Einstein's equivalence principle in quantum mechanics revisited

Einstein's equivalence principle for a uniform gravitational field states that the motion of an object in an inertial reference frame is indistinguishable from the motion of the object in the absence of this field but with respect to a suitable uniformly accelerated reference system.^1–3 Despite its importance, a discussion of this equivalence principle in quantum mechanics is generally not included in physics textbooks. Already in a 1977 article in this journal, Eliezer and Leach complained: “To what extent the equivalence principle in classical mechanics has an analog in quantum mechanics is a problem to which no attention seems to have been given in the literature.”⁴ In fact, three years earlier, prompted by an experimental proposal of Overhauser and Colella⁵ to test the equivalence principle in an experiment on quantum interference with neutrons, I obtained the analytic solution of the time dependent Schrödinger equation for this problem. My solution appeared in footnote 6 of a paper by Collela et al.,^6,7 where their proposed neutron diffraction experiment was first carried out. Although this solution has now been re-derived several times (see Refs. 8–14), it still remains largely ignored in the physics teaching literature. To fill this gap, a derivation from first principles is presented here. For information about more recent work in neutron interferometry, see Ref. 15. Experiments with atomic beams have also been carried out to test the equivalance principle and general relativity,^16,17 but this work will not be discussed here.

For the classical treatment of the equivalence principle with the gravitational field directed along the −z axis, we have Newton's equation of motion

where m_i is the inertial mass, m_g is the gravitational mass, and g is the (local) acceleration due to gravity. The equality of these two masses, m_i = m_g, which has been verified¹⁸ to 2 parts in 10¹³, implies that the mass terms drop out of this equation. Changing to a reference system with coordinate $z′$ undergoing a constant acceleration a along the negative z-axis

results in

Hence, when $a=mgg/mi$⁠, the particle motion with respect to this accelerated reference system—in an elevator falling freely after its suspension is broken—is free motion at a constant velocity determined by its initial value.

For the quantum treatment of this problem, the Schrödinger equation for a particle under the action of a constant gravitational field along the −z axis is¹⁹ 

where the potential $V(z)=mggz$⁠. Even with the equality of inertial and gravitational mass (m_i = m_g), this equation still depends on the value of this mass; it does not disappear as in the classical equation of motion given in Eq. (1). But it will be shown that this mass dependence does not imply a violation of Einstein's formulation of the equivalence principle. Changing to space and time coordinates $z′,t′$ in the accelerated coordinate system given in Eq. (2), the Schrödinger equation for ψ in these variables becomes

Let

where $ϕ$ satisfies the Schrödinger equation for a free particle in the accelerated frame

and S is assumed to be a real function of the coordinates $z′$ and $t′$⁠. The requirement that $ϕ$ satisfies the Schrödinger equation for a free particle in an accelerated frame of reference for which a = g (when the inertial and gravitational masses are equal) is the mathematical formulation in quantum mechanics of Einstein's equivalence principle in classical mechanics. The assumption that S is a real function follows from the probability interpretation in quantum mechanics that requires $|ψ|2=|ϕ|2$⁠, but a proof is given below.

Using Eq. (6) for ψ in Eq. (5), we find for the left-hand side

and for the right-hand side

with A = B. This condition requires that to satisfy Eqs. (4) and (7), the remaining coefficients of $ϕ$ and $∂ϕ/∂z′$ in the quantity A–B must vanish. The coefficient of $∂ϕ/∂z′$ is

which implies that

where $f(t′)$ is a function of $t′$ only. The vanishing of the coefficient of $ϕ$ requires that

which is the same condition in classical mechanics for uniform motion in the prime coordinate system [Eq. (3)] and that

Integrating this equation with the boundary condition f(0) = 0 yields

Hence, the solution of the Schrödinger equation in the accelerated system with coordinates $z′,t′$ is

and the corresponding solution in the system with coordinates z, t, stationary in a gravitational potential V = m_ggz, is

where the relation between the gravitational constants g and the relative coordinate acceleration a is given by Eq. (12). The equality of inertial and gravitational mass, m_i = m_g, then leads to the gravitational equivalence principle in quantum mechanics with a = g. Substituting Eq. (16) for ψ into Eq. (4), it can be readily shown that ψ then satisfies the Schrödinger equation for a particle moving in a potential V = m_iaz.

To complete the proof of this equivalence, consider the form of the Schrödinger equation for a free particle in an inertial frame with respect to an accelerated frame according to the change of coordinates in Eq. (2). It can be shown by a transformation similar to Eq. (6) that in this accelerated frame the Schrödinger equation acquires a potential $V(z)=miaz$⁠. For the case that a = g, an observer in the accelerated frame would then find that m_i = m_g, where m_g is the gravitational mass.

In 1974, Overhauser and Collela proposed an interference experiment with neutrons scattering from a lattice to check experimentally whether neutrons satisfy the predictions of the equivalence principle in quantum mechanics.⁵ Their idea was to build a neutron interferometer that splits a neutron beam into two beams traveling horizontally through the same distance d but at a different height z, and to observe the interference fringes when these beams are reunited. Setting $t=d/v$⁠, where $v=2πℏ/miλ$ is the neutron's horizontal velocity and λ is the neutron wavelength, then according to Eq. (16) the phase shift is given by

Setting m_i = m_g, so that by Eq. (12) we have a = g (in accordance also with the classical formulation equivalence principle), and the phase shift given by Eq. (17) becomes equal to the expression derived by Overhauser and Colella.⁵ It should be pointed out, however, that their derivation of this phase shift was semiclassical, based on the momentum difference δp of two neutron beams differing by an amount z in height, which is valid only to first order in δp, while the quantum mechanical derivation is exact. By energy conservation, to first order this difference in momentum between the two neutron beams is

and according to quantum mechanics the momentum is

Hence, the phase shift over a horizontal distance d is

and substituting Eqs. (18) and (19) we find that $Δc=Δ$ from Eq. (17).

As a simple example of the application of Eq. (15), consider the case that $ϕ(z,t)$ is a plane wave with momentum p₀²⁰ 

which is a solution of Eq. (7). Substituting Eq. (16) for ψ [with Eq. (21) for $ϕ$] into the Schrödinger equation, Eq. (4), it can be verified that this equation is satisfied and that ψ is an eigenstate of the quantum mechanical momentum operator $−iℏ∂/∂z$

and the energy operator $iℏ∂/∂t$

with the momentum eigenvalue $p(t)=(p0−miat)$ varying linearly in time. This result is in accordance²³ with the increase in velocity v = at of a particle in classical mechanics under the action of a constant gravitational force m_ig.

A more general solution of Eq. (7) consisting of a linear superposition of these states can then also localize the particle in conformance with the uncertainty principle $ΔzΔp≥ℏ/2$⁠.

Setting the initial state to be

where $σ=Δz$⁠, we obtain the time dependent solution for a free particle [see Eq. (7)]

The absolute square of such a wave-packet then spreads in the same manner as the corresponding classical probability distribution, and substituting $z→z+at2/2$ [see Eq. (16)] this wave-packet is then centered at $z=v0t−at2/2$ corresponding to the motion of a classical particle.^21,22 Multiplying this wave-packet along the z-axis with a corresponding wave packet for free motion along the transverse x or y axis localizes the trajectory along the well known Galilean parabola found by solving the equations of classical mechanics.

In conclusion, we remark that the neutron interference experiment discussed here verifies only the validity of the Schrödinger equation for neutrons moving in the gravitational potential V(z) = mgz. An actual test of the quantum mechanical equivalence principle would require that this experiment be repeated in a region where there is no gravitational field, e.g., in outer space inside a rocket ship that is moving at a constant acceleration or inside a freely falling container dropped from some height above the surface of Earth. More practically, such a test has been carried out by Bonse and Wroblewsi,²⁷ by setting the entire neutron interferometer into harmonic oscillations, and measuring the phase shift at the end points of the oscillation as a function of the apparatus maximal acceleration a. The results are in good agreement with the theoretical relation given by Eq. (17), closing the loophole in the experiment of Collela et al.⁵ 

The author would like to thank Michael Berry, David Griffiths, Robert Johnson, Wayne Saslow, and Samuel Werner for helpful comments, and John Faulkner for a careful reading of this manuscript and helpful comments.

The wavefunction given in Eq. (16) is also the general solution for a free particle with mass m and charge e, moving in a uniform electric field of magnitude E in the −z direction, with $a=eE/m$⁠. The stationary treatment for this problem can be found in some quantum mechanics textbooks (e.g., Refs. 24–26), where it is shown that the solution of the time independent Schrödinger equation in a homogeneous field in coordinate space is an Airy function,²⁸ which in momentum space p is

where C is a constant and F is the constant force directed in the +z direction. Then its Fourier transform,

is the time dependent solution of the Schrödinger equation, Eq. (4), with the potential $V(z)=−Fz$⁠. Integrating first over the variable E, which gives a delta function $2πδ((p/F−t)ℏ)$⁠, one obtains

which for F = −ma corresponds to our solution [Eq. (16)] in the case that $ϕ$ is a constant.²⁹ 

In his popular introduction to quantum mechanics textbook,²⁶ David Griffiths writes that “Classically, a linear potential means a constant force, and hence a constant acceleration—the simplest nontrivial motion possible, and the starting point for elementary mechanics. It is ironic that the same potential in quantum mechanics gives rise to unfamiliar transcendental functions, and plays only a peripheral role in the theory.” But as we have shown here, although the transcendental Airy function appearing in the stationary solution of the Schrödinger equation in coordinate space does not have any classical analog, the time dependent quantum solution corresponds closely to the classical one.