A particle of mass m moves in the central potential U = ar². Using Bohr quantization for circular orbits, find the radius rₙ of the nth allowed orbit.
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A particle of mass m moves in the central potential U = ar². Using Bohr quantization for circular orbits, find the radius rₙ of the nth allowed orbit.

G1 Physics · Accepted Answer

Step 1: For U=ar², the radial force is 2ar. Step 2: Circular motion gives mv²/r=2ar, so mv²=2ar². Step 3: Bohr quantization mvr=nh/(2π). Eliminating v gives r⁴=n²h²/(8amπ²), hence rₙ=[n²h²/(8amπ²)]¹ᐟ⁴.