I Effective mass position dependent

[aaaa202]

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?

 

[DrDu]

Partial integration

 

[aaaa202]

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

 

[DrDu]

When exactly is an operator hermitian?

 

[aaaa202]

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?

 

[DrDu]

That's a beginner's excercise. Show your precise steps