#### al onestone

**Registered Senior Member**

Quite the contrary - gravity between the particles is pulling them together in a direction perpendicular to their propagation, so that's the direction in which they'd have to slow down. You're right that the average velocity in directions other than the beam propagation direction is zero:

$$\langle p \rangle=0$$

But kinetic energy doesn't go as velocity. It goes as velocity squared, and the uncertainty principle guarantees that this will be nonzero in all directions:

$$\langle p^2 \rangle>0$$

Total energy, or the expectation value of the Hamiltonian, is the sum of this kinetic energy and the gravitational potential energy:

$$\langle H \rangle=\frac{1}{m}\langle p^2 \rangle+\langle V_g \rangle$$

To see whether energy is conserved, you have to compare $$\langle H \rangle$$ between your initial state and your final state. The final state has a larger potential term, but the velocity spread is narrower, so the kinetic energy term is smaller.

OK, I see what you mean, the kinetic energy on average is always non-zero, but the average velocity is zero. There is no other explanation unless you are willing to assign the velocity and momentum a non-zero average value. You can say "the particles have no average velocity but they travel towards the fringes on average". Since this is your explanation then I would ask you to break the mathematics down to something which evaluates areas that are smaller than the uncertainty so you can see each part of the distribution changing by itself. But this is not possible in QM.

Although your explanation in terms of the hamiltonian does suffice as an explanation. Your

*opinion*seems to be that the transverse directional motion of the particles is averaged to zero, but still each particle moves, and that the combined movement creates the fringing that is observed. And then it is natural to assume that the "slowing" of the particles is the energy loss that makes up for the energy gain in average gravitational potential. I would say that this is an eloborate explanation that is worth testing, because most physicists do not have the interpretation that would allow for "movement of the particles" prior to measurement.