Commutation relation for L_3 and phi

In summary, the commutation relation for L<sub>3</sub> and phi in quantum mechanics is given by [L<sub>3</sub>, phi] = i&hbar;ℏsin(phi). It is derived using quantum mechanical operators for angular momentum and position, and it tells us that the order of L<sub>3</sub> and phi matters in calculations. This is closely related to the uncertainty principle, which states that certain physical properties cannot be known simultaneously with complete precision. The commutation relation can also be applied to other quantum mechanical operators and is a fundamental tool for understanding quantum systems.
  • #1
ythaaa
4
0
Hi, just wondering whether the commutation relation [tex][\phi,L_3]=i\hbar[/tex] holds and similar uncertainty relation such as involving X and Px can be derived ?

thanks
 
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  • #2
testing that commutator is simple, since [tex] L_z \sim \frac{d}{d\phi} [/tex], let the commutator act on a function [tex] F(r,\phi,\theta)[/tex] and then you\ll have it.

The same thing for [x,P_x]
 
  • #3


I can confirm that the commutation relation [\phi,L_3]=i\hbar does hold. This means that the operators for position and angular momentum in the third direction do not commute, and therefore cannot be measured simultaneously with complete precision. This is known as the uncertainty principle, and it does extend to other pairs of operators such as X and Px. The specific uncertainty relation for these operators is given by \Delta X \Delta Px \geq \frac{\hbar}{2}. This principle is a fundamental aspect of quantum mechanics and has been experimentally verified numerous times. I hope this answers your question.
 

Related to Commutation relation for L_3 and phi

1. What is the commutation relation for L3 and phi?

The commutation relation for L3 and phi is given by [L3, phi] = iℏℏsin(phi).

2. How is the commutation relation derived?

The commutation relation is derived using the principles of quantum mechanics, specifically the quantum mechanical operators for angular momentum (L3) and position (phi).

3. What does the commutation relation tell us about L3 and phi?

The commutation relation tells us that L3 and phi do not commute, meaning that their order matters when performing calculations. This is a fundamental concept in quantum mechanics and has important implications for the uncertainty principle.

4. How does the commutation relation relate to the uncertainty principle?

The commutation relation is closely related to the uncertainty principle, which states that certain pairs of physical properties, such as position and momentum, cannot be known simultaneously with complete precision. The commutation relation for L3 and phi is one example of this principle in action.

5. Can the commutation relation be applied to other quantum mechanical operators?

Yes, the commutation relation is a general concept in quantum mechanics and can be applied to other operators as well. It is an important tool for understanding the behavior of quantum systems and making predictions about their properties.

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