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![Page 1: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/1.jpg)
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Quantum Mechanics, Concepts and ApplicationsN. Zettili; Wiley 2001
Quantum mechanics. Second editionV.G. Thankappan. New Age, 1993. 9788122425000
Quantum PhysicsF. Scheck. Springer, 2007
Essential Quantum MechanicsGary E. Bowman, 2008, Oxford University Press 0199228922
Introduction to Quantum MechanicsD. Griffiths. Prentice Hall 1995. ISBN 0131244051
Principles of quantum mechanics. Second editionR. Shankar 0306447908
Quantum physicsS. Gasiorowicz
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I. Introducción1.1 La ecuación de Schrödinger1.2 Problemas unidimensionales
1.2.1 La partícula libre1.2.2 Pozos1.2.3 Barreras y tuneleo1.2.4 El oscilador armónico
II. El formalismo de la Mecánica Cuántica
III. Descripción cuántica del átomo.
IV. Interacción semiclásica átomo-radiación.
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(0) (1)
(1) (0) (0)
donde
| |
n n n
n
E E
n V n
0 0 00
ˆnH n E n
0
ˆ
ˆ ˆ
nH n E n
H H V
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2ˆ ˆˆ ˆˆ ˆ ;
2p
H V r L r pm
2
2
, 0
, 0
, 0
z
z
H L
H L
L L
![Page 8: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/8.jpg)
2 2
ˆ
ˆ 1
ˆ
z nlm nlm
nlm nlm
nlm n nlm
L m
L l l
H E
4 2
2 2 1, 2,3...
2R
n
m e ZE n
n
, , ,
1, 2,3...; 1;
mnlm nl lr R r Y
n l n m l
![Page 9: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/9.jpg)
![Page 10: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/10.jpg)
Correcciones al espectro del átomo de hidrógeno dado por la ecuación de Schrödinger:
1. Corrección del movimiento del núcleoUsando la masa reducida
2. Estructura finaa) Correcciones relativistasb) Correcciones por el acoplamiento spín-orbita
3. Estructura hiperfinaDebida a la interacción magnética entre los momentos dipolares del electrón y el protón
4. Corrimiento LambDebido a la cuantización del campo coulombiano
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Energía de Bohr
Estructura fina
Corrimiento Lamb
Estructura hiperfina
4 2mc5 2mc
2 2mc
4 2
p
mmc
m
Jerarquía de las energías en el átomo de hidrógeno
2Constante de estructura fina: / 1 / 137e c
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2
2 2 2 4 2
Energía cinética clásica: 2
Energía cinética relativista:
pT
m
T p c m c mc
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2
2 2 2 4 2
Energía cinética clásica: 2
Energía cinética relativista:
pT
m
T p c m c mc
22 2 2 4 2 2
2 42
2 4
3 2
1 1
1 11 ... 1
2 8
...2 8
pT p c m c mc mc
mc
p pmc
mc mc
p p
m m c
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2 4 2
3 2
0
2 2 4
0 3 2
Por tanto el hamiltoniano relativista será,
en primera aproximación,
ˆ2 8
Escrito de otra manera
ˆ ˆ
donde
ˆ 2 8
p p ZeH
m m c r
H H V
p Ze pH y V
m r m c
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22
3 2
Usamos ahora teoría de perturbaciones
independientes del tiempo para resolver
el problema con el potencial de perturbación
ˆ1
8 R
pV
m c
![Page 16: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/16.jpg)
222 2
0 3 2
ˆ1ˆ ; 2 8R R
pp ZeH V
m r m c
2 22 2
3 2 2
2 2
0 02
Lo escribimos como
ˆ ˆ1 1
8 2 2
1
2
R R R
R
p pV
m c m c m
Ze ZeH H
m c r r
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22
3 2
ˆ1
8 R
pV
m c
2 2
0 02
2 2
2
2 2 42
2 2
1
2
1
2
12
2
R
n nR
n nR
Ze Zenlm V nlm nlm H H nlm
m c r r
Ze Zenlm E E nlm
m c r r
Ze Z enlm E E nlm
m c r r
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2 2 42
2 2R
2 2 2 42 2
R
1 2 2 2 42 2
R
12
2
1 1 12
2
Por tanto,
1 1 12
2
n n
n n
nlm n n
Ze Z enlm E E nlm
m c r r
E Ze nlm nlm E Z e nlm nlmm c r r
E E Ze nlm nlm E Z e nlm nlmm c r r
![Page 19: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/19.jpg)
2
2 2 3
2
2
Tenemos
1 1 1
y
1 1 1
1/ 2
donde
R
nlm nlmr a n
nlm nlmr a n l
am Ze
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1 2 2 2 42 2
R
2 2 2 3
1 1 12
2
1 1 1 1 1 1 y
1/ 2
nlm n nE E Ze nlm nlm E Z e nlm nlmm c r r
nlm nlm nlm nlmr a n r a n l
1 2 2 2 4
2 2 2 3R
1 1 1 1 12
2 1/ 2nlm n nE E Ze E Z em c a n a n l
![Page 21: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/21.jpg)
1 2 2 2 4
2 2 2 3R
1 1 1 1 12
2 1/ 2nlm n nE E Ze E Z em c a n a n l
2
22 2
1 2Como
2
se tiene
n n
e ZE n E
a n a Ze
21 2 2 2 2 4 2
2 2 2 2 3R
222 2 2 4 2
2 2 2 2 3R
2 2
2 2R R
1 2 2 12
2 1/ 2
2 1 2 11 2
2 1/ 2
4 41 4 3
2 1/ 2 2 1/ 2
nnlm n n n
n
n n
EE E Ze n E Z e n E
m c Ze n Ze n l
EZe n Z e n
m c Ze n Ze n l
E En n
m c l m c l
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6 2R
52 6
R
Tenemos que
13.6 eV ; 5.11 10 eV /
así que
13.6 eV10
5.11 10 eV
n
n
E m c
E
m c
1
2R
1 43
2 1/ 2n
nlm n
E nE E
m c l
![Page 23: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/23.jpg)
2
rel 2R
43
2 1/ 2nE n
Em c l
rel
Destruye la degeneración accidental del caso coulombiano.
0 para todo y
Dado , cuanto menor es el valor de mayor es la corrección relativista
La corrección es más importante para los nivel
E n l
n l
25
rel 2
es 1s y 2s
110
40.001%, pero detectable
vE E E
c
![Page 24: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/24.jpg)
Correcciones al espectro del átomo de hidrógeno dado por la ecuación de Schrödinger:
1. Corrección del movimiento del núcleoUsando la masa reducida
2. Estructura finaa) Correcciones relativistasb) Correcciones por el acoplamiento spín-orbita
3. Corrimiento LambDebido a la cuantización del campo coulombiano
4. Estructura hiperfinaDebida a la interacción magnética entre los momentos dipolares del electrón y el protón
![Page 25: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/25.jpg)
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Cuando se aplica un campo magnético externo al átomo de hidrógeno (y a todos los átomos), líneas espectrales bien definidas se desdoblan en múltiples líneas cercanamente espaciadas.
Se debe a la interacción del campo magnético externo con el momento dipolar magnético asociado con el momento angular orbital.
![Page 27: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/27.jpg)
Quantum Mechanics I. A. Galindo y P Pascual
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Si tenemos una distribución de
corrientes , se define el
momento magnético
12
J
r JdVc
Jackson. Classical electrodynamics. Primera edicion Wiley 1962, capítulo 5
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1 12 2
1 1 12 2 2
i i ii
i i i i i ii i
i ii i i i i i
i ii i i
J v q r r v
r q r r v dV q r r r v dVc c
q qq r v r p L
c c m c m
12
r JdVc
![Page 34: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/34.jpg)
B
B B
En el caso del electrón
2donde hemos introducido el magnetón de Bohr
2 2
y
1
l
l
e LL gmc
e emc m
g
12
ii
ii
qL
c m
![Page 35: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/35.jpg)
B
B
Dado que la energía de interacción
está dada como (Jackson)
tenemos
ˆy si entonces
l
z
lz
U B
gU L B
B B k
gU BL
B B B 2 2 2l
e L e eL gmc mc m
![Page 36: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/36.jpg)
(0) (1)
(1) (0) (0)
donde
| |
n n n
n
E E
n V n
0 0 00
ˆnH n E n
0
ˆ
ˆ ˆ
nH n E n
H H V
![Page 37: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/37.jpg)
B
B
BB
ln z
lz
ll
gE nlm BL nlm
gB nlm L nlm
gBm g mB
Blz
gV BL
BnE m B
![Page 38: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/38.jpg)
L
Introduciendo la frecuencia de Larmor
2 R
eBm c
LnE m
BB B ; l
l n z
gV g mB E nlm BL nlm m B
![Page 39: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/39.jpg)
4 14
1 0
EE B J
c c t
BE B
c t
qF qE v B
c
E B A
![Page 40: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/40.jpg)
2
y
1
2
qU q r A r v
c
qL T U mv A v q
c
4 14
1 0
EE B J
c c t
BE B
c t
qF qE v B
c
E B A
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qU q r A r v
c
xx
x x x x x x xx y z
x x xx x y z
dAq qF q v A
c dt x c xdA A A A A A Adx dy dz
v v vdt x dt y dt z dt x y z
A A Aq qF v v v q v A
c x y z x c x
q qF v A q v A
c c
q
v A
F
v
v
A
qc
v A
A
qF qE v B
c
jj j
d U UF
dt q q
![Page 42: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/42.jpg)
qF qE v B
c
E B A
4 14
1 0
EE B J
c c t
BE B
c t
21
2
qL T U mv A v q
c
![Page 43: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/43.jpg)
2
En el caso del campo electromagnético
el hamiltoniano que se obtiene haciendo
la transformación de Legendre
2
qp A
cH q
m
![Page 44: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/44.jpg)
2
1 ˆˆ ,2
eH p A r t e r
m c
2
2La fuerza de Lorentz: , ,
d r vm e E r t B r t
dt c
Las ecuaciones de Hamilton: k kk k
H Hq p
p q
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2
0
ˆ
2
p
mH
ˆˆ ˆ ( , )e
p p A r tc
22
0
ˆ 1 ˆˆ ,2 2
p ep A r t e r
m m c
H H
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2
1 ˆˆ ,2 R
eH p A r t e r
m c
2
2 22 2
2
1 ˆ ˆ ˆˆ ,2
ˆ ˆ ˆ2 2 2
R
R R R R
e e ep A r t i A i A
m c c c
ie ie eA A A
m m c m c m c
![Page 47: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/47.jpg)
2 22 2
2
ˆ2
ˆ ˆ2 2R R RR
ieieA
eA A
m m cm c m c
2
2
´
´
´ 0
B A
A A
A A
A A
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2 22 2
2
ˆ2
ˆ ˆ2 2R R RR
ieieA
eA A
m m cm c m c
0A
2 22 2
2
ˆ ˆ2 2R R R
ie eH A A
m m c m c
![Page 49: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/49.jpg)
Si tenemos un campo magnético
uniforme, entonces podemos poner
12
A B r
![Page 50: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/50.jpg)
12
A B r
Como
tenemos
12
C D C D D C D C C D
A B r B r r B r B B r
![Page 51: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/51.jpg)
12
A B r
A B r r B r B B r
0 0 3 B r Br B r B
, , , ,
3
r x y zx y z
yx zx y z
![Page 52: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/52.jpg)
12
A B r
A B r r B r B B r
0 3 0 r B B r BBr
Ecuaciones de Maxwell.
No hay monopolos magnéticos.
![Page 53: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/53.jpg)
12
A B r
A B r r B r B B r
3 0 0 r B B r Br B
El campo es uniforme
![Page 54: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/54.jpg)
12
A B r
A B r r B r B B r
3 0 0 r B r B B r B
, ,x y z
x y z x
B r B B B x y zx y z
B B B x Bx y z
![Page 55: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/55.jpg)
12
A B r
A B r r B r B B r
3 0 0 r B r B B r B
1 32
A B B B
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2 22 2
2
ˆ ˆ2 2RR R
eH
ieA A
c m cmm
12
ˆ ˆ2R R
A B r
ie ieA r Bm c m c
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ˆ ˆ ˆˆ ˆ
ˆ ˆ ˆ ˆ ˆˆ
A B C B C A C A B
i r B r B p B r p
r p B L B B L
Así que el segundo término del Hamiltoniano es
ˆ ˆ2R R
ie ieA r B
m c m c
ˆ ˆ ˆ ˆ2 2R R R
ie ie eA r B B L
m c m c m c
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2 22 2
2
ˆ2 2
ˆ ˆ ˆ2
ˆ
R R
R R
R
eH A
ieA
m m c
ie eA B L
m c
m
m
c
c
2 22 2
2
ˆˆ ˆ2 2 2R R R
e eH B L A
m m c m c
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2 2 22
2 2
El tercer término del hamiltoniano
( es a lo largo de ) se puede
poner como:
ˆ ˆ2 8R R
B Z
e eA r B
m c m c
2 22
22 ˆ ˆ
2ˆ
22 RR R
eH B L
m cc
eA
m m
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2
2 2 2
ˆ ˆ
ˆˆ ˆ
ˆ ˆ , ,0
0 0 1
ˆ
r B r Bk
i j k
x y z yi xj y x
r B B x y
2ˆr B
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2 2 2 22
2 2 22 2 2
El tercer término del hamiltoniano
( es a lo largo de )
se puede poner como:
ˆ ˆ2 8 8R R R
B Z
e e e BA r B x y
m c m c m c
2 22
22 ˆ ˆ
2ˆ
22 RR R
eH B L
m cc
eA
m m
![Page 62: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/62.jpg)
2 22 2
2
2 2 20
2 22 2 2 2 2 2
02
ˆ ˆComparemos los términos y :2 8
ˆTenemos que y que , así que
ˆ ˆ / 2 y 2
/ 88
R R
RR
RR
e e BB L x y
m c m c
L x y a
eB L e m c B
m c
e Bx y e m c a B
m c
2 2 2
2 2 22
ˆ ˆ2 2 8R R R
e e BH B L x y
m m c m c
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2 2 2 2 203
2 2 92 0 0
Así que
/ 8 1 1
/ 2 4 / 548 / 9 10 Gaussr
r
e m c a BT e B B B
T e m c B c e a e a
2 2 2
2 2 22
ˆ ˆ2 2 8R R R
e e BH B L x y
m m c m c
4
53
2
Como normalmente 10 Gauss,
10
B
T
T
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2 2 2 2 203
2 2 92 0 0
Así que
/ 8 1 1
/ 2 4 / 548 / 9 10 Gaussr
r
e m c a BT e B B B
T e m c B c e a e a
2 2 2
2 2 22
ˆ ˆ2 2 8R R R
e e BH B L x y
m m c m c
12
En las estrellas de neutrones, donde
10 Gauss
este término puede ser muy importante
B
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22 9
0
Habiendo despreciado el tercer término, debemos ver
si el segundo puede ser tratado como una perturbación.
Tenemos
/ 2
Energia del potencial de Coulomb / 5 10 gauss
que en las condiciones nor
re m c BT B
e a
males es muy pequeño.
22 ˆ ˆ
2 2R R
eH B L
m m c
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2
22
1 ˆˆ ,2
ˆ ˆ2 2R R
eH p A r t e r
m c
eH e r B L
m m c
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Potencial perturbativo: 2 z
R
eV BL
m c
L
LIntroduciendo la frecuencia de Larmor 2
Escribimos el potencial perturbativo c ˆomo z
R
eB
m c
V L
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LˆPotencial perturbativo: zV L
L
L
4 2
2 2 L
ˆTenemos que ,
ˆpor tanto,
y
2
nlm nlm
Rnm
V r m r
nlm V nlm m
m e ZE
nm
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2
1
1,0,1
n
l
m
2,1, 1
2,1,0
2,1, 1
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Cuando se aplica un campo magnético externo al átomo de hidrógeno (y a todos los átomos), líneas espectrales bien definidas se desdoblan en múltiples líneas cercanamente espaciadas.
Se debe a la interacción del campo magnético externo con el momento dipolar magnético asociado con el momento angular orbital.
![Page 72: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/72.jpg)
2l
1l
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0 02 L 1 L2 2
2 1
0 0L 2 1 0 L2 2 2 2
1 2 1 2
1 1
E EE m m
n n
E EE m m E m
n n n n
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¿Por qué hay solo 3 líneas y no 15?
Por las reglas de selección:
1
0, 1
l
m
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1
0, 1
l
m
2l
1l
0 L2 21 2
1 1E E m
n n
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2
6. (El principio de la medición) Si el estado
de un sistema es
, ,
entonces la probabilidad que una medición
encuentre al sistema en el estado es
.
n nn
j
j j j
x t c x t
c c c
![Page 77: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/77.jpg)
2 22
Por lo tanto,
,j j j jw c
![Page 78: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/78.jpg)
Atomo con momento dipolar eléctrico:
Regla de selección para el átomo hidrogenoide:
0
si entero arbitrario, 1, 0, 1
i id e r
n l m r nlm
n l m
![Page 79: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/79.jpg)
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Quantum Mechanics I. A. Galindo y P Pascual
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1
2
2 1
5890 A
5896 A
p s
![Page 84: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/84.jpg)
Bbar
0
2 2e
Nj L L Nj
e eN Ng j g L g L
mc mc
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Energía de interacción:
Torca:
Fuerza:
U B
B
F U B
![Page 87: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/87.jpg)
( · ) ( · ) ( · ) ( ) ( ) AB A B B A A B B A
U B F U B B
( · ) ( · ) ( ) ( )
y por t
ˆˆ ˆ
, ,0 0
0 0
y
0,0,
ˆ0,0
anto
( · ) ( )
Tenemos
(
,
· )
x y z y z x z
z
x x y y z z z
x x z y y z z z z z z z
F B B B B B
F B B B
i j k
B B B
B
B B
B B B B k
![Page 88: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/88.jpg)
U B F U B
B
Si el campo es homogéneo
y en la dirección ,
ˆz z z
Z
F B k
![Page 89: Quantum Mechanics, Concepts and Applications N. Zettili; Wiley 2001 Quantum mechanics. Second edition V.G. Thankappan. New Age, 1993. 9788122425000 Quantum.](https://reader035.fdocuments.us/reader035/viewer/2022081416/5665b4431a28abb57c907ec9/html5/thumbnails/89.jpg)
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1
1,0, 1
l
m
ˆ ,1,1 ,1,1
ˆ ,1,0 0
ˆ ,1, 1 ,1, 1
z
z
z
L n n
L n
L n n