20101ee2_1_HW1 solution

20101ee2_1_HW1 solution - y , L z are the lengths of the...

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Solutions to HW #1 Problem 1 The de Broglie wavelength of a particle (mass m, velocity v) is given by mv h p h = = λ , where h is the Planck’s constant ( = 6.626x10 -34 Js) Therefore, λ = 6.626x10 -34 Js/{1.5x10 -3 kg x (10 x 1.609 x 5/18)ms -1 } = 9.883x10 -32 m (incredibly small!!) Problem 2 E = 6 eV = 6 x 1.602x10 -19 J = 9.612x10 -19 J We also know that E = p 2 /2m and the mass of an electron, m e = 9.11x10 -31 kg Therefore, p = (2 x 9.11x10 -31 x 9.612x10 -19 ) = 1.323x10 -24 kgms -1 As in problem 1, λ = 6.626x10 -34 Js/1.323x10 -24 kgms -1 = 5.008x10 -10 m 5 A° Problem 3 (a) using periodic boundary condition which is applicable to a potential box (bulk crystal) From the solution of a particle in a box (with Born-Von Karman boundary conditions), we know that the quantized values of the kinetic energy and the various components of momentum are: z z z y y y x x x z z y y x x n L h p n L h p n L h p L n L n L n m h E = = = + + = , , 2 2 2 2 2 2 2 2 where n x , n y , n z are integers (quantum numbers) and L x , L
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Unformatted text preview: y , L z are the lengths of the sides of the box Therefore, for a state described by (1, 2, 1), we have p x = h/L x , p y = 2h/L y , p z = h/L z and E = h 2 /2m x (1/L x 2 + 4/L y 2 + 1/L z 2 ) which are evaluated as below: p x = 6.626x10-34 Js/1x10-9 m = 6.626x10-25 kgms-1 p y = 2 x 6.626x10-34 Js/1.3x10-9 m = 1.019x10-24 kgms-1 p z = 6.626x10-34 Js/1.5x10-9 m = 4.417x10-25 kgms-1 E = (6.626x10-34 Js) 2 /(2 x 9.1x10-31 ) x 1/10-18 (1/1 2 + 4/1.3 2 + 1/1.5 2 ) = 9.194x10-19 J E = 5.739eV (b) On the other hand, if we had an infinite potential well, using boundary condition for it: p x = 6.626x10-34 Js/1x10-9 m/2 = 3.313x10-25 kgms-1 p y = 2 x 6.626x10-34 Js/1.3x10-9 m/2 = 0.51x10-24 kgms-1 p z = 6.626x10-34 Js/1.5x10-9 m /2= 2.209x10-25 kgms-1 E=(6.626x10-34 Js) 2 /(8 x 9.1x10-31 ) x 1/10-18 (1/1 2 + 4/1.3 2 + 1/1.5 2 ) = 2.3x10-19 J = 1.43eV...
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20101ee2_1_HW1 solution - y , L z are the lengths of the...

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