0
$\begingroup$

I am following the Quantum Mechanics I, Galindo A., Pascual P. and in page 207 explaining the matrix representations of the Rotation Operators in the angular momentum it appears the next (obvious) equation: enter image description here

Being $\alpha,\beta,\gamma$ the Euler angles, $J$ the value for the total angular momentum (orbital+spin) and $M$ corresponding to the eigen value of $J_z$.

I don't understand the physical meaning of the equations. I mean, on the left part it is applying the rotation operator to the state vector $|JM\rangle$. As $J$ is invariant under rotations the right part of the equation should be obvious. So, what is the right part of the equation telling me? Maybe I'm getting confused with the Dirac notation. Any tip?

$\endgroup$
  • $\begingroup$ The r.h.s. is telling you that $R$ doesn't change the value of $J$, but only mixes the $M$ components. $\endgroup$ – AccidentalFourierTransform May 4 '16 at 18:02
  • $\begingroup$ So basicly the equation is telling me that a rotation applied to a state $|JM\rangle$ it is equal to some specific rotations above all $|JM'\rangle$? $\endgroup$ – Marc C May 4 '16 at 18:10
  • $\begingroup$ ...is equal to some linear combination of all $|JM'\rangle$ $\endgroup$ – AccidentalFourierTransform May 4 '16 at 18:15
  • $\begingroup$ Oh, now it makes sense: when a rotation it's applied to a state vector it can be expressed as a linear combination of $|JM'\rangle$. I was confused by the notation. Thank you. $\endgroup$ – Marc C May 4 '16 at 18:22
1
$\begingroup$

The left hand: rotate the state $|JM\rangle$ by applying a rotation $R$ on it. Right hand side:

  1. insert completeness condition $\sum_{M'} |JM'\rangle\langle JM'|$
  2. $D$ is the matrix representation of rotation matrix $R$ in basis ${|JM\rangle}$. The rotated state is expanded in terms of basis ${|JM\rangle}$ with coefficient $D$ in terms of rotation matrix.
$\endgroup$
0
$\begingroup$

You can think, on the left is a short hand notation for a (2J+1) x (2J+1) matrix R applied to a (2J+1) component vector |J,M> with the components labelled by M.

On the right, the matrix elements are explicitly shown, and the sum over M' is the matrix multiplication.

Actually on the left is an abstract rotation operator R that will rotate any J. When applied to the particular J in the ket, it results in the matrix mutiplication on the right.

You can think of the operator R as being an infinite dimensional matrix with one (2j+1)x(2j+1) block on the diagonal for each j=0, 1/2, 1, 3/2, 2,.... The infinite length ket vector has non-zero components only for j=J (ie: the (2J+1) elements M'). Therefore only the j=J block multiplication is shown on the right.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.