Wave functions for 2D potential with spin interactions

In summary, the addition of a spin-dependent term to the circular potential in a 2D system will not change the form of the wave functions, but will only alter the energy levels of the system based on the spin direction. In 3D, the full spin-orbit interaction can lead to more complex behavior due to the lack of commutation.
  • #1
BeauGeste
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So consider a 2D system with a circular potential and a spin-orbit interaction:

[tex] V(r) = V_0 \theta(r_0 - r) + c r_0 V_0 L_z S_z \delta(r-r_0) [/tex]

where θ is step function.

So the operators Lz and Sz commute with the Hamiltonian are are therefore conserved quantities. For the same reasons we can write the wave function as a product of radial and orbital parts (and spin parts too):

[tex]R(r) e^{i l \theta}[/tex]

where θ here is the polar angle and l is the orbital quantum number. A spinor can be affixed to the wave function but seems unnecessary as no transitions can occur for the spin.

My question regards adding another spin interaction to V(r) of the type [tex]b_z S_z\theta(r_0-r)[/tex] that only acts within the circular potential. Will the form of the wave functions change as a result of this addition?

My thought is that the wave functions remain the same since once again spin should be preserved so the spinors will not see any further structure. The only purpose of this new interaction will be to alter the effective potential of the circular well - the potential will be shifted either up or down depending on the direction of the spin (ms = up or down).

So is my reasoning correct? I understand that this problem becomes much more difficult in 3D when the full spin-orbit interaction is used since then you will have a lack of commutation.
 
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  • #2
Yes, your reasoning is correct. The addition of the spin-dependent term in the potential does not affect the form of the wave functions, since the spin part of the wave function remains unchanged. It merely shifts the energy levels of the system depending on the direction of the spin. In 3D, the spin-orbit interaction will cause a lack of commutation, which can lead to more complex behavior in the wave functions.
 

1. What is a wave function?

A wave function is a mathematical representation of a quantum system that describes the probability of finding a particle at a certain position in space and time. It is a complex-valued function that contains all the information about the system's physical properties.

2. What is a 2D potential?

A 2D potential is a mathematical function that describes the energy of a particle in a two-dimensional space. It is used to model the behavior of particles in various physical systems, such as atoms, molecules, and crystals.

3. What are spin interactions?

Spin interactions refer to the interaction between particles due to their intrinsic angular momentum, known as spin. This interaction is responsible for many phenomena in quantum mechanics, such as the behavior of electrons in magnetic fields.

4. How are wave functions for 2D potential with spin interactions calculated?

The wave functions for 2D potential with spin interactions are calculated using mathematical techniques such as the Schrödinger equation and the spin Hamiltonian. These equations take into account the potential energy of the system, the spin interactions, and other factors to determine the wave function.

5. What is the significance of studying wave functions for 2D potential with spin interactions?

Studying wave functions for 2D potential with spin interactions allows us to understand the behavior of particles in various physical systems. It also has practical applications, such as in the development of new materials and technologies, and in the field of quantum computing.

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