o espectro de energia do átimo de hidrogênio

7/24/2019 O espectro de energia do timo de Hidrognio

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xz


5/80


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b

c

a

b


7/80


8/80

En 13.6n2

h 6.62 1034


9/80


10/80


11/80

d2r

dt2 = k

r2r,


12/80

a

b

t2 t1

S= t2

t1(K V) dt,

K

V

L(qi, qi) =K V

q (t)

q (t) q(t)+ q

q (t1) = q(t2) = 0

S= 0

b

c

a

b


13/80

x= g

a

a

c

a

b

b

b

c

S =

t2t1

L(qi+qi, qi+qi) dt t2

t1

L(qi, qi) dt

=

t2t1

L

qiqi+

L

qiqi

dt=

t2t1

L

qiqi+

L

qi

d

dtqi

dt,

S =

t2t1

L

qiqi+

d

dt

L

qiqi

d

dt

L

qiqi

dt

= t2

t1 L

qiqi d

dt

L

qiqi +

t2

t1

d

dt L

qiqi dt

ddt

Lxi

xi

=

ddt

Lxi

xi + Lxi

ddtxi


14/80

qi

t1

t2

S=

t2t1

L

qiqi d

dt

L

qiqi

dt= 0.

L

qi d

dt

L

qi= 0


15/80

m1 m2 r 1

r 2

U(r 1,r 2) =1

2

2

a=bV(r 1,r 2),

a, b= 1, 2

K=1

2

2a=1

ma| r a|2,

| r a|2 = ijxiaxja


16/80

L=1

2

2

a=1

ma| r a|2 V(r 1,r 2)

.

xia =xia+x

ia

L = L(xia,

xi

a) L(xia, xia) L

xiaxia+

L

xiaxia.

S= 0


17/80

L = L

xiaxia+

L

xiaxia

= L

xiaxia+

d

dt

L

xiaxia

d

dt

L

xiaxia

=

L

xia d

dt

L

xia

xia+

d

dt

L

xiaxia

= 0 d

dt

L

xiaxia

= 0.

Q

xia=i

xia

L = L

xia

i

V(r 1,r 2) = V(|r 1 r 2|) V(r) rr 1 = rr 2 L = 0

Qi =

a

L

xiai.


18/80

Pi = a

L

xia

P =m1 r 1+m2 r 2.

P

P= 0

r = r1 r2

L=1

2 | r |2 V(|r |)

= m1m2

m1+m2

r

V(r)

r a

R

r = Rr


19/80

r

|r|2 = rTr = (Rr )T(Rr ) = r TRTRr.

r

RTR = I33

I

R3

R(1) =

1 0 0

0 cos 1 sen10 sen1 cos 1

R(2) =

cos 2 0 sen2

0 1 0

sen2 0 cos 2

R(2) =

cos 3 sen3 0sen3 cos 3 0

0 0 1

.

a a


20/80

a = 0

R(1) =R(0) + dRd1

1=0

1=

1 0 0

0 1 0

0 0 1

+

0 0 0

0 0 1

0 1 0

1

R(2) =R(0) + dR

d2

2=0

2=

1 0 0

0 1 0

0 0 1

+

0 0 10 0 0

1 0 0

2

R(3) =R(0) +dR

d3

3=0

3=

1 0 0

0 1 0

0 0 1

+

0 1 0

0 0 0

1 0 0

3.

xi=Rji xj

Iji +a(Ja)

ji

xj =xi+a(Ja)

jixj

xi=a(Ja)jixj.

L

xixi = mx

ia(Ja)ji xj

Ja


21/80

(Ja)ji =aij pi= mxi

Qa=aijpixj

Li= Qi

A

t =t .

L=r p


22/80

xi = xi(t)

xi(t) =x

i(t

+)

xi(t)

xi(t) +xi(t) xi(t) xi(t)

L = L

xixi+

L

xixi=

L

xixi(t) +

L

xixi(t)

=

L

xixi(t) +

L

xixi(t)

=

dL

dt.

L = ddt

Lxi

xi

d

dt

L

xixi

=

d

dtL,

d

dt

L

xixi(t) L

= 0.

E= L

xixi L= m xi xi 1

2mxixi+V(|r |) =1

2m| r |2 +V(|r |)


23/80

qi(t)

pi(t) = L

qi.

f

g

{f, g} = fxi

g

pi f

pi

g

xi

{f, g} = {g, f}

{f,gh} = g{f, h} + {f, g}h.

{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0.


24/80

{xj, pk

}=

xj

xi

pk

pi xj

pi

pk

xi=jiki = jk

{xi,

Lj}

={

xi, jkspkxs}

=jksikxs=jisxs

xi = aaijxj

xi = j{xi,Lj}.

pi=mxi

H=

L

xixi L

x=x(p)

=|p|2

2m+V(|r |)

{xi, H} = {xi,pjpj2m

+V |r |} = {xi,pjpj2m

}

= 12m

({xi, pj}pj+ pj{xi, pj}) = 1m

ijpj = 1m

pi= xi.


25/80

xi=t xi

xi=t{xi, H}

qi(t)

qi=a{

qi, Qa}

.

dqi

dt = {qi, H} dpi

dt = {pi, H}.

q

p

= t


26/80

{Li, Lj

}=

{irsxrps, jmnxmpn

}=irsjmn

{xrps, xmpn

}=irsjmn(xr{ps, xmpn} + {xr, xmpn}ps) =irsjmn(xrxm{ps, pn} +xr{ps, xm}pn++xm{xr, pn}ps+ {xr, xm}pspn) =irsjmn(xrsmpn+xmrnps) == irmjmnxrpn+irsjmrxmps=irmmjnxrpn+sirrjmxmps== (ijrn injr)xrpn+ (sjim ijsm)xmps= xnpnij xjpi+xipj xspsij ==xipj xjpi = ijkLk

ijkLk =ijkkrsxrps= (irjs isjr)xrps=xipjxjpi

{H, H} = 0.

{Li, H} =irsxrps, 1

2mpipi+V(|r |)

= irs

1

2mpipi+V(|r |), xrps

=

=irs

1

2mpipi, xrps

+irs {V(|r |), xrps} .

irs

1

2mpipi, xrps

=

1

2mirs(pi{pi, xrps} + {pi, xrps}pi) = 1

2mirs(pixr{pi, ps} +pi{pi, xr}ps

+xr{pi, ps}pi+ {pi, xr}pspi) = 0;


27/80

irs{

V(

|r

|), xrps

}=xrirs

{V(

|r

|), ps

}+irs

{V(

|r

|), xr

}ps=xrirs V(|r |)

xi

ps

pi V(|r |)

pi

ps

xi

+irs

V(|r |)

xi

xr

pi V(|r |)

pi

xr

xi

ps=xrirs

V(|r |)xi

=xrirsV(|r |)

r

r

xi=

=xrirsV(|r |)

r

xi

r=irsxrxi

1

r

V(|r |)r

= 0.

{Li, H} = 0

{Li, H} = 0 {Li, Lj} = ijkLk

{H, H} = 0.

Li H

V r2


28/80

p L

d(p L)dt

= dp

dt L +p d

L

dt =

dp

dt L= F(r)r (r p)

F(r)r

(r

p) = (rp)r

(rr )p.

d(p L)dt

= (F(r)r p)r (F(r)r r ) p

= m(F(r)r drdt

)r m(F(r)r r ) drdt

= mF(r)(dr

dtr r dr

dt).

d(rr

)

dt = 1

r2dr

dt+

1

r

dr

dt,

d(p L)dt

=

mF(r)r2

d(rr

)

dt

.


29/80

F(r) =

k

r2

d(p L)dt

=mkd(r

r)

dt .

d

dt (p

L mkr

r ) = 0

A= p L mkrr

,

H

Li Ai {Li, H} =0 {Ai, H}


30/80

{Li,

{Lj, H

}}+

{Lj,

{H, Li

}}+

{H,

{Li, Lj

}}= 0

{H, {Li, Lj}} = 0 {Li, Lj} {Li, Lj} =ijkLk

{Ai{Lj, H}}+ {Lj{H, Ai}} + {H{Ai, Lj}} = 0.

{Ai, Lj}

{Ai, Aj}

{Ai, Lj} = {imnpmLn mk xir

, Lj},

=imn{pmLn, Lj} mk{xir

, Lj}

=imnpm{Ln, Lj} +imn{pm, Lj}Ln mkxi{r, Lj} mkr {xi, Lj},

=imnnjkpmLk+imnmjrprLn mkr

ijkxk,

= (ijmk

ikmj)pmLk+ (njir

nrij)pnLr

mk

r ijkxk,

=piLj pjLi mkr

ijkxk,

=ijk(tukptLu mkxkr

),

=ijkAk.


31/80

{Ai, Aj

}=

{irsprLs

mkxi

r

, jmnpmLn

mkxj

r },

= {irsprLs, jmnpmLn} mk{irsprLs,xjr} +mk{jmnpmLn,xi

r}.

{irsprLs, jmnpmLn} =irsjmn{prLs, pmLn},=irsjmn(pr

{Ls, pmLn

}+

{pr, pmLn

}Ls),

=irsjmn(prpm{Ls, Ln} +pr{Ls, pm}Ln+pm{pr, Ln}Ls++ {pr, pm}LnLs),=irsjmnsnoprpmLo irsjmnmstprLnpt+irsjmnrntpmLspt,=irs(joms jsom)prpmLo irs(nsjt ntjs)prLnpt++irs(jtmr jrmt)pmLspt,=irs(prpsLj

prpoLojs)

irs(prLspt

prLtptjs)+

+irs(prLspj pmLspmjr),= irsprpoLojs irsprLspt+irsprLtptjs +irsprLspj irspmLspmjr ,= irjprpoLo irsprLspt+irjprLtptirsprLspj ijspmLspm,= ijspmLspm,=

ijsp

2Ls.


32/80

{irsprLs,

xj

r}=

{irsprLs,

xj

r},

=irs(prxj{Ls,1r} +pr{Ls, xj}1

r+xj{pr,1

r}Ls+ {pr, xj}Ls

r ),

=irspr

r{Ls, xj} +irsxj{pr,1

r}Ls+irs{pr, xj}Ls

r ,

= irsprr

jsuxu+irsxjxr

r3Ls irsrj Ls

r

{jmnpmLn,xir} =jmnpm

r invxv jmnxi xm

r3Ln+jmnmi

Ln

r .

= irsprr

jsuxu+irsxj xrr3

Ls irsrj + jmnpmr

invxv jmnxi xmr3

Ln+jmnmi

= (iujr ijru)prxur

+irsxjxrLs

r3 ijs Ls

r + (jvmi ijmv)pmxv

r jmn xixmLn

r3 +jin

Ln

r

= pjxir

+prxr

r ij+ irs

xjxrLs

r3 ijs Ls

r +

pixj

r pmxm

r ij jmn xixmLn

r3 +jin

Ln

r

= pjxir

+irsxjxrLs

r3 ijs Ls

r +

pixj

r jmn xixmLn

r3 +jin

Ln

r

=1

r(pixj pjxi) + 2ijk Lk

r +

1

r3(irsxjxrLs jmnxixmLn)

= 3ijkLk

r + 1

r3 (irsxjxrLs jmnxixmLn)= 3ijk

Lk

r +

1

r3(xjxir p xjpir2 xixjr p +xipjr2)

= 3ijkLk

r +

1

r(xjpi+xipj)

= 3ijkLk

r1

rijkLk

= 2ijkLk

r ,


33/80

{Ai, Aj} = ijkp2Lk+ 2ijk Lkr

= 2mHijkLk.

{Li, Lj} =ijkLk{Li, H} = 0{Li, Aj} =ijkAk{Ai, H} = 0{Ai, Aj} = 2mHijkLk

E =|E|

Ai=

Ai2m|E| .

{Li, Lj} =ijkLk {Li, H} = 0 {Li, Aj} =ijk Ak {Ai, H} = 0 {Ai, Aj} =ijkLk {H, H} = 0

Li Ai


34/80

Mi=1

2(Li+ Ai) Ni=

1

2(Li

Ai),

{Mi, Mj} = {12

(Li+ Ai),1

2(Lj+ Aj)}

=1

4{Li, Lj} +1

4{Li, Aj} +1

4{Ai, Lj} +1

4{Ai, Aj}

=1

4ijkLk+

1

4ijk Ak+

1

4ijk Ak

1

4+ijkLk =

1

2ijkLk+

1

2ijk Ak=ijkMk

{Ni, Nj

}=

{

1

2(Li

Ai),

1

2(Lj

Aj)

}=

1

4{Li, Lj} 1

4{Li, Aj} 1

4{Ai, Lj} +1

4{Ai, Aj}

=1

4ijkLk 1

4ijk Ak 1

4ijk Ak

1

4+ijkLk =

1

2ijkLk 1

2ijk Ak =ijkNk

{Mi, Nj} = {12

(Li+ Ai),1

2(Lj Aj)}

=1

4{Li, Lj} +1

4{Li, Aj} 1

4{Ai, Lj} 1

4{Ai, Aj}

=14

ijkLk+14

ijk Ak 14

ijk Ak 14 ijkLk= 0.

{Mi, Mj} = ijkMk {Ni, Nj} = ijkNk {Mi, Nj} = 0,


35/80

| M|2 = MiMi |N|2 = NiNi

Mi Ni

| M|2 = | N|2

| M|2 | N|2 =14|(Li+ Ai)|2 1

4|(Li Ai)|2

=14

(Li|2 + Ai2 + 2 L| A|2) (Li|2 + Ai2 2L| A|2)

= L A= p L L mkr

r L= 0

| M|2 + | N|2

| M|2 + | N|2 =14|(Li+ Ai)|2 +1

4|(Li Ai)|2

=1

4(Li|2 + Ai2 + 2 L|A|2) + (Li|2 + Ai2 2L|A|2)

=1

2(L2 +

A2) =

mk2

4|E| .

|M

|2


36/80


37/80

xixi pipi {, } 1i

[, ]

[xi,pj] =iij.

H=|p2|

2m+V(r).

H =H

Li = ijk xj pk.

Li =Li


38/80

A = (p

L

mk

r

r

) = (p

L)

(mk

r

r

) = (ijkpjLk)

(mk

r

r

) =ijkLkpj

mkr

r A = A

A=1

2(p L L p) mk r

r

[H, Ai] = 0,

[H, Li] = 0.

[Ai, Aj] = 2imijkHLk [Ai, Lj] = iijkAk

[Li, Lj] = iijkLk.


39/80

|E|

Ai= Ai2m|E|

[Ai, Aj] = iijk Ak Ai, Lj

= iijk Ak

[Li, Lj] = iijkLk.

Mi=1

2

Li+ Ai

Ni =

1

2

Li Ai

.

[Mi, Mj] = iijkMk

[Ni, Nj] = iijkNk

[Mi, Nj] = 0

[Mi, H] = 0

[Ni, H] = 0.

| M|2

| N|2

j(j+ 1)2

q(q+ 1)2

j, q= 0, 1

2, 1, . . .


40/80

|M

|2

|N

|2 =LiAi= 0

| M|2 = | N|2

| M|2 + | N|2

| = 2j(j+ 1)2|.

|L|2

2

2 mk

2

4|E| .

Ej = mk2

22(2j+ 1)2, j = 0,

1

2, 1, . . .


41/80


42/80


43/80

R

3 3

RTR= I33

A

R3

A(x,y,z) = (x, y cos zsen, y sen+zcos ).

u R3

u

= [r1, r2,...,rn] K K

= [r1, r2,...,rn] K u K

u= 1u1 + 2u2 + ... + nun

[1, 2,...,n]

u


44/80

[u] =

1

2

n

.

[u] =

x

y

z

.

[u] =X

m n

T :Rm Rn

T :K K

= [u1, u2,...,un] K = [w1, w2,...,wn]

K

A(v1),...,A(vn) K

A(v1) =11w1+...+m1wm

A(vn) =1nw1+...+mnwm

[A]


45/80

A

[A] =

11 . . . 1n

21 . . . 2n

n1 . . . nn

.

ij A

[A] =

11 12 13

21 22 23

31 32 33

.

A(u) =

11 12 13

21 22 23

31 32 33

x

y

z

A(u) = (11x +12y+13z, 21x+22y+23z, 31x+32y+33z)


46/80

x = 11x+12y+13z

y cos zsen = 21x+22y+23zy sen +zcos = 31x+32y+33z.

11= 1, 22=33 = cos 23 = 32 = sen,

A(x,y,z) =

1 0 0

0 cos sen0 sen cos

A(x,y,z) = (x, yz)

(x , y , z )

(x, y, z)

[u] =

x

y

z

.

x

y

z

=

1 0 0

0 cos sen0 sen cos

x

y

z

,


47/80

yz

R =

1 0 0

0 cos sen0 sen cos

.

xz

R3

B(x,y,z) = (x cos +zsen,y,x sen+zcos )

R3

R3

= [r1, r2,...,rn]

= [r1, r2,...,rn]

u= 1u1 + 2u2 + ... + nun [1, 2,...,n]

[u] =

1

2

n


48/80

[u] =

x

y

z

[u] =X

T :Rm Rn

T : K

K

= [u1, u2,...,un] = [w1, w2,...,wn]

A(v1),...,A(vn)

B(v1) =11w1+...+m1wm

B(vn) =1nw1+...+mnwm

[B]

[B] =

11 . . . 1n

21 . . . 2n

n1 . . . nn


49/80

ij

[B] =

11 12 13

21 22 23

31 32 33

B(x,y,z) = (x cos +zsen,y,

x sen+zcos )

B(u) =

11 12 13

21 22 23

31 32 33

x

y

z

B(u) = (11x+12y+13z, 21x+22y+23z, 31x+32y+33z)

(x cos +zsen,y,x sen+zcos ) = (11x+12y+13z, 21x+22y+23z, 31x+32y+33z)

x cos +zsen=

11x+

12y+

13z

y = 21x+22y+23z

x sen+zcos = 31x+32y+33z


50/80

11 = 33= cos , 22 = 1 13 = 31= sen

B(x , y , z ) =

cos 0 sen

0 1 0

sen 0 cos

B(x,y,z) = (x, yz)

[u] =

x

y

z

x

y

z

=

cos 0 sen

0 1 0

sen 0 cos

x

y

z


51/80

R =

cos 0 sen

0 1 0

sen 0 cos

R3

C(x,y,z) = (x cos y sen, x sen+y cos , z)

R3

R3

= [r1, r2,...,rn]

= [r1, r2,...,rn] u= 1u1 + 2u2 + ... + nun [1, 2,...,n]

[u] =

1

2

n


52/80

[u] =

x

y

z

[u] =X

T :Rm Rn

T : K

K

= [u1, u2,...,un] = [w1, w2,...,wn]

A(v1),...,A(vn)

B(v1) =11w1+...+m1wm

C(vn) =1nw1+...+mnwm

[C]

[C]=

11 . . . 1n

21 . . . 2n

n1 . . . nn


53/80

ij

[C]=

11 12 13

21 22 23

31 32 33

C(x,y,z) = (x cos

y sen, x sen+y cos , z)

C(u) =

11 12 13

21 22 23

31 32 33

x

y

z

C(u) = (11x +12y+13z, 21x+22y+23z, 31x+32y+33z)

(x cos y sen, x sen+y cos , z) = (11x+12y+13z, 21x+22y+23z, 31x+32y+33z)

x cos

y sen=

11x +

12y+

13z

x sen+y cos =21x+22y+23z

z= 31x+32y+33z


54/80

11 = 22 = cos , 12 = 21= sen 33 = 1

C(x,y,z) =

cos sen 0sen cos 0

0 0 1

C(x,y,z) = (x, yz)

[u] =

x

y

z

x

y

z

=

cos sen 0sen cos 0

0 0 1

x

y

z


55/80

R=

cos

sen 0

sen cos 0

0 0 1


56/80

2

v =d(rr)

dt =

dr

dtr+r

dr

dt = rr+r

dr

d

d

dt = rr+r

d

dt.


57/80

E=1

2

(rr)2 +

r

d

dt

2 Gm1m2

r .

L= r p= r v tangencial =rr ddt

=r2d

dt

E=1

2r2 +

L2

2r2Gm1m2

r .

r

dr

dt =

2

E L

2

2r2+

Gm1m2

r

1/2.

r()

ddtdrdt

=L2

r2

2

E L2

2r2 +

Gm1m2

r1/2 .


58/80

d=L2

r2Er2 L2

2r2+ Gm1m2

r

1/2

1

2r2

1/2

dt= L

2 1

r2

E L2

2r2 + Gm1

m2

r dr

u= 1r

du= 1r2

d =

L

2 1

E L2u2

2

+Gm1m2u du

d= du

2EL2 u2 + 2 Gm1m2

L2 u

1/2.

r0= L2

Gm1m2

d = du2E

L2u2+2ur0

1/2 = du

2E

L2u2+ 2u

r0+ 1r20

1r20

1/2 = du

2E

L2+ 1

r20

u 1

r0

2

1/2

=

r0du

r0

2EL2

+ 1r20

u 1r0

21/2 =

r0du2Er

20

L2 +1(ur01)2

1/2 .

2 = 1 +

2Er20

L2

d= r0du

(2

(ur0 1)2

)1/2

.


59/80

r0u= 1

cos

r0du=

sen

d= sen

(2 ( cos )2)1/2 = sen

(1 cos2 ))1/2 =d.

=

r0u= 1

cos

r0u= 1 cos( ) = 1 cos 1u

= r0

1 cos .

r= 1

u

r= r0

1 cos .

L= (Gm1m2r0)1/2

E= Gm1m2

2r0(2 1).

r= r0

1 cos =r0+r cos

r2 =x2 +y2

x= r cos

y= r sen


60/80

x2 +y2 = (r0+x) =r20+ 2r0x+

2x2.

Ax2 +Bxy+C y2 +Dx+Ey+F = 0.

B2 4AC < 0

B2 4AC = 0

B2 4AC >0

= 0

r= r0 x2 +y2 =r20

0< 0


61/80

= 1

x2 +y2 =r20+ 2r0x +2x2

y2

2r0x

r20 = 0

4.0.1 = 0

>1

x2 +y2 =r20+ 2r0x +2x2 (1 2)x2 +y2 2r0x r20 = 0

(1 2)

4.(1 2).1> 0


62/80

b c= d di=

j,k=1

ijkbjck

a (b c) =ijkajkrsbrcs=ijkkrsajbrcs

ijkkrs = irjs isjr

(irjs

isjr)ajbrcs=asbics

arbrci= (ac )bi

(ab)ci


63/80

ijklmn = 2il

ijkijk = 6


64/80

{f, g} = fxi

g

pi f

pi

g

xi=

f

xi

g

pi+

f

pi

g

xi

= {g, f}

{f,gh} = fxi

(gh)

pi f

pi

(gh)

xi

= f

xi

g

h

pi+

g

pih

f

pi

g

h

xi+

g

xih

= f

xi

g

pi

h

f

pi

g

xi

h+gf

xi

h

pi g

f

pi

h

xi

{f,gh} =g{f, h} + {f, g}h


65/80

{f,

{g, h

}}+

{g,

{h, f

}}+

{h,

{f, g

}}=

= f

xi

{g, h}pi

fpi

{g, h}xi

+ g

xi

{h, f}pi

gpi

{h, f}xi

+ h

xi

{f, g}pi

hpi

{f, g}xi

= f

xi

{g, h}pi

fpi

{g, h}xi

+ g

xi

{h, f}pi

gpi

{h, f}xi

+ h

xi

{f, g}pi

hpi

{f, g}xi

= f

xi

pi

g

xi

h

pi g

pi

h

xi

f

pi

xi

g

xi

h

pi g

pi

h

xi

+

g

xi

pi

h

xi

f

pi h

pi

f

xi

+

gpi

xi

h

xi

f

pi h

pi

f

xi

+

h

xi

pi

f

xi

g

pi f

pi

g

xi

h

pi

xi

f

xi

g

pi f

pi

g

xi

= 0


66/80

r p

[xi, rj] =xixj xjxi= 0

[pi, pj] =pipj pjpi= 0

[xi, pj](f) = xi

if

xj

+

i

xif

xj

= i

xi

f

xj xi f

xj fxi

xj

=

= iij

[xi, pj](f) = xiif

xj

+

ix

ifxj

=

xi

if

xj

ix

ifxj

=

= [pj, xi](f)


67/80

[Lk, xi](f) = [xipj

xjpi, xi](f) = [xipj, xi](f)

[xjpi, xi](f) =

= xi[pj , xi](f) + [xi, xi]pj(f) xj[pi, xi](f) [xj, xi]pi(f) == xi

i(xif)

xj+ixi

(f)

xj

xj

i(xf)

xi+i

(f)

xi

=

= xii

xi (f)

xj f(xi)

xj+xi

(f)

xj

xji

f(xi)

xi xi (f)

xi+xi

(f)

xi

= ixj

[Lk, pi](f) = [xipj xjpi, pi](f) = [xipj, pi](f) [xjpi, pi](f) =xi[pj , pi](f) + [xi, pi]pj(f) xj[pi, pi](f) [xj, pi]pi(f) = [xi, pi]pj(f) = i

xi

pjf

xi xipjf

xi

=

= ipj

[Lk, Li](f) = [Lk, xjpk xkpi](f) = [Lk, xjpk](f) [Lk, xkpi](f) == xj[Lk, pk] + [Lk, xj]pk xk[Lk, pj] [Lk, xk]pj = ixipk+ixkpi== ikijLj


68/80

AiAi=

ijkpjLk mk xir

irsprLs mk xi

r

=ijkpjLkirsprLs

ijkpjLkmk

xi

rmk

xi

r

irsprLs+m2k2

ijkpjLkmk xir mk xirirsprLs= 2mkr |L|2

AiAi= | A|2 =ijkpjLkirsprLs 2mkr |L|2 +m2k2

= (jrks jskr)pjLkprLs 2mk

r |L|2 +m2k2= |p|2|L|2 (p L)2 2mk

r |L|2 +m2k2

= 2m

|p|22m

kr

|L|2 +m2k2

= 2mH|L|2 +m2k2

| A|2 2mH|L|2 =m2k2


69/80

H= |E|

| A|2 + 2m|E||L|2 =m2k2

Ai=

Ai2m|E|

2m|E|||A|2 + 2m|E||L|2 =m2k2

|A|2 + |L|2 = mk2

2|E|


70/80

A=

1

2(p L L p) mk r

r

ijk [pk, Lj ] =ijkpkLj ijkLjpk

Ai =

12(ijkpjLk ijkLjpk) mk xir

=

12(

ijkpjLk+ijk [pk, Lj] ijkpkLj) mk xir

[Lj, pk] = [pk, Lj ] = ijklpl

Ai = 1

2

(ijkpjLk

iijkjklpl

ijkpkLj)

mk

xi

r = 1

2

(2ijkpjLk

i2ilpl)

mk

xi

r =

(ijkpjLk ipi) mk xi

r


71/80

AiAi= |A|2 =

(ijkpjLk ipi) mk xir

(irsprLs ipi) mk xi

r

=

= ijkpjLkirsprLs ijkpjLkipi ijkpjLkmk xir

+

ipiirsprLs+ipiipi+ipimk xir

+

mk xir

irsprLs+mkxi

ripi+m

2k2 =

= |p|2|L|2 + 2|p|2 2mkr |L|2 2mk

r

2 +m2k2

H= |p|

2

2m k

r

|A|2 = |p|2|L|2 +2|p|2 2mkr |L|2 2mk

r

2 +m2k2 = 2mH|L|2 +2|p|2 2mkr

2 +m2k2 =

= 2mH

|L

|2 + 2mH2 +m2k2 = 2mH(

|L

|2 + 2) +m2k2

|A|2 2mH(|L|2 + 2) = +m2k2

H= |E|

|A|2 + 2m|E|(|L|2 +2) = +m2k2

Ai= Ai2m|E|


72/80

2m

|E

|

|A

|2

+ 2m

|E

|(

|L

|2 +2) = m2k2

|A|2 + (|L|2 + 2) = m2k2

2m|E|

|A|2 + |L|2 = mk2

2|E| 2


73/80

[H, A] = [p2

2mk

r,1

2(p L L p) mk r

r] = 0

[p2

2m,1

2p L] = 1

2m

1

2ijk [p

2, pjLk] = 1

2m

2ijk([p

2, pj]Lk+ pj[p2, Lk])

p2

Lk

[p2, Lk] = 0

[p2

2m,1

2p L] = 0

[ p2

2m,1

2L p]


74/80

p2

2m,

mk

r

r=

k

2p2,r

r =

k

2

([pj,xi

r

]pj+ pj[pj ,xi

r

]) =

k

2

([pj, xi]pi

r

+xi[pj,1

r

]pj

+pj[pj , xi]1

r+pj[pj ,

1

r]xi) = iij 1rpj+ ixixj 1r3pj ipjij 1r +ipjxj 1r3 xi

p

2

2m,mk r

r

= ipi 1

r+1

rpi

+i

xi xj

r3pj+ pj xj

r3xi

k

r

,

2

p

L =

k2 1r , p

L= ijkk2 1r , pjLk= ijk

k2 1r , pj Lk =

= ki2r3

ijkxjLk = ki2r3

x L

k

r,1

2L p

=

ki

2r3ijkxjLk =

ki

2r3L x

[H, A] = [p2

2mk

r,1

2(p L L p) mk r

r] =

k2

2r3x L k

2

2L x 1

r3

x L

L x


75/80

x

L =ijkxjLk =ijkxjkrsxrps= (irjs

isjr)xjxrps=xjxipj

xjxjpi =

= (x (x p) r2p)

L x =ijkLjxk=ijkjrsLjxk= (kris ksir)xrpsxk== (p r2

(p

x ) x )

[H, A] =k2

2r3x L k

2

2L x 1

r3 =

k2

2r3(x (x p) r2p) k

2

2r3(p r2 (p x ) x ) = 0

[Lm, Ai] =

Lm,

(ijkpjLk ipi) mk xir

= ijk [Lm, pjLk] i[Lm, pi] mk

Lm,

xi

r

=

= pjijk [Lm, Lk] +ijk [Lm, pj]Lk i[Lm, pi] mk Lm,

xi

r = i

jnopnLoipr mk xr

r

= imirAr


76/80

[Ja, Jb] =iabcJc

J2

J2

J2f=f Jzf= f

J=Jx iJy


77/80

Jz

[Jz, J] = [Jz, jx]

[Jz, iJy] =iJy

i(

iJx) =

J

J2

[J2, J] = 0

J2

Jz Jf

J2(Jf) =JJ2f=Jf=(Jf)

Jz

Jz(Jf) = (LzJ JJz)f+JJzf= Jf+J(f) = (+ )(Jf)

J+

J

J+ft= 0

ft J+ft= 0 Jz j

Jzft=jft


78/80

J2

JJ = (Jx iJy)(Jx iJy) =J2 J2z i(iJz) J2 =JJ+J2z Jz

J2ft = (JJ+J2z Jz)ft = (0 + 2j2 +2j)ft= 2j(j+ 1)ft

= 2j(j+ 1)

Jfb= 0

j

Jz

Jzfb= jfb

J2fb= (J+JJ2z Jz)fb= 2j (j 1)fb


79/80

j (j

1) =j (j+ 1)

j =j + 1

j = j

Jzf=

mf

J2f= 2j(j+ 1)f


80/80

o espectro de energia do átimo de hidrogênio

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