Angular Momentum Operator

In summary, the operators used for the x and y components of angular momentum are Lx = y.Pz - z.Py and Ly = x.Pz - z.Px, and they do not commute, as shown by the commutator [Lx, Ly] = ihLz. This demonstrates an uncertainty relation between Lx and Lz.
  • #1
leviathanX777
42
0
The operators used for the x and y components of angular momentum are:

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7B%5Cpartial%7D%7B%5Cpartial%7Bx%7D%7D%20%20-%20x%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%7Bz%7D%7D).jpg


Show that Lx and Lz obey an uncertainty relation




2. No relevant equations.




The Attempt at a Solution



I'm going on that the assumption that if LxLy - LyLz does not equal zero then they don't commute and have an uncertainty relation. However I can only get this equal to zero and don't know how to show the uncertainty rrelation if I achieve one.
 
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  • #2
leviathanX777 said:
I'm going on that the assumption that if LxLy - LyLz does not equal zero then they don't commute and have an uncertainty relation. However I can only get this equal to zero and don't know how to show the uncertainty rrelation if I achieve one.

if you mean:

[Lx, Ly] = LxLy - LyLx

then it does not equal to zero, angular moment is the cross product: r x p

so Lx = y.Pz - z.Py Ly = x.Pz - z.Px

where x and y and z are position operators and Px, Py and Pz are momentum operators, stick those into your commutator and try again, you should end up with

[Lx, Ly] = ihLz

where h is the reduced Planck constant. and Lz is the Angular momentum operator for z axis
 
  • #3
It would also help if you showed us your calculation of the commutator so we can see where your error is.
 
  • #4
Ah I got it solved in the end. Just made a minor mistake. Thanks!
 
  • #5



The uncertainty relation for Lx and Lz can be shown by using the commutator relationship between these two operators. The commutator of two operators A and B is defined as [A,B] = AB - BA. In this case, we can calculate the commutator of Lx and Lz as [Lx,Lz] = LxLz - LzLx.

Using the definition of the angular momentum operators, we can write Lx and Lz in terms of the position and momentum operators as follows:

Lx = yp - zp
Lz = xp - yp

Substituting these expressions into the commutator, we get:

[Lx,Lz] = (yp - zp)(xp - yp) - (xp - yp)(yp - zp)

Expanding this expression and using the commutator relation for the position and momentum operators, [xp, yp] = iħ, we get:

[Lx,Lz] = iħ(yp - zp) - iħ(xp - yp) = iħ(xp + yp - zp - yp)

Simplifying further, we get:

[Lx,Lz] = iħ(xp - zp)

This result shows that Lx and Lz do not commute, as [Lx,Lz] is not equal to zero. This means that these two operators do not have well-defined simultaneous values, and thus obey an uncertainty relation. This relation can be written as:

ΔLxΔLz ≥ ħ/2

where ΔLx and ΔLz represent the uncertainties in the measurements of Lx and Lz, respectively. This relation is similar to the well-known Heisenberg uncertainty principle for position and momentum.
 

What is the Angular Momentum Operator?

The Angular Momentum Operator is a mathematical operator used in quantum mechanics to describe the angular momentum of a quantum system. It is denoted by the symbol L and is a vector operator that has both a magnitude and a direction.

How is the Angular Momentum Operator defined?

The Angular Momentum Operator is defined as the cross product of the position vector r and the momentum vector p. In mathematical notation, it can be written as L = r x p.

What are the components of the Angular Momentum Operator?

The components of the Angular Momentum Operator are the x, y, and z components, denoted by Lx, Ly, and Lz, respectively. These components represent the angular momentum along the x, y, and z directions.

What is the physical significance of the Angular Momentum Operator?

The Angular Momentum Operator is used to describe the rotational motion of a quantum system. It is related to the total angular momentum of the system, as well as the individual angular momenta of its constituent particles. It also plays a crucial role in determining the energy levels of atoms and molecules.

How is the Angular Momentum Operator measured or observed?

The Angular Momentum Operator cannot be directly measured or observed, as it is a mathematical construct. However, its effects can be observed through experiments and measurements of other physical quantities, such as the energy levels and transitions of atoms and molecules.

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