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HW1_solutions - Homework 1 1 Problem Streetman Sixth Ed...

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Unformatted text preview: Homework 1 1. Problem: Streetman, Sixth Ed., Problem 2.2: Show that the third Bohr postulate, Eq. (2-5) (that is, that the angular momentum p θ around the polar axis is an integer multiple of the reduced Planck constant ¯ h , so p θ = n ¯ h ) is equivalent to an integer number of de Broglie waves fitting within the circumference a Bohr circular the orbit. Solution: (a) The ‘easy’ way : Let’s write the electron angular momentum along the z axis, L φ as: L φ = mωr 2 = mυr , (1) where m is the mass of the electron, r is the radius of the orbit, ¯ h is the reduced Planck constant, ¯ h = h/ (2 π ) , υ is the magnitude of the electron velocity, and ω = υ/r is the angular velocity. Let’s write Bohr’s postulate as: L φ = n ¯ h . (2) Then, substituting Eq. (1) into Eq. (2): mυr = n ¯ h = nh 2 π . (3) Let’s multiply both sides of this equation by 2 π/ ( mυ ) : 2 πr = nh mυ . (4) Since deBroglie’s assumption implies λ = h/ ( mυ ) , Eq. (4) is simply 2 πr = nλ , (5) which is what the problem asked. ECE344 Fall 2009 1 (b) The ‘hard’ way : Let’s write the third Bohr postulate as in Eq. (3): mυr = n ¯ h , (6) Now, as done in class, let’s consider the magnitude of the attractive electrostatic force between the electron and the nucleus and set it equal to the magnitude of the centrifugal force so that the electron remains in a stable orbit: e 2 4 π r 2 = mω 2 r . (7)...
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HW1_solutions - Homework 1 1 Problem Streetman Sixth Ed...

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