The Weak Equivalence Principle, or WEP for short, states that under identical initial conditions, the motion of particles **of different masses** in a given gravitational field is identical. Or in other words, there are no physical effects that depend on the mass of a point particle in an external gravitational field. This is just the equivalence between the inertial and gravitational mass.

Below I'll present some results that are on either sides of the camp (that WEP is or isn't violated). From what I read, the consensus on this matter is that the equivalence principle in the general sense in violated. However, there are special cases in which the principle holds true for quantum mechanics (one such case in given below).

Let us take the Schrodinger equation for a particle of inertial mass $m_i$ and gravitational mass $m_g$, that is falling toward mass $M$.

$$i\hbar\partial_{t}\psi=-\frac{\hbar^2}{2m_{i}}\nabla^{2}\psi - G\frac{m_{g}M}{r}$$

It is obviously that even for $m_{i}=m_{g}$ the mass does not cancel out of the equations of motions. This fact is even more apparent for $m_{i}=m_{g}=m$ in a uniform gravitational field in the $x$ direction, of acceleration $g$

$$i\hbar\partial_{t}\psi=-\frac{\hbar^2}{2m}\partial_{x}^{2}\psi +mgx\psi$$

whose solution will depend parametrically on $\hbar/m$. At this point we could say that the wave functions, the propagators and probability density distributions violate the WEP. Also, Rabinowitz in the 90's examined the possibility of gravitationally bound atoms, and he found that the mass, $m$, remains in the quantized equations of motions (although the mass cancels out in the classical equations of motions). We would expect $m$ to cancel out when averaging over states with large quantum numbers, but that puts them effectively in the classical continuum.

However, P.C. Davies proposes in this article the following experiment:

Consider a variant of the simple Galileo
experiment, where particles of different mass are projected vertically in a uniform gravitational
ﬁeld with a given initial velocity $v$. Classically, it is predicted that the particles will return a
time $2v/g$ later, having risen to a height $x_{max}=v^{2}/
2g$. But quantum particles are able to tunnel
into the classically forbidden region above xmax. Moreover, the tunnelling depth depends on
the mass. One might therefore expect a small, but highly signiﬁcant mass-dependant ‘quantum
delay’ in the return time. Such a delay would represent a violation of the equivalence principle.

At the end of section $3$ he proves that the expectation value for the turn-around time of a quantum particle is identical, when the measurement is performed far from the classical turning point. In this sense, the WEP holds for a quantum particle.

This result suggest that a uniform gravitational potential—which applies locally to
any non-singular gravitational ﬁeld—has a special property in relation to quantum mechanics,
namely that the expectation time for the propagation of a quantum particle in this background
is identical to the classical propagation time. This may be taken as an extension of the
principle of equivalence into the quantum regime (for a broader discussion of what is entailed
by a ‘quantum equivalence principle’). This special property seems
to depend on the form of the potential; it does not apply in the case of a sharp potential
step, or an exponential potential.

Finally I would like to point out this article, where the authors compute low corrections to the cross-section for the scattering of different quantum particles by an external gravitational field (taken as an external field in linearized gravity). They show that to first order, the cross-sections are spin-dependent. In the second order, they are dependent on energy as well. So, they prove that the equivalence principle is violated in both cases.

This post imported from StackExchange Physics at 2014-03-30 15:22 (UCT), posted by SE-user vnb