HiLets say I have a Hamiltonian which is invariant in e.g. the

In summary, the conversation discusses the relationship between symmetries in the Hamiltonian and conserved quantities, specifically in terms of spin. Noether's theorem states that for a group of continuous symmetries in the Lagrangian, there is a corresponding conserved quantity. In terms of spin indices, a spin-1/2 particle's spin squared and spin z are conserved if they commute with the Hamiltonian. However, the Hamiltonian of a quantum field is required to be a Poincare' scalar and therefore carries no spinor indices. In general, spin is not conserved but the helicity of a massive particle or the polarization of a massless particle is.
  • #1
Niles
1,866
0
Hi

Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this?


Niles.
 
Physics news on Phys.org
  • #2


Noether's theorem says that for a group of continuous symmetries in the Lagrangian, there is a related conserved quantity. I'm not sure if there's an analog for this for the Hamiltonian...
 
  • #3


Yes consider an operator that "measures" the spin of the particle. The Sz operator for example. If this operator commutes with the Hamiltonian then it is a conserved quantity. Specifically in terms of the spin indices, for a spin-1/2 particle, then S^2 is conserved as s=1/2 and S_z is conserved as m=1/2 or m = -1/2.
 
  • #4


Whoops...I completely forgot about that lol. Sorry.
 
  • #5


The Hamiltonian of a quantum field is 'ab initio' required to be a Poincare' scalar, in particular a Lorentz scalar. Thus it carries no spinor indices whatsoever.

In general, the spin is not conserved, but the helicity of the massive particle or the polarization of the masseless particle is.
 

Related to HiLets say I have a Hamiltonian which is invariant in e.g. the

What is a Hamiltonian?

A Hamiltonian is a mathematical expression that describes the total energy of a system in classical mechanics or quantum mechanics. It is named after the physicist William Rowan Hamilton.

What does it mean for a Hamiltonian to be "invariant"?

When a Hamiltonian is invariant, it means that it remains unchanged under certain transformations. In the case of quantum mechanics, it means that the Hamiltonian remains unchanged under unitary transformations.

What is "e.g." in the context of a Hamiltonian?

"E.g." stands for "exempli gratia" in Latin, which translates to "for example". In this context, it means that the Hamiltonian is invariant in at least one particular way, but there may be other ways in which it remains invariant as well.

What is the significance of invariance in a Hamiltonian?

Invariance in a Hamiltonian is significant because it indicates that the system described by the Hamiltonian has certain symmetries, which can help in understanding and solving the system's dynamics.

How is a Hamiltonian used in physics?

A Hamiltonian is used in physics to describe the total energy of a physical system and its evolution over time. It is a fundamental concept in classical mechanics and quantum mechanics, and is used in a variety of fields including astrophysics, particle physics, and condensed matter physics.

Similar threads

A Secular Approximation of Dipole-Dipole Hamiltonian
    • Quantum Physics
Replies
2
Views
2K
I Hamiltonian of a particle in a magnetic field
    • Quantum Physics
Replies
10
Views
2K
I How is Graphene's Hamiltonian rotationally invariant?
    • Quantum Physics
Replies
2
Views
2K
I Invariance of a system under symmetry operations
    • Quantum Physics
Replies
14
Views
953
I Does an electron have a quantum phase frequency?
    • Quantum Physics
2
Replies
36
Views
3K
I Interpreting Unitary Time Evolution
    • Quantum Physics
Replies
10
Views
808
A Time-dependent unitary transformations of the Hamiltonian
    • Quantum Physics
Replies
9
Views
4K
B Is a Spin 2 Particle the Key to Understanding Gravitation in Quantum Mechanics?
    • Quantum Physics
Replies
26
Views
2K
I Spin conservation in the Dirac equation
    • Quantum Physics
Replies
5
Views
2K
A Effect of time reversed hamiltonian acting on a state?
    • Quantum Physics
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top