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Unformatted text preview: PHY251 Homework Set 6 Reading: Chapter 6 and 7 Homework: Chapter 6, Question 9 Problems 2, 9,18 Chapter 7, Problems 16,20 Hints and Solutions Question VI.9 Suppose you have a box a few centimeters wide and an electron with the energy of a few electron volts (i.e. of the same order of magnitude as an electron in a Bohr orbit). In what range would the nvalues for the electron in the box lie? Hints: No hints Solution: The wavelength of the electron with eV energy would be of order the Bohr radius, i.e. 0.1 nm = 1010 m. If the box is cmsized, i.e. 102 m, then such a wavelength would fit 10 8 times in the box, i.e. n 2245 10 8 . Problem VI.2 Consider the normalized wave function of the form ψ ( x ) = C exp{ x ² / 2 a ² }. Calculate the expectation values (a) 〈 x 〉 , (b) 〈 x ² 〉 , (c) 〈 p 〉 , and (d) 〈 p ² 〉 for this wave function. Give a physical justification of your results for (a) and (c). Hints: The value of the normalization constant C should not matter. Use the definition of the expectation value, and the operator identity: p =  ih ( ∂ / ∂ x ) . Solution: The value of C is such that 1 = ∞ ∫ ∞  ψ ( x ) 2 dx =  C  2∞ ∫ ∞ exp{ x ² / a ² } dx =  C  2 a √ π (see Appendix B2). 〈 x 〉 = ∞ ∫ ∞ x  ψ ( x ) 2 dx = ∞ ∫ ∞ x  C  2 exp{ x ² / a ² } dx , which is an integral of an odd function of x , over an interval symmetric with respect to x =0: therefore, the result is 0: 〈 x 〉 = 0. The result makes sense physically: the wavefunction is perfectly symmetric around x =0, and therefore the average should be zero as well....
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 Spring '01
 Rijssenbeek
 Physics, Atom, Energy, Work, wave function, C2 exp, /x) ., C2 x² exp

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