Showing functions are eigenfunctions of angular momentum.

In summary, the conversation discusses verifying that three functions, cos(θ), sin(θ)eiφ, and sin(θ)e−iφ, are all eigenfunctions of the operators L2 and Lz. The concept of eigenfunctions and eigenvalues is explained, and the question is resolved by realizing that a function with no dependence on the angle φ will have a zero eigenvalue for Lz, as it will have no component of angular momentum along the z axis. The importance of understanding the physical interpretation of mathematical concepts is emphasized.
  • #1
Robsta
88
0

Homework Statement


Verify by brute force that the three functions cos(θ), sin(θ)e and sin(θ)e−iφ are all eigenfunctions of L2 and Lz.

Homework Equations



I know that Lz = -iћ(∂/∂φ)
I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged apart from a multiplicative factor (the eigenvalue)

The Attempt at a Solution



So, Lzcos(θ) = -iћ(∂/∂φ)cos(θ)
But I think that equals zero. There's no component of cos(θ) in the φ direction. There different variables. So I think that when the differential operator acts on it, it makes 0. But then the function isn't left unchanged. Can somebody help me resolve this in my head?
 
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  • #2
You have ##L_z\psi = 0\psi## so ##\psi## is an eigenfunction of ##L_z## with eigenvalue ...
 
  • #3
Eigenvalue 0. Does that mean that any function orthogonal to the phi direction is an eigenfunction then? Because its differential Thanks for your help!
 
  • #4
What does the eigenvalue of 0 mean - physically? What does ##L_z## measure?
 
  • #5
Lz measures the angular momentum about the vertical (z) axis. The eigenvalue is the actual amount of ang. momentum. I've managed to crack this question now, thanks a lot for your help :)
 
  • #6
Put more precisely: since phi is the angle the total angular momentum makes with the z axis, then any wavefunction with no phi dependence will also have no component of angular momentum along the z axis. Hence, zero eigenvalue.

Notice the emphasis on physics (well, geometry) rather than mathematics (calculus) - yes it comes out zero because of the maths, but the Universe does not care about what our calculations yield. The maths is just a model - it is trying to describe Nature. Look for the truth in Nature.

Similarly ...
The "z axis" does not have to be vertical - you are not dealing with gravity here - it is determined by the orientation of the apparatus doing the measuring. It can as easily (and more usually) be along the direction of motion or the direction of an applied field. It's just a common label for an "axis of interest". You should abandon ideas about the orientation of the Cartesian axes.

The phi direction is not mentioned in the problem ... it is not actually a direction since knowing phi does not tell you where to look, but describes a set of infinitely many directions from the z axis. It is important to distinguish between a direction (component of a vector) and the dependence that the magnitude of the component has.

i.e. ##\vec B = (0, kx^2, 0)## has a y direction that depends on the x component of position.

 

Related to Showing functions are eigenfunctions of angular momentum.

1. What is angular momentum?

Angular momentum is a physical quantity that describes the rotational motion of an object around an axis. It is a vector quantity, meaning it has both magnitude and direction, and is conserved in a closed system.

2. What are eigenfunctions?

Eigenfunctions are special functions that, when acted upon by a linear operator, produce a scalar multiple of themselves. In the context of angular momentum, eigenfunctions are functions that describe the rotational motion of an object around an axis and have specific properties that make them useful in quantum mechanics.

3. How are eigenfunctions related to angular momentum?

In quantum mechanics, eigenfunctions of angular momentum are solutions to the Schrödinger equation that describe the rotational motion of particles. These eigenfunctions have specific values for the angular momentum operator and can be used to determine the energy levels and other properties of a system.

4. How do you show that a function is an eigenfunction of angular momentum?

To show that a function is an eigenfunction of angular momentum, you must demonstrate that it satisfies the Schrödinger equation and has specific values for the angular momentum operator. This can often be achieved through mathematical calculations and by comparing the function to known eigenfunctions of angular momentum.

5. Why is it important to show that a function is an eigenfunction of angular momentum?

Determining the eigenfunctions of angular momentum is important in quantum mechanics because it allows us to understand the properties and behavior of particles in rotational motion. By showing that a function is an eigenfunction of angular momentum, we can better understand the energy levels and other characteristics of a system, which has practical applications in fields such as materials science and quantum computing.

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