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DeBroglie Waves and Propagation of Mass

This starts a new series of posts that will hopefully inspire discussion among folks taking, teaching, and/or using quantum mechanics.  If you're reading this on Google+, the equations are referenced in the album attached to the post.

Did you know that DeBroglie came up with the concept of matter waves considering relativistic invariance? I didn't until quantum I lecture yesterday. Does anyone know how the reasoning went? I can see a way to make sense of it. If you look at energy being equal to, (eq. 1)
and think about frequency as the reciprocal variable of time in a Fourier transform, then, if you begin to consider momentum, (the other three components of the energy momentum four vector of special relativity), you soon thereafter could arrive at, (eq. 2)
where k is the reciprocal variable of the x, y, and z space coordinates in a Fourier transform.

In the same lecture, it was mentioned that the simple form of a sound wave could propagate energy but not mass.  For a wave that is capable of propagating mass, we need to go to complex space and a wave that looks like, (eq. 3)
Can anyone shed more light on why this is the case?  This wave equation has a form very similar to a Fourier transform and contains both Fourier reciprocal variables mentioned above.  Is that why the complex form is required? Does the earlier comment about special relativity belie the need for a complex component ala the component of i introduced with time in the Minkowski metric?


Picture of the Day:
From 1/16/13

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