From the Schr.C3.B6dinger equation to c1 time-dependence Transition of state

-



1 schrödinger equation c1 time-dependence

1.1 energy operator in schrödinger equation
1.2 unperturbed hamiltonian
1.3 extract c1(t) time dependence





from schrödinger equation c1 time-dependence

the schrödinger equation written :








(
−




ℏ

2



2
m








∂

2



∂

x

2






+
v
(
x
)
)

Ψ
(
x
,
t
)
=
i
ℏ




∂
Ψ
(
x
,
t
)


∂
t






{\displaystyle \left(-{\dfrac {\hbar ^{2}}{2m}}{\dfrac {\partial ^{2}}{\partial x^{2}}}+v(x)\right)\psi (x,t)=i\hbar {\dfrac {\partial \psi (x,t)}{\partial t}}}



energy operator in schrödinger equation

the time derivative in right part of schrödinger equation reads:







i
ℏ




∂
Ψ
(
x
,
t
)


∂
t




=
i
ℏ

(

ψ

0


exp
⁡

(
−
i





e

0


t

ℏ



)


(



c

0



′

(
t
)
−
i




e

0


ℏ




c

0


(
t
)
)

+

ψ

1


exp
⁡

(
−
i





e

1


t

ℏ



)


(



c

1



′

(
t
)
−
i




e

1


ℏ




c

1


(
t
)
)

)



{\displaystyle i\hbar {\dfrac {\partial \psi (x,t)}{\partial t}}=i\hbar \left(\psi _{0}\exp \left(-i{\dfrac {e_{0}t}{\hbar }}\right)\left({c_{0}} (t)-i{\dfrac {e_{0}}{\hbar }}c_{0}(t)\right)+\psi _{1}\exp \left(-i{\dfrac {e_{1}t}{\hbar }}\right)\left({c_{1}} (t)-i{\dfrac {e_{1}}{\hbar }}c_{1}(t)\right)\right)}








i
ℏ




∂
Ψ
(
x
,
t
)


∂
t




=
i
ℏ

(

Ψ

0



(



c

0



′

(
t
)
−
i




e

0


ℏ




c

0


(
t
)
)

+

Ψ

1



(



c

1



′

(
t
)
−
i




e

1


ℏ




c

1


(
t
)
)

)



{\displaystyle i\hbar {\dfrac {\partial \psi (x,t)}{\partial t}}=i\hbar \left(\psi _{0}\left({c_{0}} (t)-i{\dfrac {e_{0}}{\hbar }}c_{0}(t)\right)+\psi _{1}\left({c_{1}} (t)-i{\dfrac {e_{1}}{\hbar }}c_{1}(t)\right)\right)}



unperturbed hamiltonian

on right part, total hamiltonian sum of unperturbed hamiltonian (without external electric field) , external perturbation. allows substitute eigenvalues of stationary states in total hamiltonian. write:










h
^



Ψ
(
x
,
t
)
=

(

e

0



c

0


(
t
)

Ψ

0


(
x
,
t
)
+

e

1



c

1


(
t
)

Ψ

1


(
x
,
t
)
+
e
ϵ
(
t
)
x
Ψ
(
x
,
t
)
)



{\displaystyle {\hat {h}}\psi (x,t)=\left(e_{0}c_{0}(t)\psi _{0}(x,t)+e_{1}c_{1}(t)\psi _{1}(x,t)+e\epsilon (t)x\psi (x,t)\right)}



using schrödinger equation above, end with







e
ϵ
(
t
)
x
Ψ
(
x
,
t
)
=
i
ℏ
(

c

1

′

(
t
)

Ψ

1


(
x
,
t
)
+

c

0

′

(
t
)

Ψ

0


(
x
,
t
)
)


{\displaystyle e\epsilon (t)x\psi (x,t)=i\hbar (c_{1} (t)\psi _{1}(x,t)+c_{0} (t)\psi _{0}(x,t))}



extract c1(t) time dependence

we use bra–ket notation avoid cumbersome integrals. reads :







e
ϵ
(
t
)
(

c

1


(
t
)
x

|


Ψ

1


(
x
,
t
)
⟩
+

c

0


(
t
)
x

|


Ψ

0


(
x
,
t
)
⟩
=
i
ℏ
(

c

1

′

(
t
)

|


Ψ

1


(
x
,
t
)
⟩
+

c

0

′

(
t
)
x

|


Ψ

0


(
x
,
t
)
⟩
)


{\displaystyle e\epsilon (t)(c_{1}(t)x|\psi _{1}(x,t)\rangle +c_{0}(t)x|\psi _{0}(x,t)\rangle =i\hbar (c_{1} (t)|\psi _{1}(x,t)\rangle +c_{0} (t)x|\psi _{0}(x,t)\rangle )}



then multiply



⟨

Ψ

1



|



{\displaystyle \langle \psi _{1}|}

, end following







e
ϵ
(
t
)
(

c

1


(
t
)
⟨

Ψ

1



|

x

|


Ψ

1


⟩
+

c

0


(
t
)
⟨

Ψ

1



|

x

|


Ψ

0


⟩
)
=
i
ℏ

(

c

1

′

(
t
)
⟨

Ψ

1



|


Ψ

1


⟩
+

c

0

′

(
t
)
⟨

Ψ

1



|


Ψ

0


⟩
)



{\displaystyle e\epsilon (t)(c_{1}(t)\langle \psi _{1}|x|\psi _{1}\rangle +c_{0}(t)\langle \psi _{1}|x|\psi _{0}\rangle )=i\hbar \left(c_{1} (t)\langle \psi _{1}|\psi _{1}\rangle +c_{0} (t)\langle \psi _{1}|\psi _{0}\rangle \right)}



the 2 different levels orthogonal,



⟨

Ψ

1



|


Ψ

0


⟩
=
0


{\displaystyle \langle \psi _{1}|\psi _{0}\rangle =0}

. working normalized wave functions,



⟨

Ψ

1



|


Ψ

1


⟩
=
1


{\displaystyle \langle \psi _{1}|\psi _{1}\rangle =1}

.


finally,







e
ϵ
(
t
)

(

c

1


(
t
)
⟨

Ψ

1



|

x

|


Ψ

1


⟩
+

c

0


(
t
)
⟨

Ψ

1



|

x

|


Ψ

0


⟩
)

=
i
ℏ

c

1

′

(
t
)


{\displaystyle e\epsilon (t)\left(c_{1}(t)\langle \psi _{1}|x|\psi _{1}\rangle +c_{0}(t)\langle \psi _{1}|x|\psi _{0}\rangle \right)=i\hbar c_{1} (t)}



this latter equation expresses time variation of c1 time. crux of our calculation, since then, can deduce expression differential equation obtained.







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