A localized wave function is referred to as a wave packet. The most common wave packet has a Gaussian envelope (shape). The Schrödinger equation determines the time evolution of this wave packet. Although the square of the area of the wave packet (total probability) remains constant, the wave packet spreads in position space as shown in this model of the analytic solution. Why doesn't the wave packet change in momentum space?
Griffiths Quantum Mechanics p67.
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