Wheeler's delayed choice experiment In Wheeler's delayed choice experiment (http://en.wikipedia.org/wiki/Wheeler%27s_delayed_choice_experiment)
When the experimental apparatus does not contain a final interfering mirror, the photon is said to have been observed to travel as a particle because... there is no interference?
Even if the photon were to be traveling as a wave, I don't see why there would be interference without the mirror.
I.e., to take this passage from wikipedia: "Observing that photons show up in equal numbers at the two detectors, experimenters generally say that each photon has behaved as a particle from the time of its emission to the time of its detection, has traveled by either one path or the other, and further affirm that its wave nature has not been exhibited."
But just because the wave nature has not been "exhibited", does not mean it's not there. This seems like touching an apple with your eyes closed and saying "the visual properties of this apple haven't been exhibited, so it must be transparent to visible light".
I know that I'm wrong, but I can't figure out where I'm wrong. In my QM classes, experiments like this made more sense when one of the paths was blocked.
I think the problem I'm having is that I can't think of any specifically particle-like properties that are being exhibited in this experiment.
 A: I think this experiment might be clearer if you imagine performing this experiment one particle at a time (rather than say with a bright laser beam with an average of many photons).
If you don't have the second beam-splitter to recombine the two paths, then you will either get a click at the detector in one path or the other never both. If this was a classical wave, there would always be energy in both paths. Repeating this many times simply yields some particles randomly being detected in one path, and the rest detected in the other (hence the "show up in equal numbers at the two detectors" for a 50/50 beam-splitter).
On the other hand interference is not something that is associated with classical particles, so recombining the beam to demonstrate interference is typically considered to be a "wave-like" behavior.
A: Even if the photon were to be traveling as a wave, I don't see why there would be interference without the mirror.
I share your sentiment. I think the clue to the trick can be found in the Wikipedia article:
"Detection of a photon is a destructive process because a photon can never be seen in flight. When a photon is detected it "appears" in the consequences of its demise, e.g., by being absorbed by an electron in a photomultiplier that accepts its energy which is then used to trigger the cascade of events that produces a "click" from that device. A photon always appears at some highly localized point in space and time."
An E=hc/λ photon is a wave. We call them particles, but they're waves, not billiard-balls. We can diffract photons, they have a wavelength. Visible light photons have a wavelength of circa 500nm. Shortwave photons have a wavelength of 10m or more. However when we detect a photon, we detect all of it, whereupon it's localized. Now, where have we seen something like this before? In the optical Fourier transform. See Steven Lehar's web page:  

A simple lens can perform a Fourier transform in real-time. So in the standard double slit experiment, you see a pattern of dots on the screen. But when you detect a photon or electron at one slit, you convert it into something localized, so it goes through that slit only. Then when you detect it at the screen, you convert it into something localized, so you see a dot on the screen. Only now there's no interference pattern, because your photon went through one slit only. No mysticism is required, and no many-worlds multiverse either.  
But just because the wave nature has not been "exhibited", does not mean it's not there.
I agree.
I know that I'm wrong
I disagree.  
I think the problem I'm having is that I can't think of any specifically particle-like properties that are being exhibited in this experiment.
I think the problem you're having is that you refuse to accept quantum mysticism. Welcome to the club. IMHO a photon can be likened to a seismic wave, which you can detect with a pointed stick the size of a mountain range, and mop it all up into your now-vibrating stick. Alternatively you can detect the seismic wave with lots of little sticks without mopping it up. See Aephraim Steinberg re weak measurement, have a read of the physicsworld article In praise of weakness.  
