For Mrs. Clinton, this means that at a moment when she would like to turn to the general election, she is going to have to devote time and resources on the Sanders wing of the Democratic Party. In her victory speech in New York, that seemed very much on Mrs. Clinton’s mind. “To all the people who supported Senator Sanders, I believe there is much more that unites us than divides us,” she said.

As the past half-century of campaigns has demonstrated, there is much Mrs. Clinton can do beyond a few words in a victory speech. She could offer Mr. Sanders a prime-time speaking spot at the convention. She could pick a vice president who shares the concerns that have animated Mr. Sanders’s crowds. She could take a lesson from John F. Kennedy, who made his primary archrival, Lyndon Baines Johnson, his running mate in 1960. (Given that Mr. Sanders is 74 and Mrs. Clinton is 68, that seems unlikely.)

She could also include planks in the Democratic Party platform that incorporate Mr. Sanders’s positions. If Mr. Sanders ends up with a campaign debt, Mrs. Clinton can also promise to help pay it off, as President Obama did for her back in 2008.

But there may be limits to what Mrs. Clinton herself can do to earn the support of his voters. She moved to the left throughout the primary as she adjusted to Mr. Sanders’s challenges. Slipping back to the center in preparation for a general election battle, which is something Mrs. Clinton would presumably like to do, might complicate the effort to appeal to Mr. Sanders’s voters, who are already wary of the former first lady.

An even more central question is whether Mr. Sanders, who before this campaign did not even call himself a Democrat and has seemed increasingly put off by Mrs. Clinton, will wholeheartedly endorse her should she win the nomination.

And that is only the first hurdle. Howard Dean, who lost the Democratic presidential nomination in 2004 to John Kerry, said he had no trouble endorsing Mr. Kerry over George W. Bush; getting his supporters to cross that bridge proved more difficult.

“It’s much easier for the candidate to understand that he or she has to get with the program,” said Mr. Dean, who supports Mrs. Clinton. “I had made up my mind that I was going to support John, but bringing supporters along was much tougher.”