How does a physician diagnose traumatic brain injury?

Few things: The diagnosis is based on history of hitting one's head, the presence of absence of memory loss, imaging can be helpful, and then the history of new symptoms like ha, trouble concentrating etc, that were not there before the head injury.

Varying methods: There are lots of rating scales and definitions. Some are based on initial symptoms of concussion, see: http://cpancf.Com/headinjuryclassification.Asp depending on severity and symptoms imaging studies and eegs might be done acutely. Neuropsychological testing is often the most sensitive method to determine any long-lasting effects.