Momentum and Space Entropies in the Bound State Schrodinger Equation

In general, wavefunction W(x) solutions of bound state quantum problems are obtained by solving a time-independent spatial differential Schrodinger equation. Traditionally, entropy or thermodynamic ideas do not enter as there is no temperature in the problem. Nevertheless, with Shannon’s definition of entropy, even a T=0 bound state solution has probabilities P(p)=fp*fp, P(x)=W(x)W(x) and P(p and x)= P(p/x)P(x)= fp sin(px) W(x). As a result, one may calculate at least three Shannon’s entropies for T=0 bound state solutions, namely Sp, Sx and S(x and p). Sp and Sx are not independent because W(x)= Sum over p fp sin(px). Finding entropies may appear as simply a calculational exercise at this point. For the ground state of the quantum oscillator, however, E= Sum over p Sp + Integral dx Sx, so these entropies take the place of kinetic and potential energy terms. Furthermore, for the same problem, exp(-bp*p) is of the form of an Sp maximized entropy, in the same manner as in classical statistical mechanics. Such an entropy maximization does not seem to hold for other cases. It is suggested, however, in this note that one might be able to generalize the ground state oscillator state result: E= Sum over p Sp + Integral dx Sx to thermodynamic relations. Consider changing the oscillator spring constant infinitesimally. Then E, fp, W(x), Sp and Sx will all change in a tiny way. In such a case, one might consider: dE = Constant d {Sum over p Sp + Integral dx Sx}. One might argue that changing the spring constant represents changing an external parameter which should be related to dW, a change in work. For the ground state harmonic oscillator, however, d(V d(x))= (dV)d(x) + V(d/dx d(x)) with V(x)=potential and d(x)=W(x)W(x), leads to the two terms (dV)d(x) and V(d/dx d(x) being proportional, so it is as if the change in work disappears from the problem. In this note, we suggest that for other cases, one may perhaps define an overall entropy S=Sum over p Sp + Integral dx Sx. Then, one could use the thermodynamic result: dE - Constant dS = -dW to calculate dW, an infinitesimal amount of work done. Such a recipe could perhaps be applied to closely spaced eigenstates for which one knows E(n+1) and E(n). The spatial entropies Sx are readily obtained from the wavefunctions Wn+1(x) and Wn(x) and the momentum entropies from the Fourier transforms inserted in Shannon’s entropy equation. If such a procedure has meaning, it would show thermodynamics exists in quantum mechanics, even for T=0. It would also perhaps shed light on some T>0 Schrodinger type equations that are currently being presented in the literature as Sum over p Sp + integral dx Sx is already being used as entropy