position, the probability density for finding the particle at a given momentum depends upon its state, and is given by where, again, . The expectation value for the momentum is then calculated to be zero, and the variance in the momentum is calculated to be: The uncertainties in position and momentum ( and ) are defined as being equal to the square root of their respective variances, so that: This product increases with increasing n, having a minimum value for n=1. The value of this product for n=1 is about equal to 0.568 which obeys the Heisenberg uncertainty principle, which states that the product will be greater than or equal to Another measure of uncertainty in position is the information entropy of the probability distribution Hx: where x0 is an arbitrary reference length. Another measure of uncertainty in momentum is the information entropy of the probability distribution Hp: where γ is Euler's constant. The quantum mechanical entropic uncertainty principle states that for (nats) For , the sum of the position and momentum entropies yields: (nats) which satisfies the quantum entropic uncertainty principle. Energy levels The energies which correspond with each of the permitted wavenumbers may be written as The energy levels increase with , meaning that high energy levels are separated from each other by a greater amount than low energy levels are. The lowest possible energy for the particle (its zero-point energy) is found in state 1, which is given by The particle, therefore, always has a positive energy. This contrasts with classical systems, where the particle can have zero energy by resting motionlessly. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by It can be shown that the uncertainty in the position of the particle is proportional to the width of the box. Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. The kinetic energy of a particle is given by , and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above. Higher-dimensional boxes (Hyper)rectangular walls If a particle is trapped in a two-dimensional box, it may freely move in the and -directions, between barriers separated by lengths and respectively.