Heisenberg uncertainty principle Volume

The quotient NdN2 is the ratio of the probabilities of finding the electron in the infinitesimally small volume elements dv around points 1 and 2. For example, if the value of the ratio N,/N2 is 100, the electron is 100 times more likely to be found at position 1 than at position 2. The model gives no information concerning when the electron will be at either position or how it moves between the positions. This vagueness is consistent with the concept of the Heisenberg uncertainty principle. 

It might seem, at first sight, that for the distribution functions both in direct space and in momentum spaee to become sharply localized in the high-i limit constitutes a violation of the Heisenberg uncertainty principle. However, we must remember that both r and k are scaled coordinates, and that they are not canonically conjugate to each other. The reader may easily verify that if the distribution functions are plotted as functions of R and k, then as D becomes large, the volume of phase space in which both functions are appreciable is independent of D. 

With the advent of quantum theory, the exactness of these premises was disturbed (by the Heisenberg uncertainty principle). In the quantum statistics that evolved as a result, the phase space is divided into cells, each having a volume hf, where h is the Planck constant and /is the number of degrees of freedom of the particles. This new concept led to Bose-Einstein statistics, and for particles obeying the Pauli exclusion principle, to Fermi-Dirac statistics. 

In the consideration of the momentum of a large number of particles restricted to a volume V, it is often convenient to describe the system by an assembly of points in a momentum diagram (Fig. 1). The length OA represents the magnitude of momentum of the particle A, and its direction is OA. The application of Heisenberg s uncertainty principle leads to the... 

In quantum mechanics, Heisenberg s uncertainty principle states that there is a limit to which we can know the product of the uncertainties in a coordinate and its corresponding momentum, AxApx. Thus, even in quantum mechanics, there is a minimum volume in phase space in which we can localize a particle. 

If an electron has wave-like properties, there is an important and difficult consequence it becomes impossible to know exactly both the momentum and position of the electron at the same instant in time. This is a statement of Heisenberg s uncertainty principle. In order to get around this problem, rather than trying to define its exact position and momentum, we use the probability of finding the electron in a given volume of space. The probability of finding an electron at a given point in space is determined from the function ij where is a mathematical function which describes the behaviour of an electron-wave ip is the wavefunction. 

The movement of the atom is defined by its momenta and spatial coordinates. These define what is known as the phase space. The minimum volume per representative point in phase space equals as follows from Heisenberg s uncertainty principle dx dpx h. The normalized space density element becomes... 

In view of Heisenberg s uncertainty principle, there is a minimum volume /z in phase space that may be associated with a single particle. This minimum volume is called a phase cell. In the phase space of N particles, the volume of a phase cell is. Thus, is a natural unit of volume... 

The factor 1 /(27tA) in the integral measure sets the scale of the infinitesimal phase-space volume dp dx for each phase-space component. The volume element has the physical dimension of an action (measured in units of Js) and represents the smallest phase-space volume in which a single particle state can reside. This is a consequence of Heisenberg s uncertainty principle. 

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