Wavelength of a particle

EXAMPLE 1.6 Sample exercise Calculating the wavelength of a particle... 

In order to appreciate why the wavelike properties of particles had not been noticed, calculate the wavelength of a particle of mass l g traveling at 1 m-s. ... 

The momentum of a particle is the product of its mass and speed, p = mu. Making this substitution and solving for A gives a form of the de Broglie equation that links the wavelength of a particle with its mass and speed ... 

Atomic line spectra arise because electromagnetic radiation occurs only in discrete units, or quanta. Just as light behaves in some respects like a stream of small particles (photons), so electrons and other tiny units of matter behave in some respects like waves. The wavelength of a particle of mass m traveling at a velocity v is given by the de Broglie equation, A = h/mv, where h is Planck s constant. 

Thus light, which was previously thought to be purely wavelike, was found to have certain characteristics of particulate matter. But is the opposite also true That is, does matter that is normally assumed to be particulate exhibit wave properties This question was raised in 1923 by a young French physicist named Louis de Broglie (1892-1987), who derived the following relationship for the wavelength of a particle with momentum, mv ... 

Surfaces are usually ordered on the atomic scale if, during suitable preparation, the atoms are allowed to move to find their equilibrium positions. Thus LEED, X-ray diffraction, and atom diffraction are among the most useful techniques for studies of their structure. The deBroglie wavelength of a particle is given by... 

The wavelength of a particle is the ratio of Planck s constant, and the product of the particle s mass by its frequency. 

Equation 1.19 was derived using equations applicable to the photon, which is massless and has a fixed velocity c. De Broglie postulated that the equation should also apply to particles of matter with mass m and velocity u. Substituting the expression for the momentum of a particle (p = mu) into Equation 1.19 gives the de Broglie relation for the wavelength of a particle ... 

The wavelength defined in Equation 1.20 is called the de Broglie wavelength of a particle. Equation 1.20 implies that a particle in motion can be treated as a wave and that a wave can exhibit the properties of a particle (that is, its momentum). Thus, the left side of Equation 1.20 addresses the wavelike properties of matter (wavelength), whereas the right side addresses its particle-like properties (mass). 

This equation states that the wavelength of a particle is inversely proportional to its momentum, mv, and the proportionality constant is h, Planck s constant. That is, de Broglie s equation implies that a particle of mass m acts as a wave. Only a wave, remember, can have a wavelength. 

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