How can I confirm Lz=ħ,0,-ħ using operators and eigenfunctions?

In summary, the conversation discusses the use of the eigen value equation for the z-component of the angular momentum operator, Lz, which is represented by the function LzYilm(θ,ϕ) =mħYilm(θ,ϕ). The possible eigen values for Lz are -h bar, 0, and hbar, corresponding to the possible values of m when L=1. The steps to confirm this are to apply the operator on spherical harmonics with l=1 and compare the resulting values to the eigen value equation. The conversation also provides resources for further understanding of the topic.
  • #1
kenyanchemist
24
2
Hi,
I have a question
Given the function
LzYilm(θ,ϕ) =mħYilm(θ,ϕ)
What steps can I take to confirm that
Lz=ħ,0,-ħ
 
Physics news on Phys.org
  • #2
kenyanchemist said:
I have a question
Given the function
LzYilm(θ,ϕ) =mħYilm(θ,ϕ)
What steps can I take to confirm that
Lz=ħ,0,-ħ

you are writing an eigen value equation for the z component of angular momentum operator called Lz
Ylm are spherical harmonics which are eigen functions of L^2 and Lz
if L=1 then m can take values +1, 0, -1 so the possible eigen values will be -h bar, 0, hbar

now you are asking what steps to confirm - then you can write the form of Lz and apply on spherical harmonics with l=1 and see what possible values comes out from eigen value equation.
 
  • Like
Likes mbond and kenyanchemist
  • #3
Umm... how do I go about that?! Please understand am like super new on quantum mechanics
Please show me :cry::cry:
 

FAQ: How can I confirm Lz=ħ,0,-ħ using operators and eigenfunctions?

What is Lz and why is it important in science?

Lz, also known as the angular momentum operator, is a fundamental concept in quantum mechanics that describes the rotational motion of a particle. It is important in science because it helps us understand the behavior of particles at a microscopic level and is a key factor in determining the properties of atoms and molecules.

How is Lz calculated and what are its units?

Lz is calculated by taking the cross product of the position vector and the momentum vector of a particle. Its units are expressed in terms of angular momentum, which is typically measured in joule-seconds (J·s) or h-bar (ħ).

What are the possible values of Lz and what do they represent?

The possible values of Lz are quantized and can only take on discrete values. These values represent the amount of angular momentum a particle possesses in a specific direction. In other words, they represent the amount of rotation a particle has around a given axis.

How does Lz relate to the shape of an atomic orbital?

The value of Lz determines the shape of an atomic orbital. For example, an orbital with Lz = 0 has a spherical shape, while an orbital with Lz = 1 has a dumbbell shape. This is because the value of Lz corresponds to the number of angular nodes in the orbital.

Can Lz be altered or manipulated in any way?

Yes, Lz can be altered or manipulated by applying external forces or interactions to a particle. For example, an external magnetic field can change the value of Lz and cause a particle to align its spin in a certain direction. Additionally, Lz can be manipulated through quantum mechanical operations such as rotation and reflection.

Similar threads

Back
Top