Sets of Conjugate Variables in Quantum Mechanics

We will start with a review of the usual conjugate variables in quantum mechanics (coordinate x and momentum p) and then introduce the so-called kq representation (in the signal analysis community it's known as the Zak transform). This representation consists of TWO commuting operators (quasimomentum k and quasicoordinate q), which can replace the usual coordinate representation (x) or the momentum representation (p). The kq representation is well adapted to the translational symmetry of a lattice, because the quasicoordinate q is defined with respect to the unit cell of the lattice, and the quasimomentum k is defined with respect to the unit cell of the reciprocal lattice. Then we show how to construct the kq representation in finite dimensional quantum mechanics (Hilbert space of dimension M). Further, we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane, which form sets of conjugate variables. (In such a plane the coordinate x is quantized with a step c, and the momentum p with a step 2*pi/(Mc), where Mc is the size of the phase plane in the x-direction.) The construction depends on the possibility of writing M=M1*M2 with M1 and M2 relatively prime. The eigenstates of the two pairs form mutually unbiased bases. Such bases are useful when one wishes to extract maximal information about a quantum state with a minimal number of experiments.