Chapter 30 - Chapter 30 54. Picture the Problem: The de...

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Chapter 30 54. Picture the Problem : The de Broglie wavelength of an electron is related to both the momentum and the kinetic energy of an electron. Strategy: Write the kinetic energy (equation 7-6) of the electron in terms of the momentum (equation 9-1). Then use equation 30-16 to write the momentum in terms of the de Broglie wavelength. Solution: 1. Write the kinetic energy in terms of the de Broglie wavelength: 2. Insert the wavelength: Insight: In this problem we used the classical equations for the kinetic energy and the momentum. This assumption can be checked solving equation 30-16 for the speed. The resulting is small enough that relativistic effects can be neglected. 56. Picture the Problem : Because the proton is significantly more massive than an electron, it will have a greater momentum when the proton and electron have the same speed. The de Broglie wavelength is inversely proportional to the particle momentum. Strategy: Calculate the ratio of the de Broglie wavelengths using equation 30-16, where the momentum is the mass times velocity. Solution: 1. (a) Because and for identical speeds, an electron has a longer de Broglie wavelength than a proton. 2. (b) Calculate the ratio of wavelengths: Insight: In this problem we assumed a classical speed so that we could use the classical momentum equation. However, because the relativistic factor depends only upon the speed, the ratio of the wavelengths would still be 1840 at relativistic speeds. 60. Picture the Problem : According to the Heisenberg uncertainty principle, the product of the uncertainty in position and uncertainty in momentum must be greater than Planck’s constant divided by 2 π . Strategy: The 5.0% uncertainty in speed means that the magnitude of the momentum (mass times speed) of the electron and the baseball are also uncertain by 5.0%. The minimum uncertainty in position is given by equation 30-19. Solution: 1. Solve the uncertainty principle for the uncertainty in position: 2. Set 3. Calculate the uncertainty for the baseball: 4. Calculate the uncertainty for the electron: Insight: The minimum uncertainty in the position of the baseball is smaller than an atom by a factor of 10 24 . However, the minimum uncertainty in the position of the electron is measurable, roughly the thickness of a human hair. 67. Picture the Problem : When an electron’s location is known to within an uncertainty of 0.15 nm, it will have a minimum uncertainty in momentum as required by the uncertainty principle. That minimum uncertainty requires the electron to have a nonzero kinetic energy.
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Strategy: Use equation 30-19 to calculate the minimum uncertainty p in the momentum, and then
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This note was uploaded on 09/03/2008 for the course PHYS 104 taught by Professor Dr.g during the Spring '08 term at Gwinnett Technical College.

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Chapter 30 - Chapter 30 54. Picture the Problem: The de...

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