# Calculating Minimum Resulting Uncertainty and Minimum Kinetic Energy

Part 1

Suppose optical radiation (of wavelength 2.6 × 10^-7 m) is used to determine the position of an electron to within the wavelength of the light. The mass of an electron is 9.10939 × 10^-31 kg and the Planck's constant is 6.62607 × 10^-34 J · s.

What will be the minimum resulting uncertainty in the electron's velocity?

Answer in units of m/s.

Part 2

We wish to measure simultaneously the wavelength and position of a photon.

Assume that the wavelength measurement gives lamba = 4673 x 10^-10 with an accuracy of delta lambda/lamda = 5.9 × 10^-6 .

What is the minimum uncertainty in the position of the photon? Use propagation of uncertainties and the uncertainty in lambda to calculate the uncertainty in the momentum.

Answer in units of m.

Part 3

The kinetic energy of a nonrelativistic particle can be written in terms of its momentum as

K = p ^2 /2 m .

Planck's constant is 6.63 × 10^-34 J · s.

What is the minimum kinetic energy of a proton confined within a nucleus having a diameter of 6 × 10^-16 m?

Answer in units of MeV.

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This solution provides calculations for minimum resulting uncertainty and minimum kinetic energy.

Find the minimum kinetic energy of a proton.

(a) Show that the kinetic energy of a nonrelativistic particle can be written in terms of its momentum as KE = P2 (squared)/2m. (b) Use the results of (a) to find the minimum kinetic energy of a proton confined within a nucleus having a diameter of 1.0 x 10-15 m.

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