Fock-Darwin States in Circular Hard-Wall Potential w/ Magnetic Field

[ocotlet]

Hi,
Suppose I have a 2D spinless electron bounded by a parabolic confining potential. If we add a magnetic field perpendicular to the plane the eigenstates of the system are the so called Fock-Darwin states.
However, suppose we change the boundary potential to a circular hard-wall potential. In the absence of the magnetic field I know how to find the eigenstates, which are some combination of Bessel functions of first kind. However, does anybody a method to determine the new eigenstates in the presence of the magnetic field ?
Thanks

 

[eljose79]

Hello,

Thank you for your question. In the presence of a magnetic field, the eigenstates of a 2D spinless electron in a circular hard-wall potential can still be determined using a similar approach as in the absence of the magnetic field. However, the Hamiltonian for this system will now include the additional term for the magnetic field, which can be written as H = (p - eA)^2/2m + V(r), where p is the momentum operator, e is the electron charge, A is the vector potential for the magnetic field, m is the electron mass, and V(r) is the circular hard-wall potential.

To find the eigenstates of this system, you can use the same method as in the absence of the magnetic field, which involves solving the Schrödinger equation Hψ = Eψ, where ψ is the wavefunction and E is the energy. However, in this case, the wavefunction will have a different form due to the presence of the magnetic field. This can be solved using perturbation theory, where the magnetic field term is treated as a perturbation to the Hamiltonian.

Alternatively, you can also use numerical methods such as the finite element method or the finite difference method to solve for the eigenstates in the presence of the magnetic field. These methods involve discretizing the system and solving the resulting matrix equation to obtain the eigenstates and eigenenergies.

I hope this helps. Please let me know if you have any further questions.