Quantum Theory and Symmetry

The state vectors ttn) of quantum mechanical systems are associated with Hermitian operators T with eigenvalues tn, such that 

The operator serves to extract information from the state vectors and may correspond to any physical observable such as position, momentum, angular momentum (spin or orbital) or energy. The state vector itself is not observable. In most systems the number of eigenfunctions is infinite and it is an axiom of quantum mechanics that the set of all eigenfunctions of a Hermitian operator forms a complete set. These eigenfunctions define a Hilbert space on which the operator acts. 

A well known operator is the Hamiltonian of an electron in centre-of-mass coordinates of a hydrogen-like atom 

The symmetry of a physical quantum system imposes distinctive regular structures in its associated Hilbert space. These distinctive patterns are determined purely by the group theory of the symmetry and are independent of other details of the system. The wave function ip(x) of the electron in a hydrogen atom centred at the origin, may be considered again as an illustrative example. 

Suppose that the electron is in a 2p state with angular momentum proportional to cos 6 in spherical polar coordinates. The probability density (a ) 2 of such a state would be concentrated near the z-axis, where the length of the radius vector is proportional to cos2 9. Now suppose that the whole system is physically rotated, e.g. by the application of a magnetic field (active rotation) - alternatively the axes may be thought of as rotated in the opposite direction (passive rotation). After rotation the system has a new wave function ip (x) with ip x) 2 concentrated around a displaced axis, but the value of the new wave function at a rotated point must be the same as that of the old wave function at the original point, 

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