Final answer:
The quantization of energies in an electron in a wire is caused by the wave nature of matter. The restriction on the wavelengths the electron can have leads to quantization of its energies. The quantized energies can be determined using the de Broglie wavelength and derived using the wave equation for the system.
Step-by-step explanation:
The quantization of the possible energies an electron can have in a wire is caused by the wave nature of matter. When a particle is confined or bound to a small space, its allowed wavelengths are those which fit into that space. In the case of electrons in a wire, the restriction on the wavelengths leads to a quantization of the energies they can have.
To determine these quantized energies, we can use the relationship between the wavelength and energy of a particle, given by the de Broglie equation: λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum. Since the wavelength is quantized, we can substitute the quantized wavelength into the equation to find the quantized energy. The final expression for the quantized energies can be derived using the wave equation for the system, and it will depend on the length of the wire and the boundary conditions.