Puzzle

Puzzle

Thermaldynamic functions

As we know, combining the first and second law of thermaldynamics, one gets

dU=TdS-PdV, (1)

where T,S,P and V are temperature, entropy, pressure and volume.

In addition, we also know that S and V are extended functions, meaning:

U(nS,nV)=nU(S,V),

which leads to Euler’s equation (the deviation can be found on the Internet):

U=TS-PV.

On the other hand, we know there are other three energy related functions, say H (enthalpy), F(free energy) and G(Gibbs free energy). Their relation to internal energy U is the following:

H=U+PV
F=U-TS
G=U-TS+PV.

Therefore, with those functions, one can describe four different process, adiabatic+isometric, adiabatic+isobaric, isothermal+isometric, isothermal+isobaric.

However, if we consider Euler’s equation, we get
H=TS
F=-PV
G=0
what’s wrong here?

Periodic potential problem

When we study solid state physics, we often deal with periodic potential, which may make you think that it is trivial. Normally, we don’t really know the function form of the potential. So approximation is made and perturbation theory is invoked, e.g. in free electron approximation and tight binding approximation.

I was surprised when I tried to solve this Schodinger equation:

d2Ψ/dx2+cos(Kx)Ψ=EΨ, here hbar, m are omitted for simplicity.

We know from Bloch theorem that the wave function takes the form

Ψ=exp(-ikx)u(x),

where u(x)=u(x+2n/K).

But to it takes me only so far. I can not find the exact solution.

Does the analytical solution exist?