Spin aneb jak algebra vstupuje do kvantové mechaniky

Spin aneb jak algebra vstupuje do kvantove mechaniky
Helena Kolesova
Kruh u Jilemnice
22.5.2014
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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R
uzne pohledy na spin
1
Spin v nerelativisticke kvantove mechanice
- nema zadnou klasickou analogii, nevysvetlen!
(02KVAN)
2
Spin v relativisticke kvantove mechanice a QED
- vysvetlen spinu, urcen hodnoty magnetickeho momentu elektronu
s presnost na 11 desetinnych mst!
(02KTP)
3
Reprezentace Lieovy algebry su(2)
(02LIAG)
astice jako reprezentace Poincareho grupy
C
(exkluzivne v Kruhu u Jilemnice!)
4
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Periodicka tabulka prvk
u a co na to Pauli
Edmund C. Stoner (1924): Hladina energie elektronu ve vnejs slupce
atomu alkalickych kov
u (dana kvantovym cslem N ) se ve vnejsm
magnetickem poli step na urcity pocet r
uznych energetickych hladin.
Tento pocet odpovda poctu elektron
u v uzavrene energeticke slupce
prslusne stejnemu N .
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Periodicka tabulka prvk
u (*1869) a co na to Pauli
K vysvetlen stavby periodicke tabulky prvk
u je treba zavest ctvrte
kvantove cslo popisujc stav elektronu! (v dnesnm znacen kvantova
csla: hlavn N , vedlejs l , magneticke m a !spinove! s ).
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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S. Goudsmit a G. Uhlenbeck: Zaveden veliciny spin
Samuel Goudsmit (1971):
"The Pauli principle was published early in 1925. I am convinced that although it is one of the
most important publications in physics, who reads it now, of the younger generation, will nd
it hard to understand. Even that one will not understand it all. And I wrote a note in May
that the Pauli principle became easier to understand when introducing dierent quantum
numbers. The quantum numbers I used for Pauli's principle were ml and ms ; ms being always
the same, plus or minus 1/2."... "As a mathematician said, the change amounted to a simple
linear transformation - which is trivial, mathematically trivial of course, but not so for the
understanding and in teaching."
...
"In these days Kronig came from America and he came to Leiden; we collaborated in
spectroscopy and worked on the intensities in the Zeeman eect for which we found the exact
expressions. Of course, it was quite dierent from today; there was no quantum mechanics at
the time, don't forget that this did not yet exist! One had to guess these little formulae; one
developed a feeling for them. It is just as with veterinary and human medicine. People can tell
one where it hurts, but a veterinary doctor has to guess where it hurts. A horse or a cow
cannot tell that. And so it is with these little formulae. It is really curious ...... it was a kind of
numerology, and it is a miracle that we arrived at the correct expressions which later could be
derived by quantum mechanics."
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Ale my kvantovou mechaniku zname!
Elektron v atomu vodku: {Ĥ0 , L̂2 , L̂3 } ... UMP/
USKO
L̂j = εjkl xˆk p̂l - j-ta slozka momentu hybhosti
h
i
L̂j , L̂k = i~εjkl L̂l
Spolecne vlastn funkce ψN,l,m ∈ H = L2 (R3 , d3 x)
Ĥ0 ψN,l,m = EN ψN,l,m
L̂2 ψN,l,m = l(l + 1)~2 ψN,l,m
Lˆ3 ψN,l,m = m~ψN,l,m
R
N2
l ∈ {0, 1, . . . , N − 1}
EN = −
m ∈ {−l, −l + 1, . . . , l}
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Reprezentace Lieovske algebry su(2)
Lieovou algebrou nazveme vektorovy prostor g s binarn operac
[., .] : g × g → g s jistymi vlastnostmi, ktere v dalsm nebudeme
potrebovat :)
Baze tohoto vektoroveho prostoru: {Tj }nj=1 , uzavrenost ⇒
[Tj , Tk ] = ifjkl Tl . Lieova algebra jednoznacne urcenta csly fjkl (strukturn
konstanty). Pro su(2): fjkl = εjkl .
Def.: Necht V je vektorovy prostor. Reprezentace Liovske algebry g je zobrazen
ρ : g → gl(V ) takove, ze ∀X , Y ∈ g
ρ([X , Y ]) = [ρ(X ), ρ(Y )].
(gl(V ) je algebra linearnch operator
u na V , pro kompaktn algebry lze
volit ρ(Tj ) jako hermitovke matice).
Spinove matice jsou reprezentace su(2) na V = C2 , [Sj , Sk ] = iεjkl Sl
1
1
1
0 1
0 −i
1 0
S1 =
S2 =
S3 =
1 0
i 0
0 −1
2
2
2
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Reprezentace Lieovske algebry su(2)
Cartanova podalgebra: nejvets mnozina vzajemne komutujcch
n
(hermitovskych) generator
u {Hj }m
j=1 ⊂ {Tj }j=1 .
Mejme reprezentaci Lieovy algebry na vektorovem prostoru V .
Vahy (dle H. Georgi): vektor µ
~ vlastnch csel Cartanovych generator
u
prslusny spolecnemu vlastnmu vektoru |µi ∈ V :
Hj |µi = µj |µi
j = 1, 2 . . . m.
Reprezentace je jednoznacne urcena nejvyss vahou, vektorovy prostor,
na kterem p
usob, lze vybudovat z vektor
u |µi.
su(2): [Jj , Jk ] = iεjkl Jl
Pouze jeden Cartan
uv generator: napr. J3 .
Vahy 1-rozmerne, vl. csla J3 (r
uzne hodnoty pr
umetu spinu do osy z !)
Nejvyss vaha j m
uze nabyvat pouze polocelych hodnot (spin castice = j )
Vahy v repre. s danym j : −j, −j + 1, . . . , j ⇒ (2j + 1)-rozmerna repre.
astice se spinem 0, 1 opravdu pozdeji objeveny!
C
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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astice jako reprezentace Poincareho algebry (a la J. F.)
C
Hybnost a energie ⇔ translace v casoprostoru
Moment hybnosti ⇔ prostorove rotace
10 generator
u Poincareho algebry (µ, ν ∈ {0, 1, 2, 3}):
P µ (translace),
M µν = −M νµ (vlastn Lorentzovy transformace).
e rotace), Kj = M 0j (boosty)
Jk = 12 εklm M lm (prostorov
[P µ , P ν ] = 0
[M µν , M ρσ ] = i (g µσ M νρ − g µρ M νσ + g νρ M µσ − g νσ M νρ )
[M µν , P ρ ] = i (g νρ P µ − g µρ P ν )
Schurovo lemma
Casimir
uv operator: komutuje se vsemi generatory algebry −−−−−−−−−→
v ireducibiln reprezentaci realizovan nasobkem operatoru identity.
Ireducibiln reprezentace lze klasikovat pomoc hodnot vsech nezavislych
Casimirovych operator
u.
Casimry v Poincareho algebre: Pµ P µ a Wµ W µ , kde
1
W µ = − εµνρσ Mνρ Pσ
2
(Pauli − Lubanského vektor)
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
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Wignerova konstrukce unitarnch ireducibilnch reprezentac
Poincareovy grupy
trda
1
2
3
4
5
6
Vlastnosti 4-vektor
u
4-vektor k µ
0
µ
pµ p = 0, p = 0 ⇔ p = 0
0, 0
pµ p µ = M 2 , M > 0, p 0 > 0
M, 0
pµ p µ = 0, p 0 > 0
1, 0, 0, 1
µ
pµ p = M 2 , M > 0, p 0 < 0
−M, 0
pµ p µ = 0, p 0 < 0
−1, 0, 0, 1
pµ p µ = −M 2 < 0
0, 0, 0, M
µ
Helena Kolesova: Spin aneb jak algebra vstupuje do kvantove mechaniky
Mala grupa
SO(3, 1)
SO(3)
ISO(2)
SO(3)
ISO(2)
SO(2, 1)
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