Calculate the de Broglie wavelength of a particle accelerated by a potential of ΔΦ. The formula for de Broglie wavelength when considering an accelerated potential is λ = h / √(2meΔΦ), where h represents Planck's constant, m is the mass of the particle, e is the charge of the particle, and ΔΦ is the potential difference.
For an electron, λ can be calculated as follows:
λ = 6.626 × 10^-34 / √(2 × 9.1 × 10^-31 × 1.6 × 10^-19 × ΔΦ)
This expression provides the wavelength of the particle when subjected to varying potentials. It is experimental evidence that electrons exhibit wave-particle duality.
Now, let's delve into the theoretical evidence. One of the major drawbacks of many theories is their inability to explain the mathematical conditions for the quantization of angular momentum. According to these theories, a stationary wave must be associated with the movement of an electron in any orbit. For this to occur, the circumference of the orbit must be an integral multiple of the wavelength of the stationary wave. Mathematically, this condition is expressed as 2πr = nλ, where r is the radius of the orbit, n is an integer, and λ is the wavelength. Additionally, angular momentum quantization is expressed as mvr = nh / 2π, where m is the mass of the electron, v is its velocity, r is the radius of the orbit, and n is an integer. This quantization principle elucidates the nature of electronic particles and their properties, explaining them through the wave-particle duality phenomenon. However, it also introduces limitations, such as the inability to precisely determine the exact position of the electron, as described by Heisenberg's uncertainty principle and the Schrödinger equation.
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