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Obviously I must have made a mistake somewhere in my reasoning, but I can't see it.

Start with de Broglie: $$ P= \frac{h}{\lambda}$$ $$ h=P \lambda $$

Substitute into Einstein's energy equation for a photon (EDIT: read "particle" instead) without any potential fields or whatever.

$$E = h f$$ $$E = P \lambda f$$ $$E = mv^2 $$

But since this photon (particle) isn't in any potential fields, I would expect its energy to be just its kinetic energy - $0.5mv^2$ - instead.

What mistake did I make?

Henry1727
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The energy of a photon is not dependent on its velocity - its velocity is always the same (within the same medium).

Instead, as you have written, the energy of the photon depends on its frequency, which varies inversely with its wavelength, as the product of the two is its speed, which is, again, a constant.

hebetudinous
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