Manipulate @ ParametricPlot3D @ Evaluate @8 Sin @ theta D Cos @ phi D , Sin @ theta D Sin @ phi D , Cos @ theta D< Abs @ SphericalHarmonicY @ l, m , theta, phi DDD , 8 phi,

Theo re ti sche Physik II Inhal t 03 .04 .—12.0 7.2 01 2 Die Hom epa ge des Kurses ist http://www.physik.uni-bielefeld.de/~yorks/qm12 0. Orga / M ot iva tion .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 1. W ell enm echani k .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 2 1.1 Ei nf ¨u hrung .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 2 1.2 F reie T ei lche n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 6 1.3 Sch r¨odi ng er Gleic hung .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 8 1.4 W ahrsch einl ichk eit sinterp reta tion .. .. .. .. .. .. .. .. .. . 9 1.5 Z eitunabh ¨a n g ig e Schr¨ o d inger Gle ichu ng .. .. .. .. .. .. 12 1.6 He isen b erg’sch e Unsc h ¨ar ferelat io n .. .. .. .. .. .. .. .. .. 14 2. 1D Pro b leme .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 17 2.1 T eilch en im P ot en ti a lt opf .. .. .. .. .. .. .. .. .. .. .. .. .. 17 2.2 1D p erio di sche s P o tential .. .. .. .. .. .. .. .. .. .. .. .. .. 20 2.3 Str euun g a m P otentialto pf .. .. .. .. .. .. .. .. .. .. .. .. . 22 2.4 T u nnel effekt .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 24 3. All g emein er F o rmali smu s d er QM .. .. .. .. .. .. .. .. .. 27 3.1 Z ust ¨a n de / Obse rvabl en / Erw artungsw ert e .. .. .. .. . 27 3.2 Kommu tato ren und Qu a n tis ierun g .. .. .. .. .. .. .. .. . 30 3.3 Z eitent w ick lun g .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 33 3.4 Sta tis ti sche r Op er ato r .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 35 3.5 Mess p rozess in der QM .. .. .. .. .. .. .. .. .. .. .. .. .. .. 38 3.6 Ha rmonis cher O szill a to r .. .. .. .. .. .. .. .. .. .. .. .. .. . 41 4. Symm et rie n in d er QM: Ku g el symme trie, Drehi mpu ls 43 4.1 Grupp en und Generat o ren .. .. .. .. .. .. .. .. .. .. .. .. . 44 4.2 Dreh imp uls / Ei g en w ert e .. .. .. .. .. .. .. .. .. .. .. .. .. 46 4.3 Ortsda rstellu ng des Bahnd re him puls es

~ L .. .. .. .. .. .. 49 4.4 Spi n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 52 4.5 Add ition von D re him puls en .. .. .. .. .. .. .. .. .. .. .. .. 55 5. W asse rs toffat om .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 58 5.1 Z w eik¨ orp erp roblem ; Radi a lg leich ung .. .. .. .. .. .. .. .. 58 5.2 En erg iesp ekt ru m .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 61 6. N ¨ah er u ngsmetho de n .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 64 6.1 Ra ylei g h- Ritz V aratio n sp rin zi p .. .. .. .. .. .. .. .. .. .. . 64 6.2 Z eitunabh ¨a n g ig e St ¨orungstheo ri e .. .. .. .. .. .. .. .. .. . 66 6.3 Anw end ungen, a n ha rm o ni sch er Oszil lat o r .. .. .. .. .. . 68 6.4 St¨ o run g st he o rie f¨ur en ta rt ete Zus t¨and e .. .. .. .. .. .. . 72 6.5 Anw end ungen; H-F ei nstruktur .. .. .. .. .. .. .. .. .. .. . 75 7. Id entisc he T eilc hen; P aul ip rinzi p .. .. .. .. .. .. .. .. .. . 79 8. Aus bli ck / ”M ¨arche n” .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 86

Ÿ Kugelflaechenfunktionen − Visualisierung

In[1]:=

Manipulate @ ParametricPlot3D @ Evaluate @8 Sin @ theta D Cos @ phi D , Sin @ theta D Sin @ phi D , Cos @ theta D< Abs @ SphericalHarmonicY @ l, m , theta, phi DDD , 8 phi,

-

Pi, Pi < , 8 theta, 0, Pi < , PlotRange

®

88

-

.5, .5 < , 8

-

.5, .5 < , 8

-

1.1, 1.1 << , Mesh

®

False, PlotPoints

®

8 36, 18 < , MaxRecursion

®

ControlActive @ 0, 2 D , ViewAngle

®

.246, ImageSize

®

8 500, 377 < , Axes

®

False, SphericalRegion

®

True, Boxed

®

False D , 88 l, 2, "l" < , 0, 7, 1, Appearance

®

"Labeled" < , 88 m , 0, " m " < ,

-

l, l, 1, Appearance

®

"Labeled" <D

Out[1]=

l 2 m 0

35. Clebs ch-Gor dan co efficients 1 35. CL EBSCH -GORD AN COEF F ICIEN T S ,S PH ERICAL H A RM ON ICS, AND d FUNC TI O N S No te: A squa re -r o o t sig n is to b e under sto o d o v er every co efficien t, e.g. ,f o r

−

8 / 15 read

−

8 / 15 . Y

^{0 1}

= 3 4 π cos θ Y

^{1 1}

=

−

3 8 π sin θe

iφ

Y

^{0 2}

= 5 4 π 3 2 cos

2

θ

−

1 2 Y

^{1 2}

=

−

15 8 π sin θ cos θe

iφ

Y

^{2 2}

= 1 4 15 2 π sin

2

θe

2iφ

Y

−m

=(

−

1)

m

Y

m∗

j

1

j

2

m

1

m

2|

j

1

j

2

JM

=(

−

1)

J−j1−j2

j

2

j

1

m

2

m

1|

j

2

j

1

JM

d

^m,0

= 4 π 2 +1 Y

^m

e

−imφ

d

^{j m},m

=(

−

1)

m−m

d

^{j m,}m

= d

j −m,−m

d

^{1 0},0

=c o s θd

1/2 1/2,1/2

=c o s θ 2 d

1/2 1/2,−1/2

=

−

sin θ 2

d

1 1,1

= 1+c o s θ 2 d

^{1 1},0

=

−

sin θ

√

2 d

^{1 1},−1

= 1

−

cos θ 2 d

3/2 3/2,3/2

= 1+c o s θ 2 cos θ 2 d

3/2 3/2,1/2

=

−√

3 1+c o s θ 2 sin θ 2 d

3/2 3/2,−1/2

=

√

3 1

−

cos θ 2 cos θ 2 d

3/2 3/2,−3/2

=

−

1

−

cos θ 2 sin θ 2 d

3/2 1/2,1/2

= 3c o s θ

−

1 2 cos θ 2 d

3/2 1/2,−1/2

=

−

3c o s θ +1 2 sin θ 2

d

2 2,2

= 1+c o s θ 2

2

d

^{2 2},1

=

−

1+ co s θ 2 sin θ d

^{2 2},0

=

√

6 4 sin

2

θ d

^{2 2},−1

=

−

1

−

cos θ 2 sin θ d

^{2 2},−2

= 1

−

cos θ 2

2

d

2 1,1

= 1+c o s θ 2 (2 co s θ

−

1) d

^{2 1},0

=

−

3 2 sin θ cos θ d

^{2 1},−1

= 1

−

cos θ 2 (2 co s θ +1 ) d

^{2 0},0

= 3 2 cos

2

θ

−

1 2

+1

5/2 5/2 +3/23/2 +3/2

1/5 4/5

4/5 −1/55/2 5/2 −1/2 3/5 2/5 −1 −2

3/2 −1/2 2/55/23/2 −3/2−3/2 4/5 1/5

−4/51/5 −1/2−21−5/25/2−3/5 −1/2 +1/2

+1−1/22/53/5 −2/5 −1/2 2 +2 +3/2+3/2 5/2 +5/25/2 5/23/21/2 1/2 −1/3 −1 +10

1/6+1/2 +1/2 −1/2 −3/2

+1/2 2/5 1/15

−8/15

+1/2 1/10 3/103/55/23/21/2 −1/2 1/6 −1/35/2 5/2 −5/2 1

3/2 −3/2 −3/52/5−3/2 −3/2

3/5 2/5

1/2 −1 −1

0

−1/2 8/15 −1/15 −2/5 −1/2 −3/2

−1/2 3/10 3/5 1/10

+3/2 +3/2 +1/2 −1/2

+3/2 +1/2

+2+1 +2 +10 +12/5 3/5

3/2 3/5 −2/5 −1 +10

+3/21+1+3 +11 0

3 1/3+2 2/3

2

3/2 3/2 1/3 2/3

+1/2 0 −1

1/2 +1/2 2/3 −1/3 −1/2 +1/2

1

+11 0

1/2 1/2 −1/2

0 0 1/2 −1/2

1 1−1−1/2

1 1 −1/2 +1/2

+1/2+1/2 +1/2 −1/2 −1/2 +1/2−1/2 −1

3/2 2/33/2 −3/2 1

1/3

−1/2 −1/2

1/2 1/3 −2/3

+1+1/2 +1 0

+3/2 2/33 3 3 3 3 1−1−2−32/3 1/3−22 1/3 −2/3−2

0 −1 −2

−1 0 +1

−1

2/5 8/15 1/15

2 −1 −1 −2−1 0

1/2 −1/6 −1/3

1 −1 1/10 −3/10 3/5

0

2 0 1 0

3/10 −2/5 3/10

01/2 −1/2

1/5 1/53/5

+1 +1 −100−1 +1

1/15 8/15 2/5

2 +22 +1

1/2 1/2

1 1/2

2 0

1/6 1/62/3

1 1/2 −1/20 02 2 −2 1−1−1

1 −1 1/2 −1/2

−1 1/2 1/2

0 0

0 −1

1/3 1/3−1/3

−1/2

+1 −1 −1 0

+1 00 +1−1

2 1 0 0+1

+1+1

+1

1/3 1/6 −1/2

1 +1 3/5 −3/10 1/10

−1/3 −1 0+1 0+2 +1

+2

3 +3/2

+1/2+1 1/42 2 −11 2 −2 1

−1 1/4 −1/2

1/2 1/2 −1/2−1/2 +1/2−3/2 −3/2

1/21 0

03/4 +1/2 −1/2−1/2

2 +1 3/4 3/4 −3/41/4

−1/2 +1/2

−1/4

1 +1/2−1/2 +1/21

+1/2

3/5 0 −1

+1/20

+1/23/2 +1/2

+5/2 +2 −1/2+1/2+2 +1 +1/21

2 × 1/2 3/2 × 1/2 3/2 × 1 2 × 1

1 × 1/2

1/2 × 1/2 1 × 1

Notation:JJ MM... ...

. . . . . .

m1m2 m1m2Coefficients −1/52 2/7 2/7−3/7

3 1/2 −1/2 −1 −2−2 −1

04

1/2 1/2

−33 1/2 −1/2 −21−44 −2

1/5

−27/70

+1/2

7/2 +7/27/2 +5/2 3/7 4/7 +2 +1 0

1 +2 +1 +4 1

4 4 +2 3/14 3/144/7

+2 1/2 −1/20

+2 −1 0 +1 +2

+2 +1 0 −1

32 4 1/14 1/143/7 3/7+13 1/5 −1/53/10 −3/10+12 +2 +1 0

−1 −2

−2 −1 0 +1 +2

3/7 3/7

−1/14 −1/14

+11 432 2/7 2/7

−2/71/14 1/144 1/14 1/143/73/7

3 3/10 −3/101/5 −1/5−1 −2 −2 −1 0

0 −1 −2

−1 0 +1

+1 0 −1 −2

−12 4 3/14 3/144/7−2−2−2

3/7 3/7

−1/14 −1/14

−11 1/5 −3/10 3/10−1

10 0 1/70 1/708/35 18/35 8/35

0 1/10 −1/102/5 −2/50

00 0

2/5 −2/5

−1/10 1/10

0 1/5 1/5 −1/5−1/5

1/5 −1/5

−3/10 3/10

+1

2/7 2/7−3/7

+3 1/2 +2 +1 0

1/2

+2+2 +2 +1+2+1+3 1/2 −1/2 0 +1 +2

34

+1/2 +3/2

+3/2+2+5/2 4/77/2 +3/2 1/7 4/7 2/7

5/2 +3/2 +2 +1 −10

16/35 −18/351/35

1/35 12/35 18/35 4/35

3/2 +3/2+3/2 −3/2 −1/2 +1/22/5 −2/57/2 7/2

4/35 18/35 12/35 1/35

−1/25/2

27/70 3/35 −5/14 −6/35

−1/23/2 7/2 7/2 −5/2 4/7 3/7

5/2 −5/2 3/7 −4/7 −3/2−2

2/7 4/7 1/7

5/2 −3/2 −1 −2

18/35 −1/35 −16/35

−3/2 1/5 −2/5 2/5 −3/2 −1/2

3/2 −3/2 7/2 1−7/2

−1/2 2/5 −1/5 0 0 −1 −2

2/5

1/2 −1/2 1/10 3/10−1/5 −2/5 −3/2 −1/2 +1/2

5/23/21/2 +1/2 2/5 1/5 −3/2 −1/2 +1/2 +3/2−1/10−3/10

+1/2 2/5 2/5 +1 0

−1 −2

0

+33 3 +22 +21+3/2 +3/2 +1/2+1/21/2 −1/2 −1/2 +1/2 +3/2

1/232

3 0

1/20 1/209/20 9/20

21 3 −1 1/5 1/53/5

2 3 3 1−3

−2

1/2 1/2 −3/2

2 1/2 −1/2 −3/2−2

−1 1/2 −1/2 −1/2 −3/20

1 −1 3/10 3/10−2/5 −3/2 −1/2

0 0

1/4 1/4 −1/4 −1/4

0 9/20 9/20 +1/2 −1/2 −3/2

−1/20 −1/20

0 1/4 1/4−1/4 −1/4 −3/2 −1/2 +1/2

1/2 −1/20

1 3/10 3/10 −3/2 −1/2 +1/2 +3/2

+3/2 +1/2 −1/2 −3/2

−2/5

+1+1+1

1/5 3/5 1/5

1/2 +3/2 +1/2 −1/2

+3/2

+3/2 −1/5

+1/2

6/35 5/14 −3/35

1/5

−3/7 −1/2 +1/2 +3/2

5/2

2 × 3/2 2 × 2

3/2 × 3/2

−3 Figure35.1

: T h e si gn con v en ti on is th at of W ign er ( Gr ou p T he or y , A cad em ic P ress, N ew Y ork, 1959) , a ls o u sed b y C on d o n a n d S h o rt le y ( The The o ry of A tom ic S p ec tr a , C am b ri d ge U n iv . P ress, N ew Y ork, 1953) , R ose ( Elemen ta ry The o ry of A n gu lar M omen tu m , W il ey , N ew Y o rk, 1957) , an d C oh en ( T a bles of th e C lebsch- G or dan Co efficien ts , N ort h A m eri can R o ck w el l S ci en ce C en te r, T h ou san d Oaks, C al if ., 1974) . T h e co effi ci en ts her e ha v e b een ca lcula ted using co m puter pr o g ra m s w ri tten indep enden tly b y Co hen a nd a t LBNL.