# Non central potential and energy dependence

• naruto365
In summary, if the Hamiltonian is non-central, the total energy is not solely dependent on principle quantum number n but also on l. This is because for non-spherically symmetric potentials, l is not a good quantum number. This means that if the Hamiltonian does not commute with rotations, a common eigensystem of the Hamiltonian and the L^2 operator cannot be found. In other words, the energy is not independent of l for spherical potentials that are not 1/r, such as the hydrogen atom. This is due to the fact that the 1/r potential is a special case, and the degeneracy in the hydrogen atom is a result of a hidden conserved symmetry known as the Runge-Lenz-
naruto365
Could anyone tell me where can i find more about the fact that if your Hamiltonian is non-central then the total energy is not dependent only on principle quantum number n but also on l.

If your Hamiltonian is not spherically symmetric, then l is not a good quantum number in the first place: If H does not commute with rotations, you cannot find a common eigensystem of H and the L^2 operator[1].

I guess what you mean is "why is the energy dependent on n and l for for spherical potentials which are not 1/r" (i.e., not hydrogen). If this is the case: Because there generally is no reason for all energy to be independent of l. Rather, this degeneracy in the hydrogen atom is a special case for 1/r potentials, ultimately related to a hidden additional conserved symmetry (the so called Runge-Lenz-Vector already known from classical mechanics). So it is the 1/r potential which is the special case, not the other ones.[1] of course you could still prepare a wave function in a specific l state with respect to some point, but such states would not be eigenstates.

## 1. What is a non-central potential?

A non-central potential is a type of force that depends on the relative positions of two objects, but not on the distance between them. This means that the force acting on an object can vary in different directions, unlike a central potential which only depends on the distance between objects.

## 2. How does energy dependence affect non-central potentials?

The energy dependence of a non-central potential refers to how the strength of the force changes with the energy of the system. In some cases, the force may become stronger or weaker as the energy increases, which can affect the dynamics of the system.

## 3. What are some examples of non-central potentials?

Some common examples of non-central potentials include the Coulomb potential, which describes the electrostatic force between charged particles, and the Van der Waals potential, which describes the attractive force between molecules. The gravitational force between two bodies can also be considered a non-central potential.

## 4. How are non-central potentials used in physics?

Non-central potentials are used in physics to study the interactions between objects in a system. They can be used to describe the forces between particles, the motion of celestial bodies, and the behavior of atoms and molecules. Non-central potentials are also important in many areas of quantum mechanics and atomic physics.

## 5. What are some properties of non-central potentials?

Some properties of non-central potentials include rotational symmetry, where the potential is the same in all directions, and non-spherical symmetry, where the potential varies in different directions. Non-central potentials can also have both attractive and repulsive components, depending on the distance and relative positions of the objects in the system.

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