Interferometric discrepancy between the Schrödinger and Klein-Gordon wave equations due to their dissimilar phase velocities

Abstract

The Schrödinger equation predicts interference when a beamsplitter's trajectory includes a segment where its speed exceeds the phase velocity of a free non-zero rest mass particle that is in a momentum eigenstate. Such interference is neither possible for electromagnetic waves nor for eigenstates of momentum in the non-relativistic limit of the Klein-Gordon equation since the speed of the beamsplitter cannot exceed the phase velocity of the wave. The dual behavior of reflection and transmission in this case is discussed for dielectric and diffracting beamsplitters.

Figures (2)

• Figure 1: Interference generated by one dimensional motion of a beamsplitter, BS, in the lab frame. Only segments of the wave trains are shown. The length of the arrows indicate the speeds of the various components. The source, S, emits a harmonic wave that retro-reflects from the BS when it is stationary as shown in frame (a). The BS then moves faster than these wave crests in frame (b) generating a transmitted wave that is left behind it (which is unaffected by the BS motion) and a reflected wave that moves in front of the BS much faster than the BS. The BS then comes to rest after having passed through the initially reflected sequence of wave crests (shown in frame(a)) in frame (c). The lagging wave crest sequence then interferes with new wave crests that retro-reflect from the stationary BS in frame (d). A $45$ degree beamsplitter directs these two interfering wave trains to detector, DET. For clarity, other reflections from the BS's motion are treated in the text rather than shown in the figures.
• Figure 2: Worldlines of the source, S, the detector, DET, and the beamsplitter, BS, are represented by solid lines for the interferometer illustrated in fig.\ref{['fig1']}. The worldline of each wave crest from the source is represented as a dashed line emanating from S and initially moving upward and to the right. For clarity the totality of wave crest worldlines is limited to a bundle near reflection from the static BS, while not all of the wave crests that are reflected from the moving BS (moving in front of the BS) are shown. Frame (a) illustrates wave crests moving at the speed light. This interaction for the Schrödinger equation is illustrated in frame (b) for a particle moving at a speed small enough that the BS can pass through some of its retro-reflected wave crests. Note that the ordinates between (a) and (b) differ. Interference occurs only in frame (b).