 the old quantum theory is a collection of results from the years 1900â1925 which predate modern quantum mechanics. the theory was never complete or self-consistent, but was a collection of heuristic prescriptions which are now understood to be the first quantum corrections to classical mechanics. the bohr model was the focus of study, and arnold sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called space quantization (richtungsquantelung). this allowed the orbits of the electron to be ellipses instead of circles, and introduced the concept of quantum degeneracy. the theory would have correctly explained the zeeman effect, except for the issue of electron spin. the main tool was bohrâsommerfeld quantization, a procedure for selecting out certain discrete set of states of a classical integrable motion as allowed states. these are like the allowed orbits of the bohr model of the atom; the system can only be in one of these states and not in any states in between. == basic principles == the basic idea of the old quantum theory is that the motion in an atomic system is quantized, or discrete. the system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition: : \oint\limits_{h(p,q)=e} p_i \, dq_i = n_i h where the p_i are the momenta of the system and the q_i are the corresponding coordinates. the quantum numbers n_i are integers and the integral is taken over one period of the motion at constant energy (as described by the hamiltonian). the integral is an area in phase space, which is a quantity called the action and is quantized in units of planck's constant. for this reason, planck's constant was often called the quantum of action. in order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates q_i in terms of which the motion is periodic. the periods of the different motions do not have to be the same, they can even be incommensurate, but there must be a set of coordinates where the motion decomposes in a multi-periodic way. the motivation for the old quantum condition was the correspondence principle, complemented by the physical observation that the quantities which are quantized must be adiabatic invariants. given planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a gene
History_of_physics Quantum_mechanics