Spin and quantum mechanical rotation group

Transcription

1 Chapter 3 Spin and quantum mechanical rotation group The Hilbert space of a spin particle can be explored, for instance, through a composition of Stern Gerlach experiments, ie. a composition of projection operators P (~e )whichproducestateswithspinparalleltothepolarisation vector ~e. From this exercise one concludes that the Hilbert space of a spin particle is a -dimensional vector space with complex coe cients, C spanned by complex linear combinations of the vectors { ~e ± >}. Inparticular, if ~e and ~e 0 are linearly independent polarisation vectors we have ~e > =< +~e 0 ~e > ~e 0 + > + < ~e 0 ~e > ~e 0 > (3.) The projection operator ˆP (~e )cannowbewrittenas ˆP (~e )= ~e+ >< +~e = ( ~e ) >< ( ~e ) (3.) Its matrix representation can be constructed noting that the most general pure state ( ˆP = ˆP )isgivenby ˆP = + ~ ~ (3.3) with ~ =and 0 i = {, 0 0 i, i 0 are the Pauli matrices. Itthenfollowsthat 0 } (3.4) 0 ˆP = ( + ~e ~ ) (3.5) 9

2 0CHAPTER 3. SPIN AND QUANTUM MECHANICAL ROTATION GROUP 3. Quantum mechanical rotation group With the Hilbert space of a spin particle constructed we now want to understand the action of the rotation group on this space. Let R SO(3) be a rotation in 3-dimensional Euclidean space. Then there is a correspondence D R : ˆP (~e )! ˆP (R~e) (3.6) between pure states, ˆP (~e )andelementsofso(3) which leaves expectation values invariant since tr( ˆP (~e ) ˆP ( ~ f)) = tr( ˆP (R~e) ˆP (R ~ f)) (3.7) and any linear operator acting on H = C can be expanded in terms of projectors ˆP ( ~ f), f R 3. Furthermore D R (D R ( ˆP (~e ))) = ˆP (R R ~e )=D R R ( ˆP (~e )) (3.8) This provides a representation of SO(3) on {P ( ~ f)}. We will now show that for each R SO(3) there is a unitary transformation Û(R) onh = C such that D R ( ˆP (~e )) = Û(R) ˆP (~e )Û (R) (3.9) Here * stands for the hermitian conjugate. The rotation group SO(3) is Lie group which means that any rotation R(~e, ), can be written as R(~e, )=e i ~e ~ˆK (3.0) in terms of the 3 generators of infinitesimal rotations i 0 i 0 ˆK 0 0 ia, ˆK A, ˆK i 0 0A, 0 i 0 i defining the Lie algebra, so(3), (3.) [ ˆK i, ˆK j ]=i k ij ˆK k (3.)

3 3.. QUANTUM MECHANICAL ROTATION GROUP These relation, together with (3.) furthermore imply that SO(3) has an interpretation of a di erentiable (group) manifold. In addition to the defining representation (3.), this Lie algebra has infinitely many irreducible representations and, in particular, a - dimensional representation, D in terms of the Pauli matrices (3.4) through ~Ŝ = ~ˆ (3.3) For instance, Ŝ3 measures the spin in the z-direction ~Ŝ3 ~e z ± >= ± ~ ~e z± > (3.4) where ~ has been included for dimensional reasons. Using [Ŝi, Ŝj] =i k ij Ŝk (3.5) we then conclude that a rotation R(~e, ), is implemented on C as Û(~e, )=e i ~e ~Ŝ (3.6) The group generated by these matrices is denoted by SU() since det(û) = (special) andû = Û ( unitary). In particular, Û(~e, ) ~e z + >= e i ˆ3 ~e z + >= ~e z + > (3.7) which shows that Û(~e, )isaprojective representation of R(~e, ). Due to the identity (exercise) R(~e, ) ˆ~ S = Û(~e, ) ˆ~S Û(~e, ) (3.8) there is a unique element of in SO(3) for any Û SU() (homeomorphically) with R(ÛÛ) =R(Û)R(Û) (3.9) This means that SO(3) is a representation of SU() but not vice-versa. In mathematics this property is sometimes expressed as a short exact sequence map.! Z! SU()! SO(3)! (3.0) Here the! s indicates that the image of each map is in the kernel of the following

4 CHAPTER 3. SPIN AND QUANTUM MECHANICAL ROTATION GROUP Geometric interpretation: In addition to the Lie-algebra relations, the Pauli matrices satisfy the anti-commuation relations { i, j} = i j + j i = ij (3.) so that any element Û(~e, )canbewrittenintermsof{,~} as Û(~e, )=e i ~e ~ˆ =cos( ) i(~e ~ )sin( )=:a 0 + ~a ~ (3.) with a 0 R, ~a = i~, ~ R 3,sothata 0 + ~ =. Thereforethe group generated by Û(~e, )hasthesametopology as the 3-sphere, S3. In contrast to the group manifold of SO(3), the quantum mechanical rotation group defines a simply connected group manifold.thisgroupistherefore the universal covering group of SO(3). 3. Product representations The Hilbert space of two distinguishable spin particles can be explored, for instance, with the help of two independent Stern-Gerlach experiments. The space is spanned by the product states ~e,~e > ~e > ~e >= ~e > ~e > (3.3) with scalar product < ~ f, ~ f ~e,~e >=< ~ f ~e >< ~ f ~e > (3.4) which implies statistical independence. In what follows we will assume that ~e = ~e and denote the states simply by ±, ± >. It is important to note that not all pure states in H are a product states. For instance for the state vector p ( +, > +, >) H (3.5) there is statistical correlation or entanglement between the two spin particles which was the origin of the EPR paradox (see chapter??). The SO(3) group manifold is given by the 3-sphere with antipodal points identified.

5 3.3. OBSERVABLES Observables Observables of the combined system are given by linear combinations of product observables  =     > =  >  > (3.6) for example, Ŝ3 ±, + > Ŝ 3 ± > + >= ± ~ ±, + > (3.7) Since [Ŝ i, Ŝ j ]=0thetotal spin ˆ~ S ˆ~S + ˆ~ S satisfies the Lie-algebra relations (3.5). This is a product representation denoted by D D. The representation D is irreducible since there are no invariant subspaces under the action of all possible polynomials of Ŝi. Ontheotherhand,D D is reducible. Itcanbedecomposedintoadirect sum of irreducible singlet and triplet representations with D D = D 0 D (3.8) D 0 = { p ( +, > +, >)} (3.9) D = { ++>, p ( +, > +, >) >} (3.30) More generally there is an irreducible representation D j of SU() for all values of j =0,,, 3, with dimension dim(d j)=j +and ( ˆ~ S) >= j(j +) >, 8 >D j (3.3) In addition any representation of SU() can be decomposed into into a direct sum of irreducible representations, D j.

6 4CHAPTER 3. SPIN AND QUANTUM MECHANICAL ROTATION GROUP 3.4 Clebsch Gordan Decomopsition Let D j and D j be two irreducible representations of SU() with j i = 0,,, 3, then we have D j D j = D j +j D j +j D j j (3.3) with every irreducible representations D j appearing exactly once. If { j,m > } and { j,m >} is the standard basis in D j and D j respectively, then the elements { (j,j ); j; m >} of the standard basis of D j is related to { j,m >} and { j,m >} through (j,j ); j, m >= X m,m <m,j <m,j (j,j ); j, m > j,m > j,m > with j = j + j,, j j.theclebsch Gordan coe cients (3.33) <m,j <m,j (j,j ); j, m > < m,j,m,j j, m > (3.34) are elements of a unitary matrix which can furthermore be chosen orthogonal such that j,m > j,m >= X j,m <m,j,m,j j, m > j, m > (3.35) i.e. X X j,m <m,j,m,j j, m >< m, j j,m 0,j,m 0 > = m m 0 m m 0 m,m <m,j j,m,j,m >< m,j,m,j j 0,m 0 > = jj 0 mm 0 (3.36) and <m,j,m,j j, m >= 0 if m 6= m +m or j/{j +j,, j j } (3.37)

7 3.5. IRREDUCIBLE TENSOR OPERATORS 5 Example: For j and j = we have sj j +,m> = + m + j + m s > > + j,m> = j m + j + m > > + s j m Irreducible Tensor Operators j + m + > > s j + m + j + m + > > Let D be an irreducible representations of SU() on H. A collection of j operators, T jm, m = j,, j on H will be normal components of an irreducible tensor operator T j of type j if they transform under g SU() as Û(g)T jm Û(g) = X m 0 U(g) (j) mm 0 T jm 0 (3.39) where U(g) (j) mm 0 are the matrix components, in the standard basis, of the spin j representation D j of g. Thetransformationproperty(3.39)isequivalentto [ ˆK 3,T jm ] = mt jm (3.40) [ ˆK ±,T jm ] = p j(j +) m(m +)T jm± (3.4) where the { ˆK i } generate the representation of (3.) on H. An example for a scalar operator is the Hamiltonian H. The spin operators {Ŝ3, Ŝ± = Ŝ ± iŝ} are the normal components of a vector operator of type. Let H (j ), H (j ) be subspaces of H carrying the representations D j and D j of SU() respectively with possibly further quantum numbers indicated by. ThenthematrixelementsofatensoroperatorT j in the standard basis,j,m > have the simple form <,j,m T jm,j,m >=<,j T j,j >< m,j,m,j j,m > (3.4) where <m,j,m,j j,m > are the CGK-coe cients and <,j T j,j > is the reduced matrix element. ThisistheWigner-Eckhardt-Theorem. We will make use of it when considering the interaction of atoms with the radiation field. (3.38)

8 6CHAPTER 3. SPIN AND QUANTUM MECHANICAL ROTATION GROUP Historical Notes Stern had a PhD in Chemistry and became a pupil of Einstein in Prag where he learned about Bohr s quantization rule. Stern and his colleague Max von Laue made a vow: If this nonsense of Bohr should in the end prove to be right, we will quit physics! Experimental progress was hampered by financial di culties. So Stern presented a series of public lectures with an entrance fee. He also received a cheque for some hundreds of dollars. Goldman, a founder of the investment firm Goldman Sachs and progenitor of Woolworth Co stores, had family roots in Frankfurt. Because of frequent breakdown of the apparatus the deposition of silver atoms of the plate was very thin. While looking at the plate Stern was pu ng a cheap cigar which contained a lot of sulfur. This sulfur developed the silver into silver sulfide which is more easily visible. Pauli s reaction to the result of the Stern-Gerlach experiment was: This convinced me once and for all that an ingenious classical mechanism was ruled out to explain atomic phenomena... Further Reading AgoodtextbookforthequantummechanicalrotationgroupisSakurai s Modern Quantum Mechanics, chapter3. For a concise mathematical treatment of the representation theory of the rotation group and the Lorentz group see e.g. V. S. Varadaraja, Supersymmetry for mathematicians: an introduction, Volume of Courant lecture notes, section.4 and.5, available online under A detailed discussion of the Wigner-Eckhardt-Theorem can be found in H. Rollnik, Quantentheorie, Bd.. A readable book on some aspects of group theory is Wu-Ki-Tung, Group Theory in Physics.

9 Chapter 4 Quantum mechanical Lorentz group Our analysis of the quantum-mechanical rotation group lead to the interpretation of the rotation group on R 3 as a representation of the quantummechanical rotation group SU() through the important relation A SU() =) A~e ~ A =(R(A)~e ) ~ (4.) with A = A. If we introduce a fourth Paul matrix 0 = have similarly in the relativistic four-vector notation 0 0 Aq µ µa = R(A) µ q µ (4.) with R(A) 0 =0sinceA 0 A = 0. However, if we instead consider A SL(,C), the group of all complex -matrices with unit determinant the situation changes since now A 6= A.Moreprecisely q  : Q q µ µ = 0 + q 3 q iq q + iq q 0 q 3! ÂQ A Q (4.3) The map Q! A Q is linear, satisfies A Q = A Q and detq = q µ q µ =det A Q = A q µa q µ (4.4) We we assume the convention µ = diag(...). we 7

10 8 CHAPTER 4. QUANTUM MECHANICAL LORENTZ GROUP Thus there exists a Lorentz transformation (A) SO(3, ) such that Aq µ µa =( (A)) µ q µ (4.5) In other words, the map Â! (A) isahomomorphism from SL(, C) onto the proper, orthochronous Lorentz transformations SO + (3, ) with kernel {, }. It the follows form the first isomorphism theorem in group theory that SL(, C) isisomorphictoso + (3, )/{, }. 4. Spinor representations We will see that the representation theory of SL(, C) leadstothecorrect relativistic quantum theory for spin particles. For this we first consider the -dimensional representation V with (, =, )  : V 3 u! Âu i.e. 0 u =  u (4.6) An element in the dual space, Ṽ then transforms with ÂT   : Ṽ 3 ṽ! ÂT ṽ i.e. 0ṽ =  ṽ (4.7) Consequently the scalar product ṽ u = 0 ṽ 0 u is an invariant in analogy with q µ q µ.inminkowskispace-timethemetric provides an isomorphism between V and its dual. In the spinor calculus this isomorphism is provided by the invariant tensor 0 (i ) = (4.8) 0 or = = == = (4.9) and = 0. Under an SL(, C)-transformation we have 0 =   () 0 =  ÂT = (4.0) The last identity is the result of an explicit calculation. As a consequence we have  T =  (4.)

11 4.. PARITY 9 In analogy with Minkowski space-time we can think of v and v as the contravariant and the covariant components of a spinor in V. Note, however that v u = v u which shows, in particular, that spinors cannot be represented in tems of commuting objects in a relativistic quantum theory. In addition to the representation just discussed we there is a second inequivalent representation V,thecomplex conjugate or anti-fundamental representation with (, =, )  : V 3 u! ˆĀu i.e. 0 u = ˆĀ u (4.) where denotes complex conjugation. Similarly with  =  T ˆĀ  : V 3 ũ!  ũ i.e. 0ũ = ˆĀ ũ (4.3) with invariant product ũ u and (i ) e.t.c. Thus the complex conjugate components of a spinor in V transforms like a spinor in V 4. Parity Next we will investigate how parity-symmetry is represented on spinors. For this let us now return to the object Q in (4.3). From its transformation property (4.3) we infer that Q is an element of V V,ie. Q = Q = q µ ( µ ) (4.4) In other words Q is an element of the product representation denoted by (, 0) (0, )=(, ), which, according to (4.4) must be the vector representation. Note that the two representations (, 0) and (0, )areidentical for the SU()-subgroup of SL(, C). However they are inequivalent for SL(, C) since there is no generator in SL(, C) whichtakes(, 0) to (0, ). Under a parity transformation P : q =(q 0,~q)! P q =(q 0, ~q ) (4.5) the bi-spinor Q transforms as P Q = q 0 ~q ~ = Q q µ ( P µ) (4.6)

12 30 CHAPTER 4. QUANTUM MECHANICAL LORENTZ GROUP 4.3 Weyl/Dirac equation Using the anti-commuation relations (3.) it is not hard to see that P QQ = Q P Q = q µ q µ (4.7) Weyl considered a massless electron for which the four momentum p µ satisfies p =0. Thenthetwoequations p ṽ (p) =p µ µ ṽ (p) =0 (4.8) p u (p) =p µ ( P µ) u (p) =0 (4.9) define relativistic covariant Schrödinger equations in momentum space. In position space we have accordingly i~@ µ µ ṽ (x) =0 and i~@ µ ( P µ) u (x) =0 (4.0) Note that solutions of (4.0) are NOT of the form (x). If the mass M is not vanishing we note that p ṽ (p) =Mc u (p) and p u (p) =Mc ṽ (p) (4.) imply upon using (4.7) the Klein-Gordon equation in momentum space (p + M c )ṽ (p) =0 (4.) and analogously for u (p). The two equations (4.) constiute the Dirac equation for u and ṽ. 4.4 Coupling to the electro-magnetic field We obtain gauge-invariant equation in the presence of an electro-magnetic field through the substitution i~@ µ! i~@ µ e c A µ (4.3) In or conventions the relation between A µ and the Coulomb and vector potential is given by A µ =(, ~ A).

13 4.4. COUPLING TO THE ELECTRO-MAGNETIC FIELD 3 Equivalently with A = A 0 + A ~ ~ and P A = A 0 A ~ ~ the covariant Dirac equations (i~@ e ) c A ṽ (x) =Mc u e (x) and (i~@ P A ) u (x) =Mc ṽ (x) c (4.4) are gauge invariant since under a gauge transformation A µ (x)! A µ µ (x) and u (x)! e ie ~c (x) u (x), ṽ (x)! e ie ~c (x) ṽ (x) (4.5) the form of these equations is preserved. In practice it is often convenient to express the coupled system of the -component spinors u (x) andṽ (x) asasingleequationfora4-component spinor, i.e. with = u, µ = ṽ we obtain the compact equation (i~@/ The four Dirac matrices 0 µ 0 P, A/ = A µ µ = µ 0 A P A µ µ (4.6) e A/ ) = Mc (4.7) c µ satisfy the Cli ord algebra { µ, } = µ (4.8) Under Lorentz transformations 0 x µ = µ x the 4-component spinor transforms reducibly 0 A 0 ( 0 x)=s(a) (x), S(A) = 0 (A ) (4.9) Note that the representation (4.6) of the Cli ord algebra is unique only up u to linear transformations on. In particular we can choose any set of ṽ Dirac matrices µ with { µ, } = µ (4.30)

14 3 CHAPTER 4. QUANTUM MECHANICAL LORENTZ GROUP AresultduetoPaulishowsthat µ = S (i~@/ e A/ ) = Mc =) c µs and thus e cã/ ) = Mc (4.3) Ausefulchoiceforanalysingthenon-relativisticlimitoftheDiracequation is the so-called standard basis used in problem sheet 7. The quantization of the relativistic spin / particle proceeds in close analogy with the quantization of the electro-magnetic field explained in chapter by expanding the general solution of the Dirac equation in terms of plane waves with fixed four momentum k, multipliedwith(anti-commuting) operator valued coe cients b(k) andb (k). Further Reading AdetaileddiscussionofthespinorrepresentationoftheLorentzgroupcan be found in e.g.y. Ohnuki, Unitary Representations of the Poincare Group and Relativistic Wave Equations, chapter,availableonlinewithgoogle books. A description of the quantization of the Dirac field (4.6) can be found Sakurai, Advanced Quantum Mechanics, section3.