Some 15-year olds can learn the Schrödinger equation from a textbook. But if the 15-year old is not mathematically sophisticated, you can’t expect to give them a full understanding of the Schrödinger equation in 15 minutes, but you can explain to them why the Schrödinger equation is needed and what it intuitively represents.
My attempt at an explanation would cover the following points, possibly omitting some math depending on whether the 15-year old knows any calculus:
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