C4 Lecture 3 - Jim Libby1 Lecture 3 summary Frames of reference Invariance under transformations Rotation of a H wave function: d -functions Example: e + e - →μ + μ - Angular momentum as a rotation generator Euler angles Generic translations, conservation laws and Noether’s theorem

C4 Lecture 3 - Jim Libby2 Frames of reference Consider a frame of reference O in which a generic state is described i.e. the H wave function ψ( r ) of an e - in the 2p state. If O’ is a different frame of reference connected to O by where G is a group of transformations i.e. translations, rotations or Lorentz transformations of the coordinate system The wave function in O ’ will in general be ψ´( r ´) with where ψ γ is an orthonormal basis

C4 Lecture 3 - Jim Libby3 Example: infinite 1D well For the observer O the ground state is given by Now translate x´=D(a)x=x+a. In O´ the ground state is not ψ (x´) but In analogous fashion in O´´ where the translation is D(a/2) -a a 0 2a -a/2 3a/2 ψ(x) ψ(x´) ψ(x´´) x x´x´ x´´ The physics is invariant Different eigenfunction Summation over eigenfunctions

C4 Lecture 3 - Jim Libby4 Some atomic physics We will consider the H atom in one of its excited states The eigenfunctions are: We define a new reference frame O´(x´,y´,z´) which is rotated about the y axis of the original frame by an angle β –r´=R y (β)r The Hamiltonian and L 2 are unchanged as are their respective eigenvalues n and l. In other words they commute with R y. However, the z direction has changed so m is not the same, it is now projected on a new z´- axis 2p wavefunction

C4 Lecture 3 - Jim Libby5 Rotation H wavefunction The new wave function u´ nlm (r´) can be expressed by a superposition of wavefunctions with the same n and l but with different m´ Now we will use the 2p state (n=2,l=1,m=0) as an example:

C4 Lecture 3 - Jim Libby6 H continued

C4 Lecture 3 - Jim Libby7 H continued We use the previous results to express Y 10 in terms the β and spherical harmonics in the rotated frame Comparing to the general expression we get the d coefficients In a similar fashion all d l m′m coefficients can be calculated –Somewhat labourious – neater method later Work out the probability that rotated state is in an eigenstate

C4 Lecture 3 - Jim Libby8 Tabulations of d functions from the PDG

C4 Lecture 3 - Jim Libby9 Example: e + e - →μ + μ - e+e+ e-e- Spins in the relativistic limit μ+μ+ μ-μ- Only photon exchange in relativistic limit (M μ << CoM energy<<M Z0 ) z z′z′ θ Left-handed electron annihilates with right-handed electron from helicity conservation. Therefore, final state particles have opposite helicity as well Two amplitudes must be of equal intensity, ∫|A| 2 dcosθ, because of parity conservation e+e+ e−e− μ+μ+ μ−μ− z z′z′ θ Initial state J z =+1 Final state J z′ =+1 Initial state J z =+1 Final state J z′ =−1 RH LH RHLH RH

C4 Lecture 3 - Jim Libby10 Example: e + e - →μ + μ - 1+cos 2 θ Weak, Parity violating effects distort the distribution

C4 Lecture 3 - Jim Libby11 Angular momentum operators as generators of rotations J y is the generator of rotations about the y axis. Similar results for J x and J z

C4 Lecture 3 - Jim Libby12 Angular momentum operators as generators of rotations If ψ is a solution of the Schroedinger equation is ? Requires [J y,H]=0 Unitary operator

C4 Lecture 3 - Jim Libby13 Angular momentum operators as generators of rotations We will now consider time variation of matrix elements of J y Then matrix element is invariant with time and the eigenvalues of J y are constant. This implies: –the y projection of angular momentum is conserved –wavefunction is invariant under rotations

C4 Lecture 3 - Jim Libby14 Finite rotations General case where α is a vector in the direction of the rotation axis with a magnitude equal to the angle of rotation

C4 Lecture 3 - Jim Libby15 d matrices from rotation operators

C4 Lecture 3 - Jim Libby16 d matrices from rotation operators

C4 Lecture 3 - Jim Libby17 Euler angles Generic rotations described by Euler angles Define three successive rotations: –angle α about the Z axis –angle β about the y 0 axis –angle γ about the new z axis Can be recast as first γ about original z, β about the original y and α about the original z The rotations of wavefunctions can be represented by D matrices Using angular momentum operators as generators of the rotations and (z 0 ) x0x0 y0y0

C4 Lecture 3 - Jim Libby18 Euler angles 1) γ about z 2) β about y 3) α about z

C4 Lecture 3 - Jim Libby19 Euler angles 1) -γ about current z 2) -β about current y 3) -α about current z

C4 Lecture 3 - Jim Libby20 Translations Similar analysis can be applied to translations: Assume a is infinitesimal:

C4 Lecture 3 - Jim Libby21 Translations So for finite translations: Invariance of the wave equation under translations ↔conservation of momentum Similarly, for time if Hamiltonian, H, is time independent: Invariance of the wavefunction with respect to time translations ↔ conservation of energy

C4 Lecture 3 - Jim Libby22 Symmetry Principles An invariance or symmetry principle exists for a physical system, S, and transformation, g G, if the physical laws, expressed for S by the observer O in his coordinate system, also hold good for the same system S in the coordinate system of the observer O’ Noether’s theorem: –Symmetry principle ↔ Invariance of theory ↔ Conservation law