through the opening to a one dimensional system. Moreover, it requires a quantum mechanical description of the system that results in the quantisation of conductance. Quantum mechanically, the current through the point contact is equipartitioned among the 1D subands, or transverse modes, in the constriction. It is important to state that the previous discussion does not take into account possible transitions among modes. The Landauer formula can actually be generalized to express this possible transitions , where is the transition matrix which incorporates non-zero probabilities of transmission from mode n to m. At low temperatures and voltages, unscattered and untrapped electrons contributing to the current have a certain energy/momentum/wavelength called Fermi energy/momentum/wavelength. Much like in a waveguide, the transverse confinement in the quantum point contact results in a "quantization" of the transverse motion—the transverse motion cannot vary continuously, but has to be one of a series of discrete modes. The waveguide analogy is applicable as long as coherence is not lost through scattering, e.g., by a defect or trapping site. The electron wave can only pass through the constriction if it interferes constructively, which for a given width of constriction, only happens for a certain number of modes . The current carried by such a quantum state is the product of the velocity times the electron density. These two quantities by themselves differ from one mode to the other, but their product is mode independent. As a consequence, each state contributes the same amount of per spin direction to the total conductance . This is a fundamental result; the conductance does not take on arbitrary values but is quantized in multiples of the conductance quantum , which is expressed through the electron charge and the Planck constant . The integer number is determined by the width of the point contact and roughly equals the width divided by half the electron wavelength. As a function of the width of the point contact (or gate voltage in the case of GaAs/AlGaAs heterostructure devices), the conductance shows a staircase behavior as more and more modes (or channels) contribute to the electron transport. The step-height is given by . On increasing the temperature, one finds experimentally that the plateaux acquire a finite slope until they are no longer resolved. This is a consequence of the thermal smearing of the Fermi-Dirac distribution. The conductance steps should disappear for (here ∆E is the subband splitting at