Download Exercises in Quantum Mechanics: A Collection of Illustrative by H.A. Mavromatis PDF

By H.A. Mavromatis

This monograph is written in the framework of the quantum mechanical paradigm. it truly is modest in scope in that it's constrained to a few observations and solved illustrative difficulties now not available in any of the various typical (and a number of very good) texts or books with solved difficulties which have been written in this topic. also a number of roughly normal difficulties are integrated for continuity and reasons of comparability. The wish is that the issues made and difficulties solved will provide the scholar a few extra insights and a greater seize of this attention-grabbing yet mathematically just a little concerned department of physics. The hundred and fourteen difficulties mentioned have deliberately been selected to contain not less than technical complexity whereas nonetheless illustrating the results of the quantum-mechanical formalism. touching on notation, helpful expressions are displayed in oblong bins whereas calculational info which one might need to pass are integrated in sq. brackets. Beirut HARRY A. MAVROMATIS June, 1985 IX Preface to moment version greater than 5 years have handed considering that I ready the 1st version of this mono­ graph. the current revised version is extra beautiful in structure than its predecessor, and such a lot, if no longer the entire error within the unique version (many of which have been kindly mentioned through reviewers, colleagues, and scholars) have now been corrected. also the cloth within the unique fourteen chapters has been prolonged with major additions to Chapters eight, thirteen, and 14.

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Extra info for Exercises in Quantum Mechanics: A Collection of Illustrative Problems and Their Solutions

Example text

8) and for t limG(x,x',t) t-+O ----+ ° 6(x - x'), (Eq. 13) having in this case the delta function form of Eq. 15» since for t = 0, 111 (x, 0) = ----+ as it must JW(x',O)G(x,x',O) dx'. A useful formula in calculating reflection and transmission times for wavepackets, (which is a result independent of the details of A(p') but assumes it is peaked about p' = p) involves expanding E(p') about p = p' in Eq. 2) , p,2 p2 E(p) = 2m = 2m dEep') + ~p' I ' p'=p (p - p) + .... e. assuming only values about p' = p are important) yields: 1I1(x, t) = = X _1_ V'2(ff £,7f fP'=oo Jp~'=-oo _1 exp ~ I I A(p') exp {~ (p' x _ L t + dE(~') tp _ dEep') t p')} dp' h 2m d p ,p =p dp',p =p h {_~ h (Lt _ 2m dEep') dp' I t p)} p'=p 1~:~: A(p') exp {*p' (x - d~(~') I p p p'=p Thus W(x, t) = exp (i

V(x) = Ax2, -00 < x < 00. , n= 1, 2, 3 ... , and correspond to odd-parity solutions (see Eq. 12) ), while the latter's allowed energies are En = (n - ~) htf. ' n= 1, 2, 3 ... , and both odd and even parity solutions are allowed. One can thus obtain the solutions in the latter case from the solutions in the former by letting n -+ n/2. e. 44 Chapter 3 whereas if one considers the infinite well V(x) = 00, x < -a, x > a, V(x) = 0, -a < x < a, its solutions are tf;{x) = V1/asinkx (odd parity), and tf;{x) = V1/a cos kx (even parity), where ka = mr or (n -1/2)11", 2 E = h k 2 = ~ (~)2 11"2, 2 2m 2m = 1, 2, 3...

Edefined as (-1 + 8mVo ) 1 + a2 h2 , e== tanh ax ). Thus in this case as well one obtains the solutions for the symmetric well by letting n -+ n/2 in the solutions for the well which extends only from 0 < x < 00. Momentum Space 45 The solutions for Example 3 are (n - D2/3 (A~2r/3 (2~r/3 En = (2n - ~r/3 (A~2) 1/3 (4~) 2/3, n= 1, 2, 3.... 5. 31 ) an expression which for large n, n rv (n - 1/2) agrees with the Wilson-Sommerfeld result (Eq. 13) ). 6 is just n = 1, 2, 3... 32) and that this includes both even- and odd-parity solutions.

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