Question about angular momentum and spin

• rabbit44
In summary, angular momentum is a vector quantity that measures the rotational energy of an object. Spin is a type of angular momentum intrinsic to subatomic particles. According to the law of conservation of angular momentum, it is conserved in a closed system. Angular momentum is important in various fields of physics and can be calculated using the formula L = Iω.
rabbit44
Hi quick question, is there a reason that the eigenvalues of Jz, Lz and Sz are all denoted m? Are they the same? Because I have seen them interchanged such as in addition of angular momenta.

Thanks

One call them M since they was found experimentally by interaction by EM-fields, "Magnetic Quantum Number" :-)

One should use subscript in order to differ between them:

$$m_j, \, m_s, \, m_\ell$$

for your question! The eigenvalues of Jz, Lz, and Sz are typically denoted as m because they represent the component of angular momentum, spin, and total angular momentum along a specific axis (usually the z-axis). However, they are not necessarily the same, as each of these operators (Jz, Lz, Sz) correspond to different physical quantities (angular momentum, orbital angular momentum, and spin, respectively). Therefore, their eigenvalues may differ depending on the system being studied.

In terms of interchanging them, this is typically done when discussing the addition of angular momenta. In this case, we are considering the total angular momentum of a system, which is the sum of the individual angular momenta of its components. In this context, the eigenvalues of Jz, Lz, and Sz may be interchanged because they all contribute to the overall angular momentum of the system.

I hope this helps clarify the relationship between these operators and their eigenvalues. Let me know if you have any other questions!

What is angular momentum?

Angular momentum is a physical quantity that measures the rotational motion of an object around an axis. It is a vector quantity, which means it has both magnitude and direction. In simpler terms, it is the amount of rotational energy an object possesses.

What is the relationship between angular momentum and spin?

Spin is a type of angular momentum that is intrinsic to subatomic particles, such as electrons and protons. It is not related to the physical spinning of the particle, but rather a property that describes its angular momentum. The relationship between the two is that spin is a component of the overall angular momentum of a particle.

How is angular momentum conserved?

According to the law of conservation of angular momentum, the total angular momentum of a closed system remains constant over time, unless acted upon by an external torque. This means that the angular momentum of a system cannot be created or destroyed, but can only be transferred between different objects within the system.

Why is angular momentum important?

Angular momentum is important in many areas of physics, including classical mechanics, quantum mechanics, and astrophysics. It is a fundamental quantity that helps us understand the behavior of rotating objects, from the spinning of a top to the motion of planets around the sun. It also has practical applications in engineering and technology, such as in the design of vehicles and machinery.

How is angular momentum calculated?

The formula for calculating angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. The moment of inertia is a measure of an object's resistance to rotational motion, while angular velocity is the rate at which the object is rotating. Therefore, the higher the moment of inertia and angular velocity, the greater the angular momentum.

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