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Basic Theoretical Physics: A Concise Overview P23. This concise treatment embraces, in four parts, all the main aspects of theoretical physics (I . Mechanics and Basic Relativity, II. Electrodynamics and Aspects of Optics, III. Non-relativistic Quantum Mechanics, IV. Thermodynamics and Statistical Physics). It summarizes the material that every graduate student, physicist working in industry, or physics teacher should master during his or her degree course. It thus serves both as an excellent revision and preparation tool, and as a convenient reference source, covering the whole of theoretical physics. It may also be successfully employed to deepen its readers’ insight and. | 25.1 Bound Systems in a Box Quantum Well Parity 225 Next we assume E 0 i.e. for bound states . Then we have the following solutions of the Schrodinger equation u x B exp k x B 1 exp x x a 25.3 u x C1 cos k x C2 sin k x x a 25.4 u x B exp kx B 2 exp x x a . 25.5 a Firstly we recognise that the coefficents B 1 and B 2 must vanish since otherwise it would not be possible to satisfy the condition Zœ dx u x 2 1 . -œ b The remaining coefficients are determined apart from a common factor where only the magnitude is fixed by the normalization condition from the continuity conditions for u and u at the potential steps x a. The calculation is thus much easier for symmetrical potentials V x V x since then all solutions can be divided into two different classes even parity i.e. u x u x B-1 B-2 C2 0 and odd parity i.e. u x u x B-1 B-2 C 0 . One then only needs one continuity condition i.e. the one for u u at x a. From this condition see below one also obtains the discrete energy values E En for which continuity is possible for k 0 and k 0 k E k E k E k E tan k E a for even parity tan k E a for odd parity . 25.6 25.7 These equations can be solved graphically this is a typical exercise by plotting all branches of tan k-a as a function of k-a these branches intercept the x-axis at k a n n where n is integer and afterwards they diverge to I to from to at k a 2n 1 n 2 0 . Then one can determine the intersections of this multi-branched curve with the line obtained by plotting the l.h.s. of 25.6 or 25.7 as a function of k E a. In this way one obtains the following general statements which are true for a whole class of similar problems Existence I There is always at least one bound state. This statement is true for similar problems in one and two dimensions but not in three dimensions 1. 1 In d 3 dimensions one can show see below that for so-called s-states i.e. if the state does not depend on the angular coordinates h and p the wave-function 226 25 One-dimensional Problems in Quantum
