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In the second place, its projection on an axis in space (the quantization axis, usually taken as the z-axis) is quantized, it can only take the values Hence the endpoint of the angular momentum j (the blue arrow in the figure) lies on the surface of a sphere of radius. In the first place the length of the angular momentum is quantized, it can only take the discrete valuesĪnd no other values. In quantum theory this is different.Ĭonsider a quantum system with well-defined angular momentum j, for instance an electron orbiting a nucleus. The angular velocity of the wheel and the direction of the axle are both continuously changeable-in arbitrarily small steps. The length of its angular momentum is proportional to its angular velocity (number of revolutions per unit time) and the direction of its angular momentum is along its axle. In classical mechanics the angular momentum of a body is a vector that can have any length and any direction. It is an indispensable discipline for the working physicist, irrespective of his field of specialization, be it solid state physics, molecular-, atomic,- nuclear,- or even hadronic-structure physics. To date the theory of angular momentum is of great importance in quantum mechanics. When in 1926 electron spin was discovered and Pauli proved less than a year later that spin was a form of angular momentum, its importance rose even further. In 1927, Wolfgang Pauli introduced spin angular momentum, which is a form of angular momentum without a classical counterpart.Īngular momentum theory-together with its connection to group theory- brought order to a bewildering number of spectroscopic observations in atomic spectroscopy, see, for instance, Wigner's seminal work. In this paper the orbital angular momentum and its eigenstates are already fully covered by the algebraic techniques of commutation relations and step up/down operators that will be treated in the present article. This operator is the quantum analogue of the classical angular momentum vector.Īngular momentum entered quantum mechanics in one of the very first-and most important-papers on the "new" quantum mechanics, the Dreimännerarbeit (three men's work) of Born, Heisenberg and Jordan (1926). In quantum mechanics, angular momentum is a vector operator of which the three components have well-defined commutation relations.