In the Book, He doesn't break it.
Instead he puts it back in Dumbledore's tomb.
He explains why he does that:
“I’m putting the Elder Wand,” he told Dumbledore, who was watching him with enormous affection and admiration, “back where it came from.
It can stay there.
If I die a natural death like Ignotus, its power will be broken, won’t it?
The previous master will never have been defeated.
That’ll be the end of it. ”
Dumbledore nodded.
They smiled at each other.
“Are you sure?”
said Ron.
There was the faintest trace of longing in his voice as he looked at the Elder Wand.
“I think Harry’s right,” said Hermione quietly.
“ That wand’s more trouble than it’s worth ,” said Harry.
“And quite honestly,” he turned away from the painted portraits, thinking now only of the four-poster bed lying waiting for him in Gryffindor Tower and wondering whether Kreacher might bring him a sandwich there, “ I’ve had enough trouble for a lifetime. ”
Harry Potter and the Deathly Hallows, Chapter 36.
The flaw in the plan, page 631 of 638
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Because it's powerful and dangerous and brings misery to (almost) everyone who possesses it?
Because he doesn't want the power it has, and can't give it away in good conscience?
Essentially, the Elder Wand is a magnet for trouble and violence.
Its very purpose is to make the wielder unbeatable.
Harry knows that whoever has the wand, if it is known, is a target.
He is tired of fighting, and wants peace.
He is the only one who can destroy the wand, so he does.
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As they discuss, whoever controls the wand would simply have too much power.
Maybe Harry feels that the power would corrupt him, maybe he worries that possessing the wand would make him (and by extension his loved ones)
a target of those who would steal the wand, or
maybe he just fears what would happen to the wand after he dies if he kept it.
More likely, it's some combination of all three.
He decides that an implement with such power simply shouldn't exist.
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The fate of (almost) every wizard who has had the Elder Wand has been a violent death, as the wand is taken away from it's previous owner.
Even those suspected of having a connection with the wand (Snape) seem to have the same fate met.
What point is there to having such a powerful wand if it will only lead to one's own death?
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Harry broke the wand because he did not want it to fall into the wrong hands again like it did with Voldemort and almost kill everyone in his quest to kill Harry Potter.
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Harry broke the Elder Wand because he knows that all its past owners, including Antioch Peverell (its first owner), had used it for boasting and power.
He also knows that those who are pure of heart (Harry and Dumbledore) can tame it.
Dumbledore said: I was fit only to possess the meanest of them, the least extraordinary.
I was fit to own the Elder Wand, and not to boast of it, and not to kill with it.
I was permitted to tame and to use it, because I took it, not for gain, but to save others from it.
And like Dumbledore, Harry tried to protect himself and others from too much power.
So, in order to do this, he must destroy the source of the power, the Elder Wand.
Which, in conclusion, is why he destroyed the Elder Wand.
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At the end of the film version of  Harry Potter and the Deathly Hallows: Part 2 , Harry broke the Elder Wand in two and threw the pieces away.  Why did he do that?
Why did Harry Potter break the Elder Wand?
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 The fate of (almost) every wizard who has had the Elder Wand has been a violent death, as the wand is taken away from it's previous owner. What point is there to having such a powerful wand if it will only lead to one's own death? Harry knows that whoever has the wand, if it is known, is a target.  Another reason why Harry broke the wandis  because he did not want it to fall into the wrong hands again like it did with Voldemort. Harry broke the wand in the film for a number a few reasons, some being that he did not want him or his friends to become the target for people who would seek the wands power and he also did not want the wand to fall into the wrong hands. Therefore he broke it.
Harry knows that whoever posses the wand will becom,e a target and will suffer a violent death. Therefore he broke it. Harry roke the wand, because he did not want it to fall into the wrong hands again.