Operators Commutation: Explaining P(x), L(y)

In summary, P(x) and L(y) are operators representing momentum and angular momentum with respect to x and y, respectively. The Born-Jordan commutation relations show that P(x) and P(y) commute, resulting in a simplified formula for their commutator. The underlined term in the given solution is obtained by explicitly writing out the commutator and realizing that P(z) also commutes with P(x).
  • #1
rsaad
77
0
Can someone please explain to me how do we get the following:

[P(x), L(y)]= i h(cut) P(z)

P(x) is the momentum operator with respect to x
and L(y) is the angular momentum operator with respect to y.

I have also attached the solution. I am stuck at the underlined part. I do not know how to proceed from there.
 

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  • #2
Px and Py commute, as per Born-Jordan commutation relations. Thus the term from with their commutator is 0 when you apply the general formula

[A, BC] = [A,B]C+B[A,C] with [A,B] =0
 
Last edited:
  • #3
How did you obtain the general formula that you have stated in your reply?
 
  • #4
it should be as follows:
[a,bc]=[a,b]c+b[a,c]
 
  • #5
[x,px]=ih/2∏ is the usual commutation rule,if that is what you are asking.
EDIT:if you want to know how to get that underlined term then just write the commutator explicitly and see that pz commutes with px.
 
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  • #6
You got it right in post #4. Just work it out staring from [itex][A, BC][/itex] and write out the commutator, then in the middle add zero in a fancy way.
 
  • #7
Yes, I got the answer. Thank you all for your help =)
 

FAQ: Operators Commutation: Explaining P(x), L(y)

What is the meaning of "commutation" in Operators Commutation?

In mathematics, commutation refers to the ability of two operators to switch places without changing the result. In the context of P(x) and L(y), this means that the order in which they are applied does not affect the outcome.

How do P(x) and L(y) relate to each other in Operators Commutation?

P(x) and L(y) are two different operators that represent position and angular momentum, respectively. In quantum mechanics, they are known to not commute with each other, meaning their order matters when applied to a system.

What is the significance of P(x) and L(y) in quantum mechanics?

P(x) and L(y) are two of the fundamental operators in quantum mechanics that help describe the behavior of particles on a microscopic scale. P(x) represents the position of a particle, while L(y) represents its angular momentum.

How do P(x) and L(y) affect the uncertainty principle?

The uncertainty principle states that the more precisely one property of a particle is measured, the less precisely the other can be known. P(x) and L(y) are related to this principle because they do not commute, meaning they cannot both be measured simultaneously with perfect accuracy.

Are there any other operators that do not commute with P(x) and L(y)?

Yes, there are many other operators in quantum mechanics that do not commute with P(x) and L(y). For example, the momentum operator P(y) does not commute with P(x) and L(x) does not commute with L(y).

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