Wigner-Eckart theorem

The Wigner-Eckart theorem is a theorem from representation theory and quantum mechanics . It says that the matrix element of the in the basis of the eigenfunctions of the angular momentum operator can be represented as the product of two quantities, one of which is independent of the projections of the angular momentum, and the other is the Clebsch-Gordan coefficient . The name of the theorem is derived from the names of Eugene Wigner and Karl Eckart , who developed a construction linking the symmetry of the transformation of space groups with the laws of conservation of energy, momentum and angular momentum. ^[one]

The Wigner-Eckart theorem is formulated as follows:

\langle jm|T_{q}^{k}|j'm'\rangle =\langle j||T^{k}||j'\rangle C_{kqj'm'}^{jm},

{\ displaystyle \ langle jm | T_ {q} ^ {k} | j'm '\ rangle = \ langle j || T ^ {k} || j' \ rangle C_ {kqj'm '} ^ {jm} ,}

{\ displaystyle \ langle jm | T_ {q} ^ {k} | j'm '\ rangle = \ langle j || T ^ {k} || j' \ rangle C_ {kqj'm '} ^ {jm} ,}

Where $T_{q}^{k}$ ${\ displaystyle T_ {q} ^ {k}}$ $T_ {q} ^ {k}$ - $k$ ${\ displaystyle k}$ $k$ , $|jm\rangle$ ${\ displaystyle | jm \ rangle}$ $| jm \ rangle$ and $|j'm'\rangle$ ${\ displaystyle | j'm '\ rangle}$ $| j'm '\ rangle$ are the eigenfunctions of the total angular momentum $J^{2}$ ${\ displaystyle J ^ {2}}$ $J ^ {2}$ and its z components $J_{z}$ ${\ displaystyle J_ {z}}$ $J_ {z}$ , $\langle j||T^{k}||j'\rangle$ ${\ displaystyle \ langle j || T ^ {k} || j '\ rangle}$ $\ langle j || T ^ {k} || j '\ rangle$ independent of $m$ ${\ displaystyle m}$ $m$ and $q$ ${\ displaystyle q}$ $q$ , and $C_{kqj'm'}^{jm}=\langle j'm';kq|jm\rangle$ ${\ displaystyle C_ {kqj'm '} ^ {jm} = \ langle j'm'; kq | jm \ rangle}$ $C _ {{kqj'm '}} ^ {{jm}} = \ langle j'm'; kq | jm \ rangle$ - Clebsch - Gordan addition coefficients $j^{'}$ ${\ displaystyle j '}$ $j '$ and $k$ ${\ displaystyle k}$ $k$ to receive $j$ ${\ displaystyle j}$ $j$ .

As a consequence, the Wigner-Eckart theorem tells us that the action of a spherical tensor operator of rank $k$ ${\ displaystyle k}$ $k$ on the eigenfunction of the angular momentum is the same as adding a state with angular momentum $k$ ${\ displaystyle k}$ $k$ to the original state. The matrix elements found for the spherical tensor operator are proportional to the Clebsch-Gordan coefficients that arise when two angular moments are added.

Example

Consider the average value of the coordinate $\langle njm|x|njm\rangle$ ${\ displaystyle \ langle njm | x | njm \ rangle}$ . This matrix element is the average value of the coordinate operator in the spherically symmetric basis of the eigenstates of the hydrogen atom. Finding these matrix elements is a non-trivial task. However, the use of the Wigner-Eckart theorem simplifies this task. (It’s actually possible to get a solution right away using parity .)

It is known that $x$ ${\ displaystyle x}$ Is one of the components of the vector ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ . Vectors are tensors of the first rank, so $x$ ${\ displaystyle x}$ is some linear combination $T_{q}^{1}$ ${\ displaystyle T_ {q} ^ {1}}$ where $q=-1,0,1$ ${\ displaystyle q = -1,0,1}$ . It can be shown that $x={\tfrac {T_{-1}^{1}-T_{1}^{1}}{\sqrt {2}}}$ ${\ displaystyle x = {\ tfrac {T _ {- 1} ^ {1} -T_ {1} ^ {1}} {\ sqrt {2}}}}$ where the spherical tensors ^{[2] are} defined as follows: $T_{0}^{1}=z$ ${\ displaystyle T_ {0} ^ {1} = z}$ and $T_{\pm 1}^{1}=\mp (x\pm iy)/{\sqrt {2}}$ ${\ displaystyle T _ {\ pm 1} ^ {1} = \ mp (x \ pm iy) / {\ sqrt {2}}}$ (signs must be selected according to the definition ^{[2] of a} spherical rank tensor $k$ ${\ displaystyle k}$ . Consequently, $T_{q}^{1}$ ${\ displaystyle T_ {q} ^ {1}}$ proportional only to stair operators ). therefore

\langle njm|x|n'j'm'\rangle =\left\langle njm\left|{\tfrac {T_{-1}^{1}-T_{1}^{1}}{\sqrt {2}}}\right|n'j'm'\right\rangle ={\frac {1}{\sqrt {2}}}\langle nj||T^{1}||n'j'\rangle (C_{1(-1)j'm'}^{jm}-C_{11j'm'}^{jm}).

{\ displaystyle \ langle njm | x | n'j'm '\ rangle = \ left \ langle njm \ left | {\ tfrac {T _ {- 1} ^ {1} -T_ {1} ^ {1}} { \ sqrt {2}}} \ right | n'j'm '\ right \ rangle = {\ frac {1} {\ sqrt {2}}} \ langle nj || T ^ {1} || n'j '\ rangle (C_ {1 (-1) j'm'} ^ {jm} -C_ {11j'm '} ^ {jm}).}

The expressions above give us matrix elements for $x$ ${\ displaystyle x}$ in the basis $|njm\rangle$ ${\ displaystyle | njm \ rangle}$ . To find the average value, put $n'=n$ ${\ displaystyle n '= n}$ , $j'=j$ ${\ displaystyle j '= j}$ , and $m'=m$ ${\ displaystyle m '= m}$ . Selection rules for $m^{'}$ ${\ displaystyle m '}$ and $m$ ${\ displaystyle m}$ these are: $m\pm 1=m'$ ${\ displaystyle m \ pm 1 = m '}$ for spherical tensors $T_{\mp 1}^{(1)}$ ${\ displaystyle T _ {\ mp 1} ^ {(1)}}$ . Once $m'=m$ ${\ displaystyle m '= m}$ , the Clebsch - Gordan coefficients vanish, which leads to the equality of the mean values to zero.

Notes

↑ Eckart Biography Archived March 25, 2007. - The National Academies Press.
↑ ¹ ² JJ Sakurai: “Modern quantum mechanics” (Massachusetts, 1994, Addison-Wesley).

Links

JJ Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2 .
Weisstein, Eric W. Wigner – Eckart theorem on the Wolfram MathWorld website.
Wigner – Eckart theorem .
Tensor Operators .

[Eckart_Biography-1] Eckart Biography Archived March 25, 2007. - The National Academies Press.

[autogenerated1-2] ¹ ² JJ Sakurai: “Modern quantum mechanics” (Massachusetts, 1994, Addison-Wesley).

Wigner-Eckart theorem

Example

Notes

Links

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