The wavelength of the red laser light is given as 632 nm (nanometers). To find the corresponding properties for the beam of electrons, we can make use of the de Broglie wavelength equation, which relates the wavelength of a particle to its momentum. The equation is as follows:
λ = h / p
where λ is the wavelength, h is the Planck's constant (approximately 6.626 x 10^(-34) J·s), and p is the momentum of the particle.
Since the interference pattern observed with the electron beam is exactly the same as that of the red laser light, we can assume that the wavelengths are equal:
λ(electron) = λ(laser) = 632 nm
To find the corresponding momentum of the electrons, we rearrange the equation as follows:
p = h / λ
Now we can substitute the values:
p = 6.626 x 10^(-34) J·s / (632 x 10^(-9) m)
Simplifying:
p = 1.047 x 10^(-24) kg·m/s
The frequency (f) of a particle can be determined using the equation:
f = p / m
where m is the mass of the particle. In this case, since the problem doesn't specify the mass of the electrons, we'll use the mass of an electron (m_e), which is approximately 9.10938356 x 10^(-31) kg.
Substituting the values:
f = (1.047 x 10^(-24) kg·m/s) / (9.10938356 x 10^(-31) kg)
Simplifying:
f = 1.148 x 10^6 Hz
The energy (E) of a particle can be calculated using the equation:
E = hf
Substituting the values:
E = (6.626 x 10^(-34) J·s) × (1.148 x 10^6 Hz)
Simplifying:
E = 7.606 x 10^(-28) J
Therefore, the wavelength of the electron beam is approximately 632 nm, the frequency is approximately 1.148 x 10^6 Hz, and the energy of the electrons is approximately 7.606 x 10^(-28) J.