Voltage & Its Effect on Material Energy Levels

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Applying a voltage to a material shifts the energy levels of that portion, but it also creates an electric field within the material. This electric field influences the energy levels in a more complex way, potentially requiring a new Hamiltonian to describe the system. The interaction between the voltage and the material's properties leads to changes in energy states beyond a simple shift. Understanding this relationship is crucial for analyzing electronic properties in materials. The discussion highlights the need for clarity on how electric fields and Hamiltonians affect energy levels in materials.
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If we apply a voltage V to a portion of material, does this voltage only shift the energy levels of that portion, as much as V, or it makes an electric field inside that portion which changes the levels in a different manner according to a new Hamiltonian?
 
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if we do apply a voltage across a material it sets up an electric field inside it but i don't know what do you mean by Hamiltonian.sorry for incomplete answer.
 

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