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Unformatted text preview: QUANTUM MECHANICS E XERCISES Section 35.2 The Schrdinger Equation 10. The one-dimensional wave function is related to the probability by Equation 35.2, 2 ( ) . dP x dx = Since probability (a pure number) is dimensionless, the units of must be the square root of the inverse of the units of x (a length), or 1/ 2 (meters) .- 11. I NTERPRET We are given the wave function of a particle and asked about its probability distribution. D EVELOP Since the quantity 2 ( ) x represents the probability density of finding the particle, the particle is most likely to be found at the position where the probability density 2 ( ) x is a maximum. E VALUATE (a) The maximum of 2 2 2 2 2 / ( ) x a x A e - = is at x = (calculate 2 ( )/ d dx = for corroboration). (b) The probability density 2 ( ) x falls to half its maximum value 2 2 ( (0) ) A = when 2 2 2 / 1/2 , x a e- = or 1 2 ln 2 0.589 . x a a = = A SSESS The probability distribution is shown below. Note that 2 2 2 2 2 / ( )/ x a x A e - = peaks at x = and is halved at / 0.589. x a = 12. I NTERPRET We normalize the wave function that is given graphically in Figure 35.19. The particle must be somewhere , so the normalization constant is chosen such that the integral of the wave function over all space is one. D EVELOP The probability of finding the particle within dx of x is 2 ( ) ( ). P x x = We will write an equation for the wave function, square it, and integrate. Then well set the value of A such that the integral over the entire region is one. E VALUATE For 2 , L x 2 ( ) . A L x x = Since the wave function is symmetric, well just integrate from x = to 2 L x = and multiply by 2 to obtain the integral over all space. 3 2 2 2 2 / 2 2 2 4 3 1 2 3 3 L A x A L A L dx A L L L = = = = A SSESS The actual wave function for the region 2 L x is then 3/ 2 2 3 ( ) . L x x = 13. I NTERPRET We use the normalization constant and the wave function from Exercise 12 to find the probability of finding the particle in the region 4 . L x The probability of finding a particle in a region dx is 2 . dx D EVELOP We integrate the square of the normalized wave function 3/ 2 2 3 ( ) L x x = from x = to 4 L x = to find the probability of finding the particle in the region 4 . L x 35.1 35 35.2 Chapter 35 E VALUATE 3 / 4 2 3 3 3 12 12 1 1 3 16 4 L L P x dx L L = = = A SSESS There is a one in sixteen chance of finding the particle in the region described. The region is a quarter of the entire region where the particle could be, but the wave function is smaller in this region than elsewhere so the probability is less than one in four....
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