[<< wikibooks] This Quantum World/Implications and applications/Time independent Schrödinger equation
== Time-independent Schrödinger equation ==
If the potential V does not depend on time, then the Schrödinger equation has solutions that are products of a time-independent function

ψ
(

r

)

{\displaystyle \psi (\mathbf {r} )}
and a time-dependent phase factor

e

−
(
i

/

ℏ
)

E

t

{\displaystyle e^{-(i/\hbar )\,E\,t}}
:

ψ
(
t
,

r

)
=
ψ
(

r

)

e

−
(
i

/

ℏ
)

E

t

.

{\displaystyle \psi (t,\mathbf {r} )=\psi (\mathbf {r} )\,e^{-(i/\hbar )\,E\,t}.}
Because the probability density

|

ψ
(
t
,

r

)

|

2

{\displaystyle |\psi (t,\mathbf {r} )|^{2}}
is independent of time, these solutions are called stationary.
Plug

ψ
(

r

)

e

−
(
i

/

ℏ
)

E

t

{\displaystyle \psi (\mathbf {r} )\,e^{-(i/\hbar )\,E\,t}}
into

i
ℏ

∂
ψ

∂
t

=
−

ℏ

2

2
m

∂

∂

r

⋅

∂

∂

r

ψ
+
V
ψ

{\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial }{\partial \mathbf {r} }}\cdot {\frac {\partial }{\partial \mathbf {r} }}\psi +V\psi }
to find that

ψ
(

r

)

{\displaystyle \psi (\mathbf {r} )}
satisfies the time-independent Schrödinger equation

E
ψ
(

r

)
=
−

ℏ

2

2
m

(

∂

2

∂

x

2

+

∂

2

∂

y

2

+

∂

2

∂

z

2

)

ψ
(

r

)
+
V
(

r

)

ψ
(

r

)
.

{\displaystyle E\psi (\mathbf {r} )=-{\hbar ^{2} \over 2m}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)\psi (\mathbf {r} )+V(\mathbf {r} )\,\psi (\mathbf {r} ).}