψ_II(0)
Ae^ik(0) +Be^(-ik(0) )= De^(-k(0) )
A+B=D

〖ψ'〗_I(0) =〖ψ'〗_II(0)
ikAe^ik(0) -ikBe^(-ik(0) )= -kDe^(-k(0) )
ikA-ikB= -kD
ComoA+B=D =>B=D-A
ik_I A-ik_I (D-A)= -k_II D
2ik_I A-ik_I D= -k_II D
2ik_I A= iD(〖k_I+k〗_II )
D= (2k_I A)/((〖k_I+k〗_II ) )
B= (2k_I A)/((〖k_I+k〗_II ) )-A
B= (2k_I A-A(〖k_I+k〗_II ))/((〖k_I+k〗_II ))
B= (2k_I A-〖k_I A-k〗_II A)/((〖k_I+k〗_II ) )
B= (〖k_I A-k〗_II A)/((〖k_I+k〗_II ) )
B= A(〖k_I-k〗_II )/((〖k_I+k〗_II ) )

ψ_I(x) =Ae^ikx+A(〖k_I-k〗_II )/((〖k_I+k〗_II ) ) e^(-ikx)
ψ_II(x)=Ce^kx+(2k_I A)/((〖k_I+k〗_II ) ) e^(-kx)


Barrera de Potencial

Se inicia con la ecuación de Schrödinger

〖-ℏ〗^2/2m (d^2 ψ_((x)))/(dx^2 )+Vψ_((x))=Eψ_((x))
En R_1

〖-ℏ〗^2/2m (d^2 ψ_((x)))/(dx^2)=Eψ_((x))

(d^2 ψ_(I(x)))/(dx^2 )+2mE/ℏ^2 ψ_I(x) =0
K^2=2mE/ℏ^2

ψ_I(x) =Ae^(ik_I x)+Be^(-ik_I x)
En R_2

〖-ℏ〗^2/2m (d^2 ψ_(II(x)))/(dx^2 )+V_0 ψ_(II(x))=Eψ_(II(x))

〖-ℏ〗^2/2m (d^2ψ_(II(x)))/(dx^2 )=(E-E_0 ) ψ_(II(x))

(d^2 ψ_((x)))/(dx^2 )=-2m/ℏ^2 (E-E_0 ) ψ_(II(x))

(d^2 ψ_((x)))/(dx^2 )+2m/ℏ^2 (E-E_0 ) ψ_II(x) =0

(d^2 ψ_((x)))/(dx^2 )-2m/ℏ^2 (E_0-E) ψ_II(x) =0K^2=2m/ℏ^2 (E_0-E)

(d^2 ψ_((x)))/(dx^2 )-K^2 ψ_((x) )=0

ψ_((x) )=Ce^(-k_II x)+De^(k_II x)

En R_3

〖-ℏ〗^2/2m (d^2 ψ_(3(x)))/(dx^2 )=Eψ_(3(x))

(d^2 ψ_(3(x)))/(dx^2 )+2mE/ℏ^2 ψ_3(x) =0...