andersonProbCurrent

andersonProbCurrent - 140 THE DEVELOPMENT OF WAVE MECHAN...

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In order :0 complete the square we multiply by exp [(x 2 O' 2 /2) - (x 2 O' 2/ 2)J an d ob tain i P 1/J(x) A f'" [ k 2 x 2 O' 2 _ X 2 2 O' 2 ] T = _ ~ dk exp - - + ikx + - V 27T - '" 2 0'2 2 A 2 2 f'" [I ] = _ ~ e-'" a 12 dk exp - - (k 2 - 2ikxO' 2 - x 2 d') V 27T -eec 20'2 141 lI!I THE SCHRODINGER WAVE EQUATION What must be the frequency bandwidth of the detecting and amplifying stages of a radar system operating at pulse widths of 0.1 usee? If the radar is used for ranging (dis- tance measurements), what is the uncertainty in the range? atta PROBL£M 4-17 THE DEVELOPMENT OF WAVE MECHANICS 140 A 2 ' I f'" J = _ ~ e-'" a 2 exp - - (k - i X 0'21 2 V 27T - <Xl 2 0' 2 PROBLEM 4-11 (a) Find the normalization constant N for the Gaussian wave packet, (a) Use the uncertainty principle to obtain the uncertainty in the momentum of a particle of mass m constrained to the volume of a cubical box of side a. (b) What is the corresponding uncertainty in the kinetic energy of the particle? (c) How do you interpret the answer to (b) ? (See Problem 4-2.) (b) Obtain the Fourier transform of tp(x) and verify that it is normalized. PROBLEM 4-19 - di .' .. & " Ai' 7 _n a:: Show that (dfdt)[<I>(k)] = O. PROBLEM 4-' 5 PROBLEM 4-'4 . . . ( 1 ) 1/. Note that If <I>(k) IS normalized by setting A = vrra , then If'(x) is also a normalized Gaussian function; that is, the Fourier transform of a Gaussian f~nction is also a ~aussian. The spread of the packet in coordinate space is gIven by 1/0' and illustrates the reciprocal nature of f).k and Sx as required by the uncertainty principle. Show that if the coordinate wave function is normalized at t = 0, then both the coordinate momentum wave functions remain normalized for all time. 9. TH E SCHR.ODINGER. WAVE EQUATION 12 See J. L . Powell an d B. Crasemann, Quantum M echanics. Addison-Wesley Publ Co Inc Reading, M ass. (196 1), p. 475. _ _ . ., ., PROBLEM 4.16 ,tiMIU4 For a free particle show that a wave packet in coordinate space broad ens with tim e (corresponding to increasing uncertai.nty ~ the position of the particle), while its rep- resen tation in space retains its shape changes only in phase (since momentum is conserved).
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andersonProbCurrent - 140 THE DEVELOPMENT OF WAVE MECHAN...

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