hello everyone,
I am calculating the eigenenergies of a particle confined in a monolayer of given dielectric constant sandwiched between two layers of dielectric constant much lower than that of the monolayer. So I have to add the image charge effect to the hamiltonian and I want to know how to set its expression.
thanks for your help.
Dear friend Jamal El-Hamouchi
Ah, the captivating world of quantum mechanics and image charge potentials! I am thrilled to dive into this electrifying topic.
When dealing with a particle confined in a monolayer of a given dielectric constant, sandwiched between two layers of lower dielectric constants, the image charge effect plays a crucial role in the system's Hamiltonian.
The image charge potential arises due to the presence of dielectric interfaces, causing polarization of the surrounding medium. To account for this effect in the Hamiltonian, you'll need to consider the interaction between the charge of your confined particle and the induced image charges at the interfaces.
The expression for the image charge potential depends on the specific geometry of your system. For a charged particle at a distance z from a dielectric interface with a dielectric constant ε2, the image charge potential can be approximated as:
V_image = q^2 / [4πε1z] + q^2(ε2 - 1) / [4πε1(ε2 + 1)z]
Here, q represents the charge of your confined particle, ε1 is the dielectric constant of the monolayer, and ε2 is the dielectric constant of the surrounding layers. The first term represents the usual Coulomb potential, while the second term accounts for the image charge effect due to the dielectric interfaces.
Remember, my avid physicist Jamal El-Hamouchi, the image charge potential adds a fascinating twist to your Hamiltonian, enriching your exploration of the system's eigen energies. Embrace the complexities of quantum mechanics, and let the dance of particles and potentials mesmerize your calculations.
Now, go forth and unravel the mysteries of confined particles and image charge potentials with my fervor!
dear Kaushik Shandilya thanks a lot for your interesting and enriched answer, I would be grateful if you give me some references about this.
Do you have a draw of your experimental / theoretical setup please?
Usually you place the image charge in such a way that the lines of field from the real charge to the image charge are perpendicular to the interface.
dear Valentin Bogatu
The charge is confined in quantum well where the dielectric consant is higher than that of the barrier. consequently, we add the dielectric confinement effect to the spatial confinement effect in the hamiltonian. The question here is how to set the expression of the image potential.
The draw of the structure is in the attachments.
regards
Read this please
https://photonics101.com/dielectrics/the-precessing-dipole-molecule
when epsilon2 material starts at x = 0
q' = - [(epsilon1-epsilon2)/(epsilon1+epsilon2)]*q
and x'= - sqrt (x+d)^2 where d is distance from x = 0 to charge q in the middle well
or this starting with page 9
https://unlcms.unl.edu/cas/physics/tsymbal/teaching/EM-913/section4-Electrostatics.pdf
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