The de broglie wavelength of a non-relativistic particle is given by?
Question: The de broglie wavelength of a non-relativistic particle is given by?
The de Broglie wavelength is a concept that relates the wave and particle nature of matter. It was proposed by Louis de Broglie in 1924, who suggested that any particle with momentum p has an associated wavelength λ given by:
$$\lambda = \frac{h}{p}$$
where h is Planck's constant. This equation implies that matter can behave like a wave, and that the wavelength of the wave depends on the momentum of the particle.
But how do we calculate the momentum of a non-relativistic particle? A non-relativistic particle is one that moves at speeds much lower than the speed of light c, so that we can neglect the effects of special relativity. In this case, the momentum of a particle is simply equal to its mass m multiplied by its velocity v:
$$p = mv$$
Therefore, the de Broglie wavelength of a non-relativistic particle is:
$$\lambda = \frac{h}{mv}$$
This formula shows that the de Broglie wavelength is inversely proportional to the mass and velocity of the particle. For particles moving at a given velocity, the larger the mass is, the smaller the wavelength is. For example, an electron with mass 9.11 x 10^-31 kg and velocity 2 x 10^6 m/s has a de Broglie wavelength of about 3.6 x 10^-10 m, which is comparable to the size of an atom. On the other hand, a baseball with mass 0.145 kg and velocity 40 m/s has a de Broglie wavelength of about 2.9 x 10^-34 m, which is much smaller than any observable length scale.
The de Broglie wavelength formula can be used to calculate the wavelength of any non-relativistic particle, such as atoms, molecules, or even macroscopic objects. However, it is important to note that the wave nature of matter becomes more significant when the de Broglie wavelength is comparable to or larger than the size of the object or its surroundings. For most macroscopic objects, their de Broglie wavelengths are too small to be detected by any experiment, so they behave like classical particles. For microscopic objects, such as electrons or atoms, their de Broglie wavelengths can be large enough to exhibit wave-like phenomena, such as interference or diffraction.
The de Broglie wavelength concept is one of the foundations of quantum mechanics, which describes the behavior of matter at the atomic and subatomic scales. It reveals that matter has both wave and particle aspects, and that these aspects are complementary rather than contradictory.
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