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# Quantum Physics PHY 344

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This 5 page Class Notes was uploaded by Miss Lucienne Hamill on Wednesday October 28, 2015. The Class Notes belongs to PHY 344 at Wake Forest University taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/230738/phy-344-wake-forest-university in Physics 2 at Wake Forest University.

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Date Created: 10/28/15

April 29 2002 Scattering amplitudes and phase shifts 2 2 Suppose we have a particle with momentum hk and energy E 2 H27 Apart from normal ization its wavefunction is given by you 01quot 1 If this particle is scattered far from the tar et the wavefunction take the form 8 0119 WI 0 f 97 2 739 where the scattering amplitude 9 will be discussed further below The angle 9 measures the angle of the scattered particle relative to its incident direction It turns out that scattering cross section for this process can be determined in terms of the scattering amplitude according to do 7 z 9 2 3 m f gt lt gt As we will see it is convenient to determine f9 in terms of scattering phase shifts 6 for each angular momentum quantum number l 1 0 9 Z2l1e 5 si116lPlcos 9 4 10 In this expression Bcos 9 denotes a Legendre polynomial The total diffential cross section can also be analyzed by integrating over all angles Because of the orthogonal properties of the Legendre polynomials the result simpli es do 47r 0 2 v U fd d 162 02H 1sin 6 a The scattering equations can be directly derived from a Green39s function analysis A simplier argument goes as follows First we note that a plane wave can be expressed in spherical polar coordinates centered at the target as a some of partial wave contributions XI e11 Z2l1i jlkrPlcos9 6 0 Here jlkr denotes a spherical Bessel function 4 sinkr 4 sinkr coskr H j0kr 7 j1kr 1602 7kquot I which has the asymptotic form r i k jlkrk1 Sln T 2 8 kquot Now consider a general solution to the Schrodinger equation for the particle in the potential of the target Outside the range R of the potential the Schrodinger equation is that of a free particle in spherical polar coordinates with the general solution form 1139 Acos 63quotkr sin 6mkrPcos 9 9 Here the amplitude A and the phase shift 6 are to be determined The function mUcr denotes a Neumann function 4 coskr 4 coskr sinkr 100 7m m M 2 M 10 which has the asymptotic form k 3 mk7krgtgt1 m 11 kquot The phase shift 6 depends on the form of the target potential while the amplitude is determined by requiring that the wave function take the asympotic form Evaluating the asymptotic form of the general solution 9 1 00 k 5 Prkgt1ZAls1n T 2 01340039 12 0 Or k 1 k 1 I x 391 r 7 1 7 1 r 7 1 1111 12A 2 M 2 Pcos9 13 0 From this expression we note that if we choose the amplitude A 2l1i e 14 then the expansion becomes 1 x vikr Ll261 fan Ll 1111 1291 1 1ilacos 9 15 0 21kquot which can be rearranged to from I x k7quot 0216 1 1 Cik r 11 21 21 1 r Sm 2 if P y 1 r g 1 M 2 it T 1cos9 6 Or 2 6 k 00 H 1 f 1 1111 1 0 Z211Wwaces9 CT 17 10 which determines the form of the scattering amplitude f9 according to Eq January 31 2002 Summary of angular momentum formalisms Coordinate representation of orbital angular momentum In spherical polar coordinates the operator representing the squared angular momentum L2 takes the form 1 8 8 1 82 L2 2 7 7 a 1 a sin98981n989sin298d239 l while the operator representing z component of angular momentum takes the form 8 La l7l 2 The spherical harmonic functions Ylm are eigenfunctions of both L2 and La with L2i2m ma 1 3 and LzYlm 7am Some of these spherical harmonic functions are r 1 no E a Ylil 1H i sin Qeim 7 In the process of evaluating the differential eigenvalue equations we nd that the quantum numbers l must be postive integers l 0 1 2 and m is restricted to the integer values between l 3 m g l Operator representation of general angular momentum The following derivation follows the discussion of Shankar s text Principlcs of Quantum Mcchanics 2nd edition Chapter 12 It turns out that a very similar eigenvalue structure can be derived in an operator formalism In this operator formalism we will see that additional halfinteger solutions for the angular momentum quantum numbers are also possible For this generalization we will use J2 and J3 to represent the square and 2 components of the angular momentum respectively Furthermore we will assume that we can nd the eigenvalues of these operators which we will denote by a and b for the moment J2 ab a b 8 Julia Mb 9 We can now introduce 2 other operators which will prove to be very helpful Ji E Jfr i Jy 10 We can show that these operators have the effect of incrementing or decrementing the b eigenvalue of gab by one First we note the following commutation relations lng Jil ink 11 and W J11 0 12 Later we will also need to use the result M A 47ng 13 which follows from the identity Jw Jy 17212 14 We can then show that the function 75 has eigenvalues a and b i 7 of J2 and J3 respectively Acting on 75 with J2 J2Ji ab JiJ2 ab Jimab aJi ab 15 Acting on 75 with J3 J3Ji ab ihgab Jingab ihgab Jibgab i7 bJi ab 16 This mean that we can write the function Ji ab N abi7z where N is a normalization constant determined from N2ab i 7z ab i ab ngab ab J2 J31 RJz ab a b2 1 ab 17 assuming that abggab 1 This result means that NZVa b23137lb 18 In order to make further progress we notice that since the normalization cannot be negative for a given value of a there are restrictions on the value of b In particular we can safely assume that there is a maximum value of b which we will denote by bmax From the behavior of a maximum value we know that gnaw 0 19 Now multiplying the above equation by L we nd JJ abmax 0 J J mm Jy abmax J2 J3 ma gabm a 53M mm 20 This de nes the eigenvalue 1 in terms of bmax to be a bmaxbmax We can also use Eq 18 to argue that b has a minimum value bmin and analyzing the properties of bmin using similar steps above we can also Show that a bminbmin 7i 22 Comparing Eqs 21 and 22 it is apparent that bmin bmax 23 It is now convenient to de ne bmax E M so that the eigenvalue 1 can be written a mn 9 This analysis then suggests that if we de ne a general value of the eigenvalue 6 to take the form 6 E 73m 25 the results tell us that mj can take the values j 3 m g j 23 1 different values in all for a given With these de nitions the normalized increment or decrement operation can be written 7537737 7 JJ1 m mj i Ui mj i1 26 This structure of the eigenvalues jmj is very similar to the eigenvalues of orbital angular moment lm There is one new wrinkle however The above arguments tell us that we can get from the maximum value of my j to the minimum value my j in a number of applications of the operator J Suppose that that number of applications is U This means that the sequence of values of the eigenvalue m is 333 133 23nj Ug 27 so that j U j as or U i 29 J 2 Since U must be an integer j can be an integer if U is even but can also be a halfinteger if U is oddll This means that we can use this formalism to describe orbital spin and total angular momentum

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