Modern Physics

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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 n -values 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 = 10 -10 m. If the box is cm-sized, i.e. 10 -2 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 B-2). 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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