## PHY456H1F: Quantum Mechanics II. Lecture 20 (Taught by Prof J.E. Sipe). Spherical tensors.

Posted by peeterjoot on November 23, 2011

# Disclaimer.

Peeter’s lecture notes from class. May not be entirely coherent.

# Spherical tensors (cont).

READING: section 29 of [1].

**definition**. Any operator , are the elements of a spherical tensor of rank if

where was the matrix element of the rotation operator

So, if we have a Cartesian vector operator with components then we can construct a corresponding spherical vector operator

By considering infinitesimal rotations we can come up with the commutation relations between the angular momentum operators

Note that the text in (29.15) defines these, whereas in class these were considered consequences of 2.1, once infinitesimal rotations were used.

Recall that these match our angular momentum raising and lowering identities

Consider two problems

We have a correspondence between the spherical tensors and angular momentum kets

So, as we can write for angular momentum

We also have for spherical tensors

Can form eigenstates of and (z-comp of the total angular momentum).

FIXME: this won’t be proven, but we are strongly suggested to try this ourselves.

We can check the dimensions for a spherical tensor decomposition into rank 0, rank 1 and rank 2 tensors.

latex 0$} & (1) & (\text{Cartesian vector})(\text{Cartesian vector}) \\ \text{spherical tensor rank } & (3) & (3)(3) \\ \text{spherical tensor rank } & (5) & 9 \\ \hline\text{dimension check sum} & 9 & \\ \end{array}\end{aligned} \hspace{\stretch{1}}(2.11)$

Or in the direct product and sum shorthand

Note that this is just like problem 4 in problem set 10 where we calculated the CG kets for the decomposition starting from kets .

**Example**.

How about a Cartesian tensor of rank 3?

## Why bother?

Consider a tensor operator and an eigenket of angular momentum , where is a degeneracy index.

Look at

This transforms like . We can say immediately

unless

This is the “selection rule”.

Examples.

\begin{itemize}

\item Scalar

unless and .

\item . What are the non-vanishing matrix elements?

unless

unless

\end{itemize}

Very generally one can prove (the Wigner-Eckart theory in the text section 29.3)

where we split into a “reduced matrix element” describing the “physics”, and the CG coefficient for “geometry” respectively.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

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