mp-chapt-5-sol - Chapter 5 Problem Solutions 1. Which of...

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Inha University Department of Physics Chapter 5 Problem Solutions 1 . Which of the wave functions in Fig. 5.15 cannot have physical significance in the interval shown? Why not? 3 . Which of the following wave functions cannot be solutions of Schrödinger's equation for all values of x ? Why not? (a) y = A sec x ; (b) y = A tan x ; (c) y = A exp( x 2 ); (d) y = A exp( - x 2 ) . Sol Figure (b) is double valued, and is not a function at all, and cannot have physical significance. Figure (c) has discontinuous derivative in the shown interval. Figure (d) is finite everywhere in the shown interval. Figure (f) is discontinuous in the shown interval. Sol The functions (a) and (b) are both infinite when cos x = 0, at x = ± π /2, ±3 π /2, … ±(2 n +1) π /2 for any integer n , neither y = A sec x or y = A tan x could be a solution of Schrödinger's equation for all values of x . The function (c) diverges as x ± , and cannot be a solution of Schrödinger's equation for all values of x .
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Inha University Department of Physics 5. The wave function of a certain particle is y = A cos 2 x for - π /2 < x < π /2 . (a) Find the value of A . (b) Find the probability that the particle be found between x = 0 and x = π / 4. Sol Both parts involve the integral cos 4 xdx, evaluated between different limits for the two parts. Of the many ways to find this integral, including consulting tables and using symbolic- manipulation programs, a direct algebraic reduction gives [ ] [ ] [ ] , cos cos ) cos ( cos ) ( cos cos ) cos ( ) (cos cos x x x x x x x x x 4 2 4 1 2 2 1 2 2 2 1 2 1 8 1 2 1 8 3 2 1 4 1 2 4 1 2 2 1 2 2 4 + + = + + + = + + = + = = where the identity cos 2 q = ½(1+cos 2 q ) has been used twice. (a) The needed normalization condition is [ ] 1 4 2 2 2 2 2 2 2 8 1 2 1 8 3 2 2 2 4 2 2 2 = + + = = + - + - + - + - + - / / / / / / / / / / cos cos cos p p p p p p p p p p y y xdx xdx dx A xdx A dx The integrals 2 2 4 1 2 2 2 2 2 1 2 2 4 4 and 2 2 / / / / / / / / sin cos sin cos p p p p p p p p + - + - + - + - = = x dx x x dx x are seen to be vanish, and the normalization condition reduces to . , p p 3 8 or 8 3 1 2 = = A A
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Inha University Department of Physics (b) Evaluating the same integral between the different limits, [ ] , sin sin cos / / 4 1 32 3 4 2 4 0 32 1 4 1 8 3 4 0 4 + = + + = p p p x x x dx x The probability of the particle being found between x = 0 and x = π /4 is the product of this integral and A 2 , or ( 29 ( 29 46 0 4 1 32 3 3 8 4 1 32 3 2 . = + = + p p p A 7. As mentioned in Sec. 5.1, in order to give physically meaningful results in calculations a wave function and its partial derivatives must be finite, continuous, and single-valued, and in addition must be normalizable. Equation (5.9) gives the wave function of a particle moving freely (that is, with no forces acting on it) in the + x direction as ) )( / ( pc Et i Ae - - = Ψ h where E is the particle's total energy and p is its momentum. Does this wave function meet all the
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This note was uploaded on 11/29/2010 for the course PHY 111 taught by Professor Kim during the Spring '10 term at 카이스트, 한국과학기술원.

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mp-chapt-5-sol - Chapter 5 Problem Solutions 1. Which of...

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