Átomo de Hidrógeno 1
of 17
-
Upload
tarquino666 -
Category
Documents
-
view
224 -
download
0
Transcript of Átomo de Hidrógeno 1
-
8/10/2019 tomo de Hidrgeno 1
1/17
SS 06 - 20 102: Introduction to atomic and molecular physics Lecture 1: Basic concepts about atoms and quantum physics
The Schrdinger equation in spherical coordinates.
For most realistic potentials, the Schrdinger equation in Cartesian coordinates is not soluble by using theseparation of the wave function in three independent wave functions. For example, for the electromagneticinteraction between an electron and a proton, the potential is:
Transformation to a spherical coordinate system.
)r(Vr
ke
zyx
ke)z,y,x(V
2
222
2
==
++
=
The Schrdinger equation is then:
Now, the new equation in sphericalcoordinates is separable.
A set of three new quantum numbersappears. But now, the quantumnumbers are not fully independent oneanother.
-
8/10/2019 tomo de Hidrgeno 1
2/17
SS 06 - 20 102: Introduction to atomic and molecular physics Lecture 1: Basic concepts about atoms and quantum physics
Separation of variables
-
8/10/2019 tomo de Hidrgeno 1
3/17
SS 06 - 20 102: Introduction to atomic and molecular physics Lecture 1: Basic concepts about atoms and quantum physics
Separation of variables
1. The Schrdinger equation for the hydrogen atom involves Coulomb potential ke2
/r. Since this is a centralpotential (it has spherical symmetry) the equation can be separated in different independent equationswhen it is expressed in spherical coordinates.
RADIALRADIAL
ANGULARANGULAR
RADIALRADIAL == --LL22 ANGULARANGULAR == --LL
22
0
d
)(dPsin
d
d
)(P
sinsin
L
d
)F(d
)F(
1 22
2
2
2
=
+
h
)(F)(P)r(R),,r( =
-
8/10/2019 tomo de Hidrgeno 1
4/17
SS 06 - 20 102: Introduction to atomic and molecular physics Lecture 1: Basic concepts about atoms and quantum physics
Separation of variables: Solution of angular components
-
8/10/2019 tomo de Hidrgeno 1
5/17
SS 06 - 20 102: Introduction to atomic and molecular physics Lecture 1: Basic concepts about atoms and quantum physics
Solution of the angular part of the Schrdinger equation.
2. The solution of every independent equation gives rise to a different quantum number.
0d
)(dPsin
d
d
)(P
sinsin
Lm
2
2
22
l =
+
h
Cd
)F(d
)F(
12
2
=
cAe)F( =
General solutionGeneral solution
)2F()F( +=
Azimuth boundary conditionAzimuth boundary condition
++
2
l
l
im
mc
egerintm
Ae)F( l
=
=
=
Azimuth solutionAzimuth solution
General solutionGeneral solution
This equation has asolution, only when:
The solution are
Legendre polynomial
l
2
l2
2
2
2
ml;0mL
)1l(lL
>
+=
h
h
cosx
)x(P l
ml
=
iml
mlm,l e)(cosP)(F)(P),(Y
l== lm
l
l
