The questions have prompted a sharp debate among mental health professionals about the so-called Goldwater rule adopted by the American Psychiatric Association barring members from evaluating anyone they have not personally examined, a rule generated in response to questions raised about Senator Barry Goldwater, the Republican presidential nominee in 1964. Mr. Trump is due for his annual physical examination on Friday, but the White House would not say whether it would include mental acuity tests. Some psychiatrists have said it is irresponsible to throw around medical terms without an examination. “These amateurs shouldn’t be diagnosing at a distance, and they don’t know what they’re talking about,” said Allen Frances, a former psychiatry department chairman at Duke University School of Medicine who helped develop the profession’s diagnostic standards for mental disorders.