Angular momentum and quantum spin

[touqra]

Is there an invariance relation for angular momentum similar to $$E^2 = p^2 + m^2$$ ?
How about quantum spin ?

 

[pam]

Nothing simple. rXp is a combination of three components of two four vectors.
It can be transformed but the result is complicated.
Quantum spin can be transformed using the Dirac equation, but it too is complicated except for special cases.

 

[Entropia]

Angular momentum and quantum spin are both fundamental concepts in quantum mechanics that describe the rotational properties of particles and systems. While they are related, they are not exactly the same thing.

Angular momentum is a vector quantity that measures the amount of rotational motion of a particle or system. It is defined as the cross product of the position vector and the momentum vector. In classical mechanics, angular momentum is conserved, meaning that it remains constant unless acted upon by an external torque. However, in quantum mechanics, angular momentum is quantized, meaning it can only take on discrete values.

On the other hand, quantum spin is an intrinsic property of particles that describes their angular momentum even when they are not physically rotating. It is a form of intrinsic angular momentum that is not related to the motion of the particle. Spin is also quantized and can only take on discrete values.

In terms of an invariance relation for angular momentum, there is no exact equivalent to the relation E^2 = p^2 + m^2. However, there are conservation laws for total angular momentum and for the individual components of angular momentum (such as spin). These conservation laws state that the total angular momentum of a system remains constant unless acted upon by an external torque.

Similarly, there is no invariance relation for quantum spin, but there are conservation laws for the total spin and the individual spin components. These conservation laws state that the total spin of a system remains constant unless acted upon by an external force.

In summary, while there are no invariance relations for angular momentum and quantum spin similar to E^2 = p^2 + m^2, there are conservation laws that govern their behavior and ensure their conservation in isolated systems.