Broglie wave length

Matter is classically particulate in nature, but it also manifests wave character. The wave property of matter is related to its particle nature by de Broglie s relation A = hip, where A is known as the de Broglie wave length. 

A nonrelativistic particle is moving five times as fast as a proton. The ratio of their de Broglie wave lengths is 10. Calculate the mass of the particle. 

It is however possible to discuss several special cases analytically. The zero temperature correlation length can still be observed as long as this is smaller than the thermal de Broglie wave length At which can be rewritten for K not too close to Ku as t < f/y Kt(, KtK" 1 with tu Lpl, where we defined tk via = jj-, analogously to the definition of At, and used (30). We call this domain the quantum disordered region. 

Fig. 5. The low temperature crossover diagram of a one-dimensional CDW. t and K are proportional to the temperature and the strength of quantum fluctuations, respectively. The amount of disorder corresponds to a reduced temperature tu 0.1. In the classical and quantum disordered region, respectively, essentially the t = 0 behavior is seen. The straight dashed line separating them corresponds to At 1, i.e., K 1, where At is the de Broglie wave length. In the quantum critical region, the correlation length is given by At- Pinning (localization) occurs only for t = 0, K

The real measure for the breadth of the potential barrier is the De Broglie wave length, so that the actual breadth of the barrier is, for the same geometrical dimensions, larger for deuterons (nuclei of heavy hydrogen) than for protons. 

The mass of the electron determines the de Broglie wave-length A = hjinv and so fixes the scale of the quantum phenomena in which it participates. It is because the electron is of small mass that the atom has a radius very many times greater than the nucleus. The electrons being remote, the nucleus may be treated as a point charge with a high degree of approximation, and most of chemistry thus becomes an affair of the electron patterns alone. 

On the femtosecond time scale, an entirely new domain emerges. First, a wave packet can be prepared, as the temporal resolution is sufficiently short to "freeze" the nuclei at a given intemuclear separation. Put in another way, the time resolution is much shorter than the vibrational (and rotational) motions such that the wave packet is prepared, highly localized with a de Broglie wave length of 0.1 A, with the structure frozen. Second, this synthesis is not in violation of the... 

At medium energies, where the projectile speed approaches that of light, relativistic kinematics are introduced into the Schrodinger equation in order to produce the correct de Broglie wave length and relativistic density of states. According to the traditional view, however, the use of relativistic... 

Applications of multiple scattering theory are usually made for intermediate energy projectiles with speeds comparable to that of light, c. Thus, it is necessary to adjust the kinematics of the nonrelativistic equation of motion in order to reproduce the correct de Broglie wave length and density of final states. For the Schrodinger equation (eq. 2.25)... 

The full quantum nuclear wavepacket to be compared as a reference is a Gaussian function initially prepared at a position Rq, with the initial wave number ko. The width is set to = 10//co so as to be in the same order of its corresponding de Broglie wave length. The explicit form is... 

For liquefied normal hydrogen, at a temperature of Z0.4 K, the ratio of the mean de Broglie wave length. A,to the... 

Let us consider more closely the physical significance of A. To a particle of momentum f there corresponds a wave length (de Broglie wave length)... 

Thus, the quantity A is the de Broglie wave-length of the relative motion of two molecules with relative kinetic energy e divided by the geometrical scale factor a. We may notice that the wave character of the particles as expressed by the value of A will be especially important for light molecules (small m) and weak interactions (small b ). For this reason (see Table 18.1.1). 4 is espedaTy important for helium and to a lesser extent for hydrc en. 

The classical case corresponds to the vanishing de Broglie wave length. Formula (18.1.2) permits to visualize quite easily the factors which determine quantum effects. For heavy particles A is small and we may n lect the quantum effects. Also an increase in the interaction parameter e lowers A. If we consider for example the rare gases, both factors work in the same direction because with increasing number of electrons there is an increase in the polarisability of the molecule and consequently of e. For this reason the value of A drops rapidly if we go from He to A (cf. Table 18.1.1). Table 18.1.1 gives the values of the quantum mechanical parameter a for some atoms and molecules. 

For sufficiently h%h temperatures, the thermod5mamic properties may be expanded in powers of Planck s constant h, or still better in powers of the reduced de Broglie wave length (18.1.2) (Kirkwood [1931], Uhlenbeck and Beth [1936], Slater [1931], De Boer and Bird, Ch. VI of the book by Hirschfelder, Curtiss and Bird [1954]). 

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