By Simons J.
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Extra resources for The basic tools of quantum mechanics
Suggested Extra Reading- Appendix C: Quantum Mechanical Operators and Commutation] The structure of quantum mechanics (QM) relates the wavefunction Ψ and operators F to the 'real world' in which experimental measurements are performed through a set of rules (Dirac's text is an excellent source of reading concerning the historical development of these fundamentals). Some of these rules have already been introduced above. Here, they are presented in total as follows: 1. The time evolution of the wavefunction Ψ is determined by solving the time-dependent Schrödinger equation (see pp 23-25 of EWK for a rationalization of how the Schrödinger equation arises from the classical equation governing waves, Einstein's E=hν, and deBroglie's postulate that λ=h/p) ih∂Ψ/∂t = HΨ, where H is the Hamiltonian operator corresponding to the total (kinetic plus potential) energy of the system.
Example: The px , p y and pz orbitals are eigenfunctions of the L 2 angular momentum operator with eigenvalues equal to L(L+1) h2 = 2 h2. Since L 2 and L z commute and act on the same (angle) coordinates, they possess a complete set of simultaneous eigenfunctions. Although the px , p y and pz orbitals are not eigenfunctions of L z , they can be combined to form three new orbitals: p0 = pz, p1= 2-1/2 [px + i py ], and p-1= 2-1/2 [px - i py ] that are still eigenfunctions of L 2 but are now eigenfunctions of L z also (with eigenvalues 0h, 1h, and -1h, respectively).
These eigenvalues are obtained by solving Fφj =fj φj, where the φj are the eigenfunctions of F. Once the measurement of F is made, for that subpopulation of the experimental sample found to have the particular eigenvalue fj, the wavefunction becomes φj. , by spectroscopic means). The reader is strongly urged to also study Appendix C to gain a more detailed and illustrated treatment of this and subsequent rules of quantum mechanics. 3. , functions of the variables on which F operates). As expressed through equality of the first and third elements above, Hermitian operators are often said to 'obey the turn-over rule'.
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