Effective mass position dependent

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Discussion Overview

The discussion revolves around the concept of effective mass in heterostructures, particularly focusing on how to incorporate position-dependent effective mass into the Hamiltonian. Participants explore the implications of this dependency on the Hermiticity of the kinetic energy operator.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about incorporating position-dependent effective mass into the Hamiltonian and proposes a kinetic energy operator form.
  • Another participant suggests that the proposed form may not maintain Hermiticity.
  • Further discussion includes attempts to prove the Hermiticity of the kinetic energy operator through integration by parts.
  • Questions arise regarding the conditions under which an operator is considered Hermitian.
  • Participants discuss the implications of integration by parts on the Hermiticity condition, with one participant seeking clarification on the steps involved.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the Hermiticity of the proposed operator and continue to explore different aspects of the problem, indicating ongoing uncertainty and debate.

Contextual Notes

Limitations include potential misunderstandings about the conditions for Hermiticity and the specifics of integration by parts in the context of position-dependent effective mass.

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I have some trouble understanding what to do when the effective mass in a heterostructure depends on position. Suppose for example that you have two materials, which are put together such that in one region the kinetic energy is described by one effective mass and in the other region another. How do you incorporate this into the Hamiltonian?
My first guess was to simply say that the kinetic energy is:
-ħ2/2m*(x) ∂2/∂x2
But thinking about it further I guess this does not work, since H is now no longer Hermitian necessarily. I looked up solutions for this and apparently the kinetic energy operator that does the job is:
-ħ2∂/∂x[1/m*(x)∂/∂x]
Now I am not sure how to see that this is indeed hermitian? Can anyone help me realize that?
 
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Partial integration
 
Hmm doing so I can move the outer derivative to the other side. But that does not prove it's hermitian? For that I want to prove that:
∫f ∂/∂x[1/m*(x)∂/∂x]g = ∫∂/∂x[1/m*(x)∂/∂x]f g
 
When exactly is an operator hermitian?
 
Hmm I think I wrote wrong. Hermiticity of an operator means that:
∫f*Ag = ∫(Af)*g
But I still don't see how integration by parts can give me this. It can move the outer derivative over to f but not the second?
 
That's a beginner's exercise. Show your precise steps
 

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