Wavelength linear momentum

The kinetic energy of a particle of mass m is related to its speed, v, by F.K = rmr. We can relate this energy to the wavelength of the particle by noting that the linear momentum is p = mv and then using the de Broglie relation (Eq. 7) ... 

Theory for block copolymer rheology is still in its infancy. There are no models that can predict the rheological behaviour of a block copolymer from microscopic parameters. Fredrickson and Helfand (1988) considered fluctuation effects on the low frequency linear viscoelastic properties of block copolymers in the disordered melt near the ODT. They found that long-wavelength transverse momentum fluctuations couple only to compositional order parameter fluctua-... 

The observation that the wavelength of light is linked to the particle-like momentum of a photon prompted de Broglie to postulate the likelihood of an inverse situation whereby particulate objects may exhibit wave-like properties. Hence, an electron with linear momentum p could under appropriate conditions exhibit a wavelength A = h/p. The demonstration that an electron beam was diffracted by periodic crystals in exactly the same way as X-radiation confirmed de Broglie s postulate and provided an alternative description of the electronic stationary states on an atom. Instead of an accelerated particle the orbiting electron could be described as a standing wave. To avoid self-destruction by wave interference it is necessary to assume an... 

The constant Ac = 2.425 pm is called the Compton wavelength of the electron. The wavelength shift given by Eq. (1.10) can be easily reproduced theoretically if the interaction between the radiation and the electron is considered as a collision between two particles in which the energy and the linear momentum are conserved (conservation of momentum in the incident direction and in the direction perpendicular to it). These particles are a photon of energy hv and linear momentum p = hvlc=hlX and a stationary electron of mass m which acquires velocity v (Fig. 1.2). It is then found... 

Fig. 1.6 A small indeterminacy in x as the result of appropriate superposition of waves having a variety of wavelengths information about linear momentum p = hj is entirely lost (see, for example, ref. 12).

Comparison of de Broglie s equation (see Eq. 4.20) with Bohr s equation (see Eq. 4.22) shows that the wavelength of the standing wave is related to the linear momentum, f, of the electron by the following simple formula ... 

De Broglie used the theory of relativity to show that exactly the same relationship holds between the wavelength and momentum of a photon. De Broglie therefore proposed as a generalization that any particle—no matter how large or small—moving with linear momentum p has wavelike properties and a wavelength of A = hjp associated with its motion. 

L. de Broglie (1924) suggested that it might be useful to associate wave-like behavior with the motion of a particle. He postulated that a particle with linear momentum p be associated with a wave whose wavelength A is given by... 

Alternatively, Einstein s relationship relates the wavelength of light A to the linear momentum p according to X = h/p (where h = 6.626 x 10 [Ls] is the Planck constant). De Broglie applied this to the material wave of the mass m to obtain... 

The linear momentum can be rewritten in terms of the de Broglie wavelength for this motion,... 

We have seen that, according to the de Broglie relation, a wave of constant wavelength, the wavefunction sin(2nx/X), corresponds to a particle with a definite linear momentum p = h/L However, a wave does not have a definite location at a single point in space, so we cannot speak of the precise position of the particle if it has a definite momentum. Indeed, because a sine wave spreads throughout the whole of space, we cannot say anything about the location of the particle because the wave spreads everywhere, the particle maybe found anywhere in the whole of space. This statement is one half of the uncertainty principle, proposed by Werner Heisenberg in 1927, in one of the most celebrated results of quantum mechanics ... 

An important additional conclusion is that the angular momentum of a particle is quantized. We can use the relation between angular momentum and linear momentum (angular momentum J=pr), and between linear momentum and the allowed wavelengths of the particle (A = Inrlmf to conclude that the angular momentum of a particle around the z-axis is confined to the values... 

Calculate the linear momentum per photon, energy per photon, and the energy per mole of photons for radiation of wavelength... 

Let us find the de Broglie wavelength for an object from our day-to-day world, a bowling ball rolling at 36 km h X If the mass of the ball is 8 kg, then the linear momentum of this "particle" is... 

Substituting this linear momentum into the de Broglie relation yields the wavelength ... 

The starting point of a new quantum theory of atoms was the hypothesis put forward by de Broglie (1924) that a particle moving with linear momentum p has associated with it a wave whose wavelength is given by... 

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