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- TL;DR Summary
- I recently saw an explanation for how quantum mechanics approaches classical mechanics at the limit of Planck's constant becoming 0 using the Heisenberg equation of motion but am confused about what it is about this limit that reduces the equation of motion to its classical limit.

Starting from the Heisenberg equation of motion, we have

$$ih \frac{\partial p}{\partial t} = [p, H]$$

which simplifies to $$ih \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}$$

but this just results in ## \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}## and I'm not sure where the limit of the Planck's constant was even used. Can anyone point out my mistake or help me understand?

$$ih \frac{\partial p}{\partial t} = [p, H]$$

which simplifies to $$ih \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}$$

but this just results in ## \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}## and I'm not sure where the limit of the Planck's constant was even used. Can anyone point out my mistake or help me understand?