Gradient of Pressure Present in a Quantum Bound State Particle Even Without a Potential?

In a previous note, it was suggested one can use definitions of temperature and heat flow from the Boltzmann transport equation and calculate similar quantities for a bound state quantum wavefunction. It was also suggested that one could not use the “transport equations” such as the Euler equation for the quantum bound state system. In another note, however, it was shown that the quantum bound state contained important terms of the Euler equation, namely  -d/dx ln(d(x)) and F(x)d(x) where d(x) is the density and F(x) the force. In this note, we consider the definition of pressure from the Boltzmann transport equation and calculate its gradient. We find the usual F(x)d(x) term in addition to an extra term. In setting F(x)=0, we find the extra term coincides with the gradient of pressure for a quantum particle in a box with no potential. Thus, it appears that a quantum particle, even without a potential, represents a pressure gradient which affects the form of the Schrodinger equation and causes quantum particles to differ from those of classical and classical statistical mechanics.