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argument of the delta function is zero at x− l = 0 ⇔ x = l , so ∫ ∞ −∞ cos (nπx l ) sin (nπx l ) δ (x− l) dx = cos (nπx l ) sin (nπx l ) ∣∣∣∣ x= l = cos (nπ) sin (nπ) = cos (nπ) (0) = 0 for all n . (f) −∞ < 2 < ∞ , therefore ∫ ( 16ξ4 − 48ξ2 + 12 ) δ (ξ − 2) dξ = 16 · 24 − 48 · 22 + 12 = 256− 192 + 12 = 76 (g) 0 < a0 < ∞ , thus ∫ 2√ 27 a −3/2 0 ( 1 − 2r 3a0 + 2r2 27a20 ) e−r/3a0 δ(r − a0) dr = 2√ 27 a −3/2 0 ( 1 − 2a0 3a0 + 2a20 27a20 ) e−a0/3a0 122 = 2√ 27 a −3/2 0 ( 1 − 2 3 + 2 27 ) e−1/3 = 22 81 √ 3 a −3/2 0 e −1/3 (h) Both angular parameters are between their

discussion. total shell n2 O n = 5 5s 1 5p 3 5d 5 5f 7 5g 9 25 N n = 4 4s 1 4p 3 4d 5 4f 7 16↑ M n = 3 3s 1 3p 3 3d 5 9 E L n = 2 2s 1 2p 3 4 K n = 1 1s 1 1 l = 0 l = 1 l = 2 l = 3 l = 4 Figure 13 − 3. Quantum Number, Chemical Designation Correspondence. The subscripts on the underbars of the subshells indicate the number of states 2l+ 1 in that subshell. Postscript: Since the energy, to this point, depends only on n , the quantum numbers l and m having no effect on energy, we have an n2–fold degeneracy in energy using this picture. A magnetic field removes this degeneracy

is V (x) = 1 2 kx2 . Its graph is a parabola as seen in the figure on the left. Any relative minimum in a smooth potential energy curve can be approximated by a simple harmonic oscillator if the energy is small compared to the height of the well meaning that oscillations have small amplitudes. Figure 10–1. SHO potential well. Figure 10–2. Relative potential energy minimum. Expanding an arbitrary potential energy function in a Taylor series, where x0 is the minimum, V (x) = V (x0) + dV dx ∣∣∣ x0 (x− x0) + 1 2! d2V dx2 ∣∣∣ x0 (x − x0)2 + 1 3! d3V dx3 ∣∣∣ x0 (x− x0)3

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References 1a. Lovelock, J, E., Zlatkis, A., and Becker, R. S.: Nature 1.93 (1.962) 540. 1b. Lijinsky, W.: 1.5th Mid-American Symposium on Spectroscopy, June 1.-4, 1.964, Chicago, Illinois. 2. Elmenhorst, H., and Reckzeh, G.: Beiträge zur Tabakforschung 2 (1.964) 1.80. 3. Wynder, E. L., and Hoffmann, D.: Deutsche Medizinische Wochenschrift 88 (1.963) 623. 4. Scherbak, M., Rice, R. L., and DeSouza, J. E.: 1.7th Tobacco Chemists' Research Conference, September 22.:..25, 1963, Montreal, Quebec. 5. Smoking and Health, Report of the Advisory Committee to the Surgeon

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,67 6,21 300 7,58 6,63 6,14 400 7,58 6,54 6,06 500 7,45 6,45 5,95 1000 7,35 6,29 5,71 1500 7,20 6,21 5,60 2000 7,10 6,17 5,56 15,7% 18,8% 8,61 8,61 6,58 6,36 6,03 5,78 5,69 5,41 5,50 5,21 5,40 5,05 5,26 4,93 5,16 4,79 5,08 4,71 4,85 4,50 4,79 4,39 4,72 4,25 19,5% 7,87 5,99 5,44 5,13 4,92 4,86 4,76 4,67 4,59 4,39 4,27 4,21 TABELLE 1 Virgin H20 cm3/g Starnp- fungen 0 50 100 150 200 250 300 400 500 1000 1500 2000 Stamp- fungen 0 50 100 150 200 250 300 400 500 1000 1500 2000 10,4% 11,78 10,10 9,70 9,54 9

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., Gzik-Szumiata, M., Javed, A., Morley, N. A., & Gibbs, M. R. J. (2013). Mössbauer study of vacuum annealed Fe 100− x Ga x (10 ≤ x ≤ 35) thin films. Nukleonika , 58 , 25–28. 9. Taneja, S. P. (2004). Mössbauer studies of thermal power plant coal and fly ash. Hyperfine Interact ., 153 , 83–90. 10.1023/B:HYPE.0000024715.55347.fe. 10. Vandenberghe, R. E., de Resende, V. G., & De Grave, E. (2009). Mössbauer effect study of fly and bottom ashes from an electric generating plant. Hyperfine Interact ., 191 , 11–16. DOI: 10.1007/s10751-009-9978-8. 11. Vandenberghe, R