EP 228: Quantum Mechanics Lec 33: Tensor Operators & - - PowerPoint PPT Presentation

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

• perators [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

• perators
• 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
• f 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>

• <j’,m’|T(k,q)|j,m> = <j’||T(k)||j> <j,k; m,q| j’,m’>
• RHS has two terms:

first term is dynamical term called reduced matrix element(needs experimental data) 2nd term is geometrical dependent on

• rientation given by CG coefft

Wigner- Eckart Theorem

• <j’,m’|T(k,q)|j,m> = <j’||T(k)||j> <j,k; m,q| j’,m’>
• Using the above theorem, we can see that

matrix elements of scalar operators must be

• nly diagonal(non-diagonal terms is zero)
• Given matrix elements for z-component of a

vector operator, can we obtain x-component

• f 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