Final answer:
The de Broglie wavelength of a free electron can be calculated using its wave function and the formula λ = h/p. The momentum of the electron can be calculated using the formula p = 2.10 * 10^11 * h. The kinetic energy of the electron can be calculated using the formula KE = p^2 / (2m) and converted to eV.
Step-by-step explanation:
(a) The de Broglie wavelength (λ) is related to the momentum (p) of an object by the equation λ = h/p, where h is Planck's constant. In this case, we can calculate the de Broglie wavelength of the electron using the given wave function. The wave function can be written as ψ(x) = A * e^(i * 2.10 * 10^11 * x), where A is a constant. Comparing this to the general form of the wave function, we can say that the momentum (p) is given by p = 2.10 * 10^11 * h. Substituting the value of h, we get p = 2.10 * 10^11 * 6.62607015 × 10^-34. Finally, we can calculate the de Broglie wavelength using λ = h/p.
(b) The momentum (p) of the electron can be calculated using the given wave function and the relation p = 2.10 * 10^11 * h, where h is Planck's constant. Substituting the value of h, we get p = 2.10 * 10^11 * 6.62607015 × 10^-34.
(c) The kinetic energy (KE) of an object is given by KE = p^2 / (2m), where p is the momentum and m is the mass of the object. In this case, since we are dealing with a free electron, we can assume its mass to be the rest mass of an electron, which is 9.10938356 × 10^-31 kg. We can substitute the value of p in the formula and calculate the kinetic energy in electron volts (eV) using the conversion factor 1 eV = 1.602176634 × 10^-19 J.