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Particles according to quantum physics are only "partly localised", so does it mean that Schrodinger's equation can only be applied in a confined region of space?

Also, from what I read from my text book, Schrodinger's Equation is applied to wave packets, because it has an "estimated" boundary of [tex]\Delta x[/tex] of large magnitude. If so, how can a simple harmonic quantum oscillator exist? An ideal simple harmonic motion is represented by pure sine or cosine waves, where [tex]\Delta x = \infty[/tex].