Self-verifying theories

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The Wigner–Eckart theorem is a theorem of representation theory and quantum mechanics. It states that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient. The name derives from physicists Eugene Wigner and Carl Eckart who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.[1]

The Wigner–Eckart theorem reads:

jm|Tqk|jm=j||Tk||jCkqjmjm

where Tqk is a rank k spherical tensor, |jm and |jm are eigenkets of total angular momentum J2 and its z-component Jz, j||Tk||j has a value which is independent of m and q, and Ckqjmjm=jm;kq|jm is the Clebsch-Gordan coefficient for adding j′ and k to get j.

In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank k on an angular momentum eigenstate is like adding a state with angular momentum k to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta.

Proof

Starting with the definition of a spherical tensor, we have that

[J±,Tq(k)]=(kq)(k±q+1)Tq±1(k)

which we use to then calculate

α,jm|[J±,Tq(k)]|α,jm=(kq)(k±q+1)α,jm|Tq±1(k)|α,jm.

If we expand the commutator on the LHS by calculating the action of the J± on the bra and ket, then we get

α,jm|[J±,Tq(k)]|α,jm=(j±m)(jm+1)α,jm1|Tq(k)|α,jm(jm)(j±m+1)α,jm|Tq(k)|α,jm±1.

We may combine these two results to get

(j±m)(jm+1)α,jm1|Tq(k)|α,jm=(jm)(j±m+1)α,jm|Tq(k)|α,jm±1+(kq)(k±q+1)α,jm|Tq±1(k)|α,jm.

This recursion relation for the matrix elements closely resembles that of the Clebsch-Gordan coefficient. In fact, both are of the form jaijxj=0. We therefore have two sets of linear homogeneous equations

jaijxj=0,jaijyj=0

one for the Clebsch-Gordan coefficients (xj) and one for the matrix elements (yj). It is not possible to exactly solve for the xj. We can only say that the ratios are equal, that is

xjxk=yjyk

or that xj = cyj, where c is a coefficient of proportionality independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch-Gordan coefficient j1j2;m1,m2±1|j1j2;jm with the matrix element α,jm|Tq±1(k)|α,jm, then we may write

α,jm|Tq±1(k)|α,jm=(proportionality constant)jk;mq±1|jk;jm.

By convention the proportionality constant is written as αj||T(k)||αj12j+1, where the denominator is a normalizing factor.

Example

Consider the position expectation value njm|x|njm. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate basis, which is a nontrivial problem. However, using the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using parity, although a slightly longer route will be taken.)

We know that x is one component of Template:Vec, which is a vector. Vectors are rank-1 tensors, so x is some linear combination of T1q for q = -1, 0, 1. In fact, it can be shown that

x=T11T112,

where we defined the spherical tensors[2] T10 = z and

T±11=(x±iy)/2

(the pre-factors have to be chosen according to the definition[2] of a spherical tensor of rank k. Hence, the T1q are only proportional to the ladder operators). Therefore

njm|x|njm=njm|T11T112|njm=12nj||T1||nj(C1(1)jmjmC11jmjm)

The above expression gives us the matrix element for x in the |njm basis. To find the expectation value, we set n′ = n, j′ = j, and m′ = m. The selection rule for m′ and m is m±1=m for the T1(1) spherical tensors. As we have m′ = m, this makes the Clebsch-Gordan Coefficients zero, leading to the expectation value to be equal to zero.

References

  1. Eckart Biography– The National Academies Press
  2. 2.0 2.1 J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley)

External links

  • J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2.
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  • Wigner–Eckart theorem
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