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FYSH300, Fall 2013 Symmetries in QM Translations, rotations Parity Charge conjugation FYSH300, fall 2013 Tuomas Lappi [email protected] Office: FL249. No fixed reception hours. fall 2013 Part 5: Spacetime symmetries 1 Symmetries in QM Translations, rotations Parity Charge conjugation Reminder of quantum mechanics I Dynamics (time development) is in the Hamiltonian operator H^ ^ =) Schrodinger¨ i@t = H y I Observable =) Hermitian operator A^ = A^ I A is conserved quantity (liikevakio) , iff (if and only if) d [A^; H^] = 0 () hA^i = 0 dt note: Commutation relation [A^; H^] = 0 =) state can be eigenstate of A^ and H^ at the same time I Conserved quantity associated with symmetry; Hamiltonian invariant under transformations generated by A^ (What does it mean to “generate transformations”? We’ll get to examples in a moment.) 2 Symmetries in QM Translations, rotations Parity Charge conjugation Translations and momentum Da 0 I Define the operation of “translation” Da as: x −! x = x + a 0 I A system is translationally invariant iff H^(x ) ≡ H^(x + a) = H^(x) (In particle physics systems usually are; nothing changes if you move all particles by fixed a.) I How does this operation affect Hilbert space states j i? Define ^ corresponding operator Da (Now with hat, this is linear operator in Hilbert space!) : ^ Daj (x)i = j (x + a)i I Infinitesimal translation, a small, Taylor series 1 j (x + a)i ≈ (1 + i(−ia · r) + (i(−ia · r))2 + ::: )j (x)i pˆ = −ir 2 1 = (1 + ia · pˆ + (ia · pˆ) + ::: )j (x)i = eia·pˆ j (x)i 2 Translations generated by momentum operator pˆ ia·pˆ I j (x + a)i = e j (x)i I Translational invariance =) momentum conservation; [pˆ; H^] = 0 (Note: rules out external potential V (x)!) 3 Symmetries in QM Translations, rotations Parity Charge conjugation Rotations, angular momentum Rotation around z axis: 0 x 1 0 x 0 1 0 x cos ' − y sin ' 1 0 x − y' 1 0 @ y A ! @ y A = @ x sin ' + y cos ' A ≈ @ y + x' A z z0 z '!0 z Wavefunction (scalar) rotated in infinitesimal rotation as 0 “ ^ ” (x) ! (x ) ≈ (x) + iδ'(−i) (x@y − y@x ) (x) = 1 + iδ'Lz (x) Rotation around general axis θ Direction: θ=jθj, angle θ = jθj. ^ (x) −! eiθ·L (x) Rotations generated by (orbital) angular momentum (pyorimism¨ a¨ar¨ a)¨ operator L^. 4 Symmetries in QM Translations, rotations Parity Charge conjugation Rotations as part of Lorentz group L^ rotates coordinate-dependence the wavefunction. You know that particles also have spin=intrinsic angular momentum. Total angular momentum is J^ = L^ + S^ I J^ is one of the generators of the Lorentz group =) conserved in Lorentz-invariant theories ^ I L^,S not separately conserved I For S 6= 0 particles also spin rotates in coordinate transformation: ^ (x) −! eiθ·J (x) 2S+1 I Spectroscopic notation LJ L often denoted by letter: L = 0 : S; L = 1 : P; L = 2 : D ::: Spin S of composite particle = total J of constituents: examples ++ 4 ∆ = uuu has S = 3=2: S-wave (L = 0), quark spins S3=2 j "ij "ij "i χc = cc¯ has Sχc = 0, resulting from P-wave orbital state L = 1 and 3 quark spins j "ij "i for total S = 1, but S L =) J = 0. P0 5 Symmetries in QM Translations, rotations Parity Charge conjugation QM of angular momentum Recall from QMI ^ ^ ^ 2 ^ I Commutation relations: [Ji ; Jj ] = i"ijk Jk ; [Jˆ ; Ji ] = 0. ^ 2 I Eigenstates of J3 & Jˆ are denoted jj; mi with ^ 2 J3jj; mi = mjj; mi Jˆ jj; mi = j(j + 1)jj; mi I Permitted range m = −j; −j + 1;::: j ^ ^ ^ I Ladder operators J± = J1 ± iJ2 raise and lower the z-component of the spin: p ^J+jj; mi = j(j + 1) − m(m + 1)jj; m + 1i =) 0, if m = j p ^J−jj; mi = j(j + 1) − m(m − 1)jj; m − 1i =) 0, if m = −j 6 Symmetries in QM Translations, rotations Parity Charge conjugation Coupling of angular momentum Recall from QMI I How to add two angular momenta Jˆ = Jˆ1 + Jˆ2? I State (tensor) product jj1; m1i ⊗ jj2; m2i ≡ jj1; m1ijj2; m2i ≡ jj1; m1; j2; m2i. I Want to express jj1; j2; J; M = m1 + m2i in terms of jj1; m1ijj2; m2i. I This is done via Clebsch-Gordan coefficients: 2 j j 3 X1 X2 jj1; j2; J; Mi = 4 jj1; m1ijj2; m2ihj1; m1; j2; m2j5 jj1; j2; J; Mi m1=−j1 m2=−j2 CG;from tables X z }| { = hj1; m1; j2; m2j j1; j2; J; Mi jj1; m1i jj2; m2i m1;m2 I Possible values J = jj1 − j2j;:::; j1 + j2. CG6= 0 only if m1 + m2 = M Not only for angular momentum Other quantum numbers have similar mathematics: SU(N) group Isospin SU(2) (like angular momentum); Color SU(3) (slightly more complicated). 7 Symmetries in QM Translations, rotations Parity Charge conjugation Properties of Clebsch-Gordan coefficients Recall from QMI One can also go the other way X jj1; m1ijj2; m2i = hj1; j2; J; Mjj1; m1; j2; m2i jj1; j2; J; Mi M;J Transformation between orthonormal bases is always unitary + Clebsch’s are real =) matrix of Clebsch’s actually orthogonal. elem JM;j1j2 in uncoupled ! coupled elem j1;m1;j2;m2 in coupled ! uncoupled z }| { z }| { hj1; j2; J; Mjj1; m1; j2; m2i = hj1; m1; j2; m2jj1; j2; J; Mi Consequence: probability to find coupled state in uncoupled is the same as probability to find uncoupled in coupled: 2 jhj1; m1; j2; m2jj1; j2; J; Mij 8 Symmetries in QM Translations, rotations Parity Charge conjugation Clebsch tables 36. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS, AND d FUNCTIONS JJ... Notation: Note: A square-root sign is to be understood over every coefficient, e.g.,for 8/15 read 8/15. MM... − − 1/2 ×1/2 1 m1 m2 +11 0 0 3 5/2 Y1 = cos θ 2×1/2 m m Coefficients +1/2 +1/2 1 0 0 4π +5/2 5/2 3/2 1 2 + + + + . +1/2 −1/2 1/2 1/2 1 2 1/2 1 3/2 3/2 . − + − − 3 1/2 1/2 1/2 1/2 1 Y 1 = sin θeiφ +2 −1/2 1/5 4/5 5/2 3/2 . 1 8π + + − + + −1/2 −1/2 1 − 1 1/2 4/5 1/5 1/2 1/2 +1−1/2 2/5 3/5 5/2 3/2 0 5 3 2 1 Y2 = cos θ 0+1/2 3/5 −2/5 −1/2 −1/2 3/2 4π 2 − 2 1×1/2 − +3/2 3/2 1/2 0 1/2 3/5 2/5 5/2 3/2 15 −1 +1/2 2/5 −3/5 −3/2 −3/2 +1 +1/2 1 +1/2 +1/2 Y 1 = sin θ cos θeiφ 2 2 − − +1 −1/2 1/3 2/3 3/2 1/2 − 8π 3/2 ×1/2 1 1/2 4/5 1/5 5/2 +2 2 1 − + − − 0 +1/2 2/3 −1/3 −1/2 −1/2 2 1/2 1/5 4/5 5/2 2 1 15 2 2iφ +3/2 +1/2 1 +1 +1 − Y = sin θe −21−1/2 0 1/2 2/31/3 3/2 2 4 2π + − −1 +1/2 1/3 −2/3 −3/2 3/2 1/2 1/4 3/4 2 1 +1/2 +1/2 3/4 −1/4 0 0 2×1 3 −1 −1/2 1 5/2 + 3/2 ×1 +1/2 −1/2 1/2 1/2 2 1 3 3 2 +5/2 5/2 3/2 + + + + − + − − − 2 1 1 2 2 +3/2 +1 1 +3/2 +3/2 1/2 1/2 1/2 1/2 1 1 + − − 2 0 1/3 2/3 3 2 1 +3/2 0 2/5 3/5 5/2 3/2 1/2 1/2 1/2 3/4 1/4 2 + + − + + + 1 1 2/3 1/3 1 1 1 +1/2 +1 3/5 −2/5 +1/2 +1/2 +1/2 −3/2 +1/2 1/4−3/4 −2 + − 2 1 1/15 1/3 3/5 +3/2 −1 1/10 2/5 1/2 −3/2 −1/2 1 2 +1 0 8/15 1/6 −3/10 3 2 1 + − 1×1 + 1/2 0 3/51/15 1/3 5/2 3/2 1/2 22 1 0 +1 2/5 −1/2 1/10 0 0 0 −1/2 +1 3/10 −8/15 1/6 −1/2 −1/2 −1/2 +1 +1 1 +1 +1 +1 −1 1/5 1/2 3/10 +1/2 −1 3/10 8/15 1/6 +1 0 1/2 1/2 2 1 0 003/5 0 −2/5 3 2 1 −1/2 0 3/5 −1/15 −1/3 5/2 3/2 0 +1 1/2 −1/2 0 0 0 −1 +1 1/5 −1/2 3/10 −1 −1 −1 −3/2 +1 1/10 −2/5 1/2 −3/2 −3/2 +1 −1 1/6 1/2 1/3 0 −1 2/5 1/2 1/10 −1/2 −1 3/5 2/5 5/2 − − − 00 2/3 02−1/3 1 −1 0 8/15 −1/6−3/10 3 2 3/2 0 2/5 3/5 5/2 −1 +1 1/6 −1/2 1/3 −1 −1 −2 +1 1/15 −1/3 3/5 −2 −2 −3/2 −1 1 0 −1 1/2 1/2 2 −1 −1 2/3 1/3 3 m m m − − − Y − =( 1) Y −1 0 1/2 −1/2 −2 2 0 1/3 2/3 3 j1j2m1m2 j1j2JM ℓ − ℓ ∗ | −1 −1 1 ℓ 4π m imφ −2 −1 1 J j j d = Y e =( 1) 1 2 j2j1m2m1 j2j1JM m,0 2ℓ +1 ℓ − − − − | 9 Symmetries in QM Translations, rotations Parity Charge conjugation Example Short notation for representations according to dimension: 1 Singlet, j = 0 = m 1 1 2 Doublet j = 2 ; m = ± 2 3 Triplet j = 1; m = −1; 0; 1..
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