- #1
dRic2
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Hi, I'm a bit confused about Bloch functions. This is what, I think, I understood: can someone please tell me what's wrong?
From Bloch's theorem we know that the wave-function of an electron inside a periodical lattice can be written as ##ψ_k(r)=u_k(r)e^{ik⋅r}##. We hope that far from a lattice point, i.e. far from the nucleus of an atom the ##u_k(r)## term will be more or less constant, in order to get the plane wave ##e^{ik⋅r}##.
I think this is reasonable also in view of the effective mass theorem. In fact far from a lattice point the physics should be similar to the free particle: the potential is more or less constant and the energy band takes the form of a paraboloid ##E(k)≈E(k_0)−\frac{ℏ^2k^2}{2m^*}## where ##m^*## is the effective mass ##m^*=\frac 1 {\frac 1 {ℏ^2} \left( \frac{∂^2E}{∂k^2} \right)}##. Thus the problem can be shown to be equivalent to solve the Schrodinger equation for the Hamiltonian ##\hat H=−\frac {ℏ^2} {2m^*}Δ## and the solutions are plane waves.
Now, assuming what I wrote is correct (very big assumption
) here it comes the critical part. The Bloch functions ##ψ_k(r)=u_k(r)e^{ik⋅r}## are not eigenfunctions of the momentum operator ##\hat p##. This is clear. But now my professor said that , to avoid this problem, we note that the average velocity of the electron is given by the relation:
2nd question: if I take into account the Born-von Karman boundary conditions is my reasoning still correct (assuming it was correct in the first place)?
3rd question: far from a lattice point, where Bloch functions take the form of plane waves, they are eigenfunction of the momentum operator, right ? So the momentum (and velocity) is well defined and the above equation reduces to ##p=ℏk##, right?
thanks in advance
Ric.
From Bloch's theorem we know that the wave-function of an electron inside a periodical lattice can be written as ##ψ_k(r)=u_k(r)e^{ik⋅r}##. We hope that far from a lattice point, i.e. far from the nucleus of an atom the ##u_k(r)## term will be more or less constant, in order to get the plane wave ##e^{ik⋅r}##.
I think this is reasonable also in view of the effective mass theorem. In fact far from a lattice point the physics should be similar to the free particle: the potential is more or less constant and the energy band takes the form of a paraboloid ##E(k)≈E(k_0)−\frac{ℏ^2k^2}{2m^*}## where ##m^*## is the effective mass ##m^*=\frac 1 {\frac 1 {ℏ^2} \left( \frac{∂^2E}{∂k^2} \right)}##. Thus the problem can be shown to be equivalent to solve the Schrodinger equation for the Hamiltonian ##\hat H=−\frac {ℏ^2} {2m^*}Δ## and the solutions are plane waves.
Now, assuming what I wrote is correct (very big assumption
) here it comes the critical part. The Bloch functions ##ψ_k(r)=u_k(r)e^{ik⋅r}## are not eigenfunctions of the momentum operator ##\hat p##. This is clear. But now my professor said that , to avoid this problem, we note that the average velocity of the electron is given by the relation:$$<v>=\frac 1 ℏ \frac {∂E(k)}{∂k} =\frac {<p>}m$$
1st question: where does this relation come from ? Is it from the semi-classical treatment, substituting ##p=kℏ##?2nd question: if I take into account the Born-von Karman boundary conditions is my reasoning still correct (assuming it was correct in the first place)?
3rd question: far from a lattice point, where Bloch functions take the form of plane waves, they are eigenfunction of the momentum operator, right ? So the momentum (and velocity) is well defined and the above equation reduces to ##p=ℏk##, right?
thanks in advance
Ric.
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