FYSH300, Fall 2013

FYSH300, Fall 2013

Symmetries in QM Translations, rotations Parity Charge conjugation FYSH300, fall 2013 Tuomas Lappi tuomas.v.v.lappi@jyu.fi 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..

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    21 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us