We have to first specify what you mean by amplitude of single photon. Are you asking what would be the E-field observed from a single photon? That is well defin

We have to first specify what you mean by amplitude of single photon. Are you asking what would be the E-field observed from a single photon? That is well defined, however it depends on what photon you are talking about. Is it a photon confined to a cavity made of perfect mirrors? Is it a photon with well defined "position" and poorly defined "frequency"? In all these cases you can answer "what is the E field for a single photon", and you need mostly classical electromagnetism to answer this. Are you asking about the probability amplitude of the state? This can kinda be answered in the confines of Quantum Mechanics by talking about Klein-Gordon's equation, but truly, it is a mess and not a particularly practical approach. To get useful results you probably need to switch to a more general formalism called Quantum Field Theory, and in that formalism the pop-sci handwavy talk about wave-functions breaks a bit, but it also has a well defined answer.