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Ideas of Quantum Chemistry P100

Ideas of Quantum Chemistry P100 shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field. | 956 F. TRANSLATION vs MOMENTUM and ROTATION vs ANGULAR MOMENTUM where p -ifiV is the total momentum operator see Chapter 1 . Thus for translations we have k T and K p. Rotation and angular momentum operator Imagine a function f r of positions in 3D Cartesian space think e.g. about a probability density distribution centred somewhere in space . Now suppose the function is to be rotated about the z axis the unit vector showing its direction is e by an angle a so we have another function let us denote it by U a e f r . What is the relation between f r and U a e f r This is what we want to establish. This relation corresponds to the opposite rotation i.e. by the angle -a see Fig. 2.1 and p. 58 of the coordinate system U a e f r f U-1r f U -a e r where U is a 3 x 3 orthogonal matrix. The new coordinates x a y a z a are expressed by the old coordinates x y z through1 r cos a sin a 0 sin a cos a 0 Therefore the rotated function U a e f r f x a y a z a . The function can be expanded in the Taylor series about a 0 - z . df U a e f r f x a y a z a f x y z ai ---- f X Z a dx a df dy a f dz a df da dx da dy da dz a 0 f x y z a y- - x- f dx dy_ Now instead of the large rotation angle a let us consider first an infinitesimally small rotation by angle e a N where N is a huge natural number. In such a situation we retain only the first two terms in the previous equation Va a N j f r f x y z N d dx xdy f x y z A a ih d y N ih _y dx d IV A . a IV xVf V Nib P y ypx J 1 1A positive value of the rotation angle means an anticlockwise motion within the xy plane x axis horizontal y vertical z axis pointing to us . 2 The Hamiltonian commutes with the total momentum operator 957 If such a rotation is repeated N times we recover the rotation of the function by a possibly large angle a the limit assures that e is infinitesimally small f r Jim 1 - tJz f r N œ N h I U a e f r lim Ü a e N œ N N œ exp -iaJz f exp -ae J f. h h Thus for rotations U a e exp -j ae J and therefore we have k ae and K .