andersonProbCurrent

# andersonProbCurrent - 140 THE DEVELOPMENT OF WAVE MECHAN...

This preview shows pages 1–2. Sign up to view the full content.

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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

andersonProbCurrent - 140 THE DEVELOPMENT OF WAVE MECHAN...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online