Thank you for this question. It is one I had a lot.
Some comments first. In the Islamic belief, the Quran is the word of God, not of the prophet, who is the messenger.
No, Quran was not written after Muhammad's death, it was written within his lifetime. You are maybe confusing with the Hadiths, that were, indeed, written and compiled by the 4 imams a significant time after the prophet's death. According to tradition, the prophet specifically asked that only the Quran be written, not his Hadiths, so as to avoid confusion between the two.
It was, however, assembled, ordered (not chronologically) and selected (not long) after his death. It is true that the prophet did not leave any indications or order to do so, but it was decided to assemble the Quran in one place, in one book, so that the umma can have a guide, or a constitution, like Christians and Jews did.
It is true that after the body of the Quran was decided upon, the order was given to burn any diverging writings, which implies there were diverging writings, but the counsel that was assembled basically encompassed all those who knew the Quran by heart and all living companions of the prophet. And among those, to my knowledge, there were no divergence (but my knowledge may be flawed, I encourage you to do the research and repost here).
Given the close proximity, in time and space, to the death of the prophet, and the huge numbers (as opposed to, say, 12 apostles) involved in this endeavour, it is reasonable to think that the text is, indeed, 14 centuries old and that it did not change since that time.
However, in its first days, and for several centuries, the Quran was written in Aramaic style, with no dots and diacritics (google image search will tell you what I mean). These were added on two occasions fairly recently. But since exegeses from very early on were preserved, it is also relatively fair to think that, for the most part, the diacritics and the dots were added in the right places.
There is a very interesting book called The Syro-Aramaic Reading of the Koran which I read and which I can advice to read. But it gives very small places where a divergent reading could be made. And the divergence is not critical to the overall meaning of the Quran.
Now, the fervent Muslims reading this may wonder why I didn't talk about the verse that says "We brought down the book and We are keeping it whole" and my answer is that it is not a logical nor a scientific argument, it is a matter of faith. You cannot offer, as proof that a book is whole, a citation from that very same book. After all, the Bible also has "This in truth I say" before certain verses, and the general Islamic consensus is that it has been tampered with. This is StackEx: answers should be factual above all.