zigoteau:

Q_xx = p_x p_x - (1/3)p^2

or, since there are only 5 independent components, it can be written
in terms of a traceless operator with 5 components,


Q_+/-2 = (1/2)(Q_xx - Q_yy +/- 2i Q_xy)

Q+/-1 = (1/2)(Q_xz - iQ_yz)

Q_0 = sqrt(1/3)[Q_zz - (1/2)(Q_xx + Q_yy)]


Oscillations in the xy-plane only, are the linerar combinations above,
that contain only Q_xx, Qyy and Q_xy, i.e., Q_+/-2.


You don't seem to be saying the same thing. It is perhaps tangential
to sense.

Bilge, it is not clear why Q_0 should not be symmetrical in x, y and
z.

I defined the z-axis, i.e., p_z = p cos(\theta) as the longitudinal
direction. Q_0 might look more familar if you write out the components:


Q_zz + (1/2)[Q_xx + Q_yy]

= (p_z)^2 - (1/3)p^2 - (1/2) [(p_x)^2 + (p_y)^2 - (2/3)p^2]

Since p^2 = (p_x)^2 + (p_y)^2 + (p_z)^2, p^2 - (p_z)^2 = (p_x)^2 + (p_y)^2
and you have,

Q_0 = (p_z)^2 - (1/3)p^2 - (1/2) [p^2 - (p_z)^2 - (2/3)p^2]

(3/2) (p_z)^2 - (1/2)p^2

= (1/2) [3 cos^2(\theta) - 1] p^2

which apart from a normalization, is Y_20 x p^2. (I also have the
normalization wrong on Q_0. it should be sqrt(2/3)).

Tom, distances between which points? This is presumably connected to
the metric tensor.

Since you give different answers while both referring to MTW (which I
do not have) I deduce that the subject is not discussed explicitly
anywhere in that textbook, and that this is just a tactic by both of
you to blind me with science and shut me up. Or could either or both
of you give page numbers?

Actually, I didn't refer to MTW. I just wrote down a quadrupole tensor
with the components written in terms of the projections along one of the
axes. That was basically what you had asked.

I seem to remember something about elements of tensor space which map
into each other under the operation of the set of rotations. Is there
a connection to spherical harmonics and the total angular momentum
quantum number l? Am I getting warm?

Yes, since in particular, the Q_m are the components of Y_lm for l = 2.

(Bilge) wrote in message ...

Hi, Bilge,

Thanks for your detailed reply.

Q_xx = p_x p_x - (1/3)p^2

or, since there are only 5 independent components, it can be written
in terms of a traceless operator with 5 components,


Q_+/-2 = (1/2)(Q_xx - Q_yy +/- 2i Q_xy)

Q+/-1 = (1/2)(Q_xz - iQ_yz)

Q_0 = sqrt(1/3)[Q_zz - (1/2)(Q_xx + Q_yy)]


Oscillations in the xy-plane only, are the linerar combinations above,
that contain only Q_xx, Qyy and Q_xy, i.e., Q_+/-2.


You don't seem to be saying the same thing. It is perhaps tangential
to sense.

Bilge, it is not clear why Q_0 should not be symmetrical in x, y and
z.

I defined the z-axis, i.e., p_z = p cos(\theta) as the longitudinal
direction. Q_0 might look more familar if you write out the components:


Q_zz + (1/2)[Q_xx + Q_yy]

= (p_z)^2 - (1/3)p^2 - (1/2) [(p_x)^2 + (p_y)^2 - (2/3)p^2]

Since p^2 = (p_x)^2 + (p_y)^2 + (p_z)^2, p^2 - (p_z)^2 = (p_x)^2 + (p_y)^2
and you have,

Q_0 = (p_z)^2 - (1/3)p^2 - (1/2) [p^2 - (p_z)^2 - (2/3)p^2]

(3/2) (p_z)^2 - (1/2)p^2

= (1/2) [3 cos^2(\theta) - 1] p^2

which apart from a normalization, is Y_20 x p^2. (I also have the
normalization wrong on Q_0. it should be sqrt(2/3)).

Tom, distances between which points? This is presumably connected to
the metric tensor.

Since you give different answers while both referring to MTW (which I
do not have) I deduce that the subject is not discussed explicitly
anywhere in that textbook, and that this is just a tactic by both of
you to blind me with science and shut me up. Or could either or both
of you give page numbers?

Actually, I didn't refer to MTW. I just wrote down a quadrupole tensor
with the components written in terms of the projections along one of the
axes. That was basically what you had asked.

I seem to remember something about elements of tensor space which map
into each other under the operation of the set of rotations. Is there
a connection to spherical harmonics and the total angular momentum
quantum number l? Am I getting warm?

Yes, since in particular, the Q_m are the components of Y_lm for l = 2.

Cheers,

Zigoteau.