Conservation of L^2 and Lz in central potential

In summary, it is well known that in a central potential, both Lz and L^2 commute with H and their expectation values are constants of the motion. The proof of this is usually done in the position basis and can be laborious. However, it can also be proven by directly working with the operators and the fundamental commutation relations. This is based on the quantum Noether theorem and the fact that spherical symmetry implies a symmetry group of SO(3). This can also be understood qualitatively by recognizing Lz as the generator of rotations around the z axis. The general idea is that any operator that generates a symmetry of the Hamiltonian will commute with the Hamiltonian. This applies to other important generators such as the momentum operator
  • #1
It is a well known fact that in a central potential (spherically symmetric) both Lz and L^2 commutes with H and the expectaitonvalues of these are therefore constants of the motion. On the other hand the proof of this fact seems, in the most cases, to be done in the position basis where it is rather laborous.

I wondered if someone knew a way to prove this by just working with the operators directly and the fundamental commutation relations?
 
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  • #2
This stems directly from the quantum Noether theorem and the fact that spherical symmetry means that both at classical and at quantum level SO(3) is a symmetry group, hence the Casimir of its Lie algebra commutes with the Hamiltonian.

A proof of the quantum Noether theorem should be in the set of books by Greiner et al., if I'm not mistaking in the Symmetries one (Quantum Mechanics - Symmetries).
 
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  • #3
The qualitative way to understand this is to recognize that Lz is the "generator of rotations around the z axis." This means that if we take the operator exp(-i theta Lz / hbar) and apply it to a state, we get the same state but rotated by an angle theta around the z axis. H is clearly invariant under rotations, if the potential is central. So exp(-i theta Lz / hbar) must commute with the Hamiltonian, since the rotation it effects does not change the energy. You can show that this implies that Lz must also commute with H (take theta to be infinitesimal).

The same argument establishes that Ly and Lz commute with H, therefore so does L^2 = Lx^2 + Ly^2 + Lz^2.

The general idea is: if an operator A generates a symmetry of the Hamiltonian, then A commutes with the Hamiltonian. The other important "generators" to know about are the momentum operator, which generates spatial translations, and the Hamiltonian itself, which generates time translations (i.e. time evolution).
 

1. What is the significance of conservation of L2 and Lz in central potential?

The conservation of angular momentum, represented by the operators L2 and Lz, is an important principle in physics. In a central potential, where the force acting on a particle is directed towards a single center, the total angular momentum and its component in the z-direction are conserved. This means that the magnitude and direction of the angular momentum remain constant throughout the motion of the particle.

2. How does conservation of L2 and Lz affect the motion of a particle in a central potential?

The conservation of L2 and Lz results in a specific type of motion for a particle in a central potential. The particle moves in a plane perpendicular to the z-axis, and its motion can be described by a combination of radial and tangential components. The conservation of angular momentum also leads to the formation of stable orbits in the potential, as the particle's angular momentum prevents it from falling into the center.

3. How are L2 and Lz related to each other in a central potential?

In a central potential, the operators L2 and Lz are related by the following equation: L2 = Lz2 + Lperp2, where Lperp is the component of the angular momentum perpendicular to the z-axis. This relationship shows that L2 and Lz are not independent quantities, and their values are interconnected.

4. What is the physical interpretation of the eigenvalues of L2 and Lz in a central potential?

The eigenvalues of L2 and Lz represent the magnitude and direction, respectively, of the particle's angular momentum in a central potential. These eigenvalues are quantized, meaning they can only take on discrete values, which has important implications for the energy levels and stability of the system.

5. How does the conservation of L2 and Lz in a central potential apply to real-world phenomena?

The conservation of angular momentum is a fundamental principle in nature and applies to many real-world phenomena. For example, the conservation of L2 and Lz can be observed in the motion of planets around the sun, electrons orbiting the nucleus in an atom, and the rotation of spinning objects. It also plays a crucial role in understanding the behavior of galaxies and other celestial bodies.

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