What's the difference? Well, using a double-headed arrow to represent the horizontal direction emphasizes that in a quantum mechanical context, there's no distinction between left and right. They each have the same ...
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I chose to motivate things with classical waves, where it makes sense for this arrow to represent a unit vector in the right direction, rather than the more abstract idea of a horizontal state vector, I chose to stick with the single-headed notation throughout, though this runs slightly against convention.
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