Observing electron interference through an ordinary macroscopic two-slit experiment would be extremely difficult and would require very slowly moving electrons. (Slow for an electron means < 10^3 m/s.)
How slowly would electrons have to move through slits spaced 6.53 μm apart in order for the first angular displacement from the central interference maximum to where we observe no electrons to be 12.3 degrees? (Answer in m/s.)
To find out how slowly the electrons would have to move through the slits, we can use the formula for the angular spacing of interference fringes in a two-slit experiment:
θ = λ / d,
where θ is the angular spacing, λ is the wavelength of the electrons, and d is the distance between the slits.
In this case, we know the angular spacing (θ = 12.3 degrees) and the distance between the slits (d = 6.53 μm or 6.53 × 10^(-6) m). We need to solve for the wavelength (λ).
First, we need to convert the angular spacing from degrees to radians:
θ_radians = θ * (π / 180).
θ_radians = 12.3 * (π / 180) ≈ 0.214 radians.
Now we can rearrange the formula to solve for λ:
λ = θ * d.
Substituting the given values:
λ = 0.214 * 6.53 × 10^(-6) ≈ 1.396 * 10^(-6) meters.
Since we know the velocity (v) of the electrons is less than 10^3 m/s, we can use the equation for the velocity of a wave:
v = λ * f,
where f is the frequency of the wave. For electrons, we know their velocity is close to the speed of light (c), so we can approximate the frequency as:
f ≈ c / λ,
where c is the speed of light (3 × 10^8 m/s).
Substituting the values:
f ≈ (3 × 10^8) / (1.396 × 10^(-6)) ≈ 2.152 * 10^14 Hz.
Now, we can find the velocity (v) by multiplying the frequency (f) by the wavelength (λ):
v = λ * f.
v ≈ (1.396 × 10^(-6)) * (2.152 × 10^14) ≈ 3.0 × 10^8 m/s.
Therefore, the electrons would need to move at a velocity of approximately 3.0 × 10^8 m/s or slower for the observed angular displacement to be 12.3 degrees.