mp-Chapt-6-sol - Chapter 6 Problem Solutions 1. Why is it...

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Inha University Department of Physics sol Whether in Cartesian ( x, y, z ) or spherical coordinates, three quantities are needed to describe the variation of the wave function throughout space. The three quantum numbers needed to describe an atomic electron correspond to the variation in the radial direction, the variation in the azimuthal direction (the variation along the circumference of the classical orbit), and the variation with the polar direction (variation along the direction from the classical axis of rotation). Chapter 6 Problem Solutions 1. Why is it natural that three quantum numbers are needed to describe an atomic electron (apart from electron spin)? 3. Show that sol For the given function, is a solution of Eq. (6.14) and that it is normalized. o a r e a r R 2 2 3 0 10 2 / / ) ( - = and 2 2 5 0 10 , / / o a r e a R dr d - - =
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Inha University Department of Physics 10 2 2 2 2 5 10 2 2 2 1 2 1 2 1 R a r a e a r r r a dr dR r dr d r o o a r o o o - = - - = - / / This is the solution to Equation (6.14) if l =0 ( as indicated by the index of R 10 ), 2 2 2 2 2 4 or 4 2 2 me h a h me a o o o o e p pe = = , which is the case, and 1 2 2 2 8 or 1 2 E a e E a E m o o o = - = - = pe , h again as indicated by the index of R 10 . To show normalization, , / - - = = 0 2 0 2 2 3 0 2 2 10 2 1 4 du e u dr e r a dr r R u a r o o where the substitution u =2 r / a o has been made. The improper definite integral in u is known to have the value 2 and so the given function is normalized.
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Inha University Department of Physics 5. In Exercise 12 of Chap. 5 it was stated that an important property of the eigenfunctions of a system is that they are orthogonal to one another, which means that sol From Equation (6.15) the integral, apart from the normalization constants, is m n dV m n = - 0 y y * l l m m m m d l l Φ Φ for 2 0 f p * Verify that this is true for the azimuthal wave functions of the hydrogen atom by calculating l m Φ , * f f p f f p d e e d l l l l m i im m m - = Φ Φ 2 0 2 0 It is possible to express the integral in terms of real and imaginary parts, but it turns out to be more convenient to do the integral directly in terms of complex exponentials: [ ] 0 1 2 0 2 0 2 0 = - = = - - - p f p f p f f f f ) ( ) ( ) ( l l l l l l m m i l l m m i m i im e m m i d e d e e The above form for the integral is valid only for m l m l ’, which is given for this case. In evaluating the integral at the limits, the fact that e i2 p n = 1 for any integer n ( in this case ( m l ’ – m l )) has been used.
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Inha University Department of Physics 7. Compare the angular momentum of a ground-state electron in the Bohr model of the hydrogen atom with its value in the quantum theory. 9.
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mp-Chapt-6-sol - Chapter 6 Problem Solutions 1. Why is it...

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