how do you convince yourself that electromagnetic waves carry energy and momentum?

Photons will not be able to give out light if they didn't had both energy and momentum. The characteristic colour of certain metals and compounds is based on this concept only. Natural phenomenon such as aurora borealis are also based on it. Even a tube light will not be able to produce light if the electromagnetic waves did not carry both momentum and energy. 

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Simple, If they did not carry momentum and energy photoelectric effect would not take place i.e they cannot knock off electrons from metal surface .

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