EP 228: Quantum Mechanics Lec 33: Tensor Operators & Wigner-Eckart Theorem Recap: CG coeffts for addn of two spin 1/2

• Uncoupled basis

• Maximum m=1 and s=jmax = 1. Hence this coupled state Recap:CG coeffts for addn of two spin ½ contd

• m = 0 can be obtained by acting lowering operator on LHS and RHS of

• Recall ,

• LHS: Recap:CG coeffts for addn of two spin ½ contd

• Action of lowering operator on RHS of

• =

• Recall LHS . • Equating

• Recap: CG coeffts for addn of two spin ½ contd

• Acting lowering operator on LHS and RHS of

• What will we get? • Check whether you get this

• How to determine Recap: CG coeffts for addn of two spin ½ contd

• The state must be orthogonal to

• , ,

• In particular, m=0 state requires the same uncoupled basis. Hence,

• Now we will look at tensor operators Scalar operators

• Under rotation, we know how state vectors transform. How will linear operator A transform? • A A’ = U†(θ) A U(θ) • When do we call a operator scalar? • Scalar do not change under rotations • From the similarity transformation, we can conclude that all scalar operators will commute with angular momentum

operators [A, Ji ] =0 • We know components of a vector transform like like the position vector components under rotations. Vector operators

• Under rotation, we know how state vectors transform. How will linear operator A transform? • A A’ = U†(θ) A U(θ) • We know components of a vector transform like the position vector components under rotations.

• What is the commutator of [Ji, rj ]? • The same holds for any vector operators- • Examples: position vector, linear momentum, angular momentum, vector potential etc • Examples of scalar operators: dot product of two like (r.p) , radial component r= √(r.r) Irreducible spherical Tensor operators • We denote T(k,q) as irreducible tensor operator of rank k where q takes –k,-k+1,….+k • Scalar operator will be T(0,0). • Vector operator will be T(1,q) where q can be 1,0,-1. • Position vector {-iy± (-x)}/√2=T(1, ±1), z=T(1,0)

• We can check [ Jz , T(k,q)] = qT(k,q) • [J±, T(k,q)]=? Irreducible spherical Tensors like Spherical harmonics Irreducible spherical Tensor operators • T(k,q) irreducible tensor operator of rank k where q takes –k,-k+1,….+k resembles state vector|k,q> • Just like we take tensor product of

states |k1 , q1 > |k2 , q2 > giving uncoupled basis, we can take tensor product of two irreducible tensors giving reducible tensor ReducibleTensor operators • We can take tensor product of two vectors –for

example moment of inertia tensor Iij (reducible). How many components does this have? • This has 9 components (like uncoupled basis) • We can break it three pieces (like coupled basis): (i) trace of I (behaves like scalar k=0) (ii) antisymmetric matrix I (behaves like vect k=1) (iii) symmetric traceless matrix I- how many components does this have?

Recall |1, q1> |1,q2> is 9 diml LVS. The coupled basis |j,q> will allow j=0,1,2

Commutator with angular mom Irreducible spherical tensor operators • Using the same CG coeffts, we can divide the tensor product of two vectors. • A(k, q) B(r,s) = ∑ CG T(a,q+s) where a is an element of angular momentum addition k + r. • (i) trace of I (behaves like scalar k=0) • (ii) antisymmetric matrix I (behaves like vector k=1) • (iii) symmetric traceless matrix I( behaves like rank 2 tensor with 5 components) Wigner- Eckart Theorem • Selection rule for matrix elements of spherical tensor operators T(k,q) where states are angular momentum states |j,m> • = • RHS has two terms: first term is dynamical term called reduced matrix element(needs experimental data) 2nd term is geometrical dependent on orientation given by CG coefft Wigner- Eckart Theorem • = • Using the above theorem, we can see that matrix elements of scalar operators must be only diagonal(non-diagonal terms is zero) • Given matrix elements for z-component of a vector operator, can we obtain x-component of the vector operator? • Yes. Using CG coeffts, we can write the answer • Quadrapole moment tensor is rank 2 tensor? Applications in nuclear physics • Protons, neutrons and other particles have isospin I satisfy algebra like angular momentum J • Isopin does not interact with angular momentum. • Rotational invariance requires any scattering process or decay process to obey angular momentum conservation • In strong nuclear interactions, scattering • process and decay process conserve • isospin. Applications in nuclear physics • Using Wigner-Eckart theorem, the ratios of scattering amplitudes or decay rates can be computed for strong nuclear process. • Determine the ratio of the decay rates of rho meson and Pi meson whose I=1: Hope you enjoyed the semester