Uncertainty relation coordinate-momentum

The second view is based on the uncertainty relation between coordinate and momentum. In the region of a classically allowed barrier, the de Broglie wavelength of an electron is... 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

The widths of the coordinate and the momentum distribution are inversely related to each other as required by the uncertainty relation. The wider the coordinate distribution the narrower is the momentum distribution and vice versa. Figure 5.2 depicts two examples. 

The final state density can be derived firom Heisenberg s uncertainty relation. The space coordinates shall be denoted by x, y, z, and the momentum coordinates by p, py, p. Then, according to the uncertainty relation (in the form used by Fermi (1934))... 

In 1927 Werner Heisenberg showed from quantum mechanics that it is impossible to know simultaneously, with absolute precision, both the position and the momentum of a particle such as an electron. Heisenberg s uncertainty principle is a relation that states that the product of the uncertainty in position and the uncertainty in momentum of a particle can be no smaller than Planck s constant divided by 4tt. Thus, letting Ax be the uncertainty in the x coordinate of the particle and letting be the uncertainty in the momentum in the x direction, we have... 

The time-energy uncertainty relation is even more mysterious than that of position and momentum, but it has been verified by many experiments. Time is not a mechanical variable that can be expressed in terms of coordinates and momenta and does not correspond to any quantum mechanical operator, so we cannot calculate a standard deviation of a time. The standard interpretation of the time-energy uncertainty relation is that if At is the time during which the system is known to be in a given state (the lifetime of the state) then there is a minimum uncertainty A in the energy of the system that is given by... 

These facts are given general expression by the Heisenberg uncertainty principle, which we may state in the form the product of the uncertainty in a coordinate and the uncertainty in the conjugate momentum is at least as large as h/4n. (By the conjugate momentum of a coordinate we mean the component of momentum along that coordinate.) In Cartesian coordinates we can state the uncertainty principle by the relations... 

The matter may be regarded from the point of view of the uncertainty principle. The behaviour of particles which is defined by the wave equation is equivalent to an indefiniteness in what may be known of their dynamical coordinates, lip and q are the momentum and position coordinates, Heisenberg s principle states that both cannot be known simultaneously except with a range of uncertainty given by the relation ApAg = h, approximately. In the temperature... 

Solution The momentum and coordinate uncertainties are related by eq. (7.2.1). Let the atom have a linear size I, then the electron will be somewhere in the limits Ax=ll2. The uncertainty principle can then be written as ... 

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