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For an electron, if the uncertainty in momentum is twice the uncertainty in position, find the uncertainty in velocity. (Given: h-bar = h/(2*pi))

1. (1/(2m)) * sqrt(h-bar)
2. 1/(2m * sqrt(h-bar))
3. h-bar/(4*pi*m)
4. (1/m) * sqrt(h-bar)

Correct answer: (1/(2m)) * sqrt(h-bar)

Solution

Let deltaₚ = uncertainty in momentum, deltaₓ = uncertainty in position. Given: deltaₚ = 2*deltaₓ. Heisenberg principle (minimum): deltaₚ * deltaₓ >= h-bar/2. Using equality: deltaₚ * deltaₓ = h-bar/2. Substituting deltaₚ = 2*deltaₓ: 2*(deltaₓ)² = h-bar/2 => deltaₓ = sqrt(h-bar/4) = sqrt(h-bar)/2. deltaₚ = 2*deltaₓ = sqrt(h-bar). delta_v = deltaₚ/m = sqrt(h-bar)/m = (1/m)*sqrt(h-bar). This matches option D. But wait: 'h' in the problem likely means h-bar itself (since the problem says 'Given: h = h/2pi', meaning their symbol h IS h-bar). If their h IS h-bar, then the delta_v = sqrt(h)/m = (1/m)*sqrt(h-bar). However the option listed as D is (1/m)*sqrt(h-bar) and option A is (1/(2m))*sqrt(h-bar). The answer should be (1/m)*sqrt(h-bar), matching option D. But the options use 'h' to mean h-bar per the problem statement.

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