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Unformatted text preview: Duke University Department of Physics Physics 143 Fall Term 2007 PRACTICE QUIZ 2 SOLUTIONS Problem 1: Consider a helium atom which has had one of its electrons stripped. What formula will describe the frequencies of the spectral lines for this He + ion? Find an expression for the energy required to remove the remaining electron in its ground state. Solution: The nucleus has Z = 2. The allowed energy levels are therefore E n = − 1 2 m e ( 2 αc n ) 2 . The allowed transition frequencies are f = E n − E m h = 4 m e ( αc ) 2 2 h ( 1 n ′ 2 − 1 n 2 ) The energy required to remove the remaining electron (the ionization en ergy) corresponds to a final state of n → ∞ , and n ′ = 1, so 1 2 m e (2 αc ) 2 . Problem 2: A 1.00 g marble is constrained to roll inside a tube of length L = 1 . 00 cm. Modeling this as a 1D infinite square well, find the value of the quantum number n if the marble is initially given an energy of 1.00 mJ. Calculate the energy required to promote the marble to the next available energy state. Solution: E n = n 2 ¯ h 2 π 2 2 mL 2 Plugging in and solving for n gives n = 4 . 5 × 10 28 . The energy required to get to the next state is E n +1 − E n = (( n + 1) 2 − n 2 ) ¯ h 2 π 2 2 mL 2 = (2 n + 1) ¯ h 2 π 2 2 mL 2 ∼ 2 n ¯ h 2 π 2 2 mL 2 and plugging in, get this to be 5 × 10 − 32 J. Problem 3: An electron of momentum p and mass m e is at distance r in a circular orbit around a stationary proton. The system has kinetic energy K = p 2 / 2 m e and a potential energy U = − ke 2 /r . The average position of the electron is at the proton but the uncertainty of its position is approximately equal to the radius r of its orbit. The electron’s average momentum will be zero, but the uncertainty in its momentum is approximately equal to its momentum...
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 Fall '07
 Scholberg
 Physics, Electron, Kinetic Energy, Photon, Uncertainty Principle

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