Why did the government investigate Robert J Oppenheimer?

Answer 1

It's pretty convoluted, but the short answer is that they thought he might be a Soviet spy.

Oppenheimer had, when younger, had some friends and acquaintances who were communists. He had drifted apart from most of these, but was still on fairly good terms with at least one of them. A Soviet agent approached his friend, and that friend spoke with Oppenheimer about it. Very briefly and casually; it was something along the lines of "So-and-so was wondering if there anything you can do to make sure the Soviets are kept in the loop, perhaps unofficially?" and Oppenheimer's response was something along the lines of "No, that's not how to do these things."

The problem was that when Oppenheimer reported this to his superiors, he didn't report it accurately and tried to hide his own involvement and that of his friend. He just said that "some people" had been approached on behalf of the Soviets. So the US started a big spy hunt for these Soviet agents, and Oppenheimer's responses to their questions weren't adding up. In fact, he wound up making things a lot worse for his friend than was actually justified, because the Government already suspected both Oppenheimer and his friend based on their (known) past association with communists, and Oppenheimer trying to cover it up made it look as if their involvement had been much more sinister than it really was.