Why Is There a Difference in Orbital Angular Momentum Calculation Methods?

In summary, the conversation discusses the application of the Lagrangian operator, represented by $L$. The speaker realizes that they are applying something they do not fully understand, leading to a discussion about the different ways to represent $L_x$. The speaker also mentions recalling other exercises where it was correct to combine the del components in a different way.
  • #1
ognik
643
2
I got to here in a simple exercise (orb. ang. momentum cords), realized I was applying something I didn't understand ...

$L = -i \begin{vmatrix}\hat{x}&\hat{y}&\hat{z}\\x&y&z\\\pd{}{x}&\pd{}{y}&\pd{}{z}\end{vmatrix}$

I 'know' it equates to $L_x =-i \left( y\pd{}{z} - z\pd{}{y} \right) $ - but I could just as well (?) have $L_x =-i \left( \pd{y}{z} - \pd{z}{y} \right) $?

(I seem also to recall other exercises where it was right to combine the del components the 2nd way above)
 
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  • #2
by definition - answered in other post
 
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FAQ: Why Is There a Difference in Orbital Angular Momentum Calculation Methods?

What does "order with del in cross product" mean?

"Order with del in cross product" refers to the way that the gradient operator (del) is applied to a vector in a cross product. This order is important because it affects the direction of the resulting vector.

Why is the order of del important in a cross product?

The order of del is important in a cross product because it determines the direction of the resulting vector. If del is applied to the first vector, the resulting vector will be perpendicular to both the first and second vectors. However, if del is applied to the second vector, the resulting vector will be in the opposite direction.

How do you determine the order of del in a cross product?

The order of del in a cross product is determined by the right-hand rule. When using the right-hand rule, the first vector is represented by the index finger, the second vector by the middle finger, and the resulting vector by the thumb. The direction of del should follow this same pattern, with del representing the middle finger.

What happens if the order of del is switched in a cross product?

If the order of del is switched in a cross product, the resulting vector will be in the opposite direction. This is because the right-hand rule is no longer being followed, and the resulting vector will be represented by the opposite finger (in this case, the index finger).

How is "order with del in cross product" used in physics and engineering?

"Order with del in cross product" is used in physics and engineering to calculate the direction of a resulting force or torque in a system. It is also used in electromagnetism to determine the direction of the magnetic field produced by a current-carrying wire. Additionally, it is used in fluid mechanics to determine the direction of the velocity of a fluid flow.

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