a) The x-coordinate of an electron is measured with an uncertainty of 0.20 mm. What is the xcomponent
of the electron’s velocity, vx, if the minimum percentage uncertainty in a simultaneous
measurement of vx is 1.0%.
b) Repeat part (a) for a proton.
Answer Details
Δx = 0.200mm = 0.200*10-3m
the mass of electron is: m=9.1 × 10-31kg
Δpx = mΔvx = mvx*1.0% = mvx /100
since
Δx Δpx = Δxmvx /100
the Planck's constant h = 6.626*10-34m2kg/s
57.9
a) Well, an electron's velocity is difficult to pinpoint, just like trying to catch a sneeze. With a percentage uncertainty of 1.0%, it's like trying to hit a bullseye with a water balloon. So, the x-component of the electron's velocity, vx, would be highly uncertain just like the weather forecast for a clown parade. Let's hope for scattered laughter and not a chance of boos!
b) Ah, now we're dealing with protons, those positively charged little rascals. Just like trying to predict their behavior, determining the x-component of a proton's velocity with uncertainty is like juggling watermelons covered in banana peels. Again, a percentage uncertainty of 1.0% means we're in for a wild ride on a unicycle. The x-component of the proton's velocity, vx, will leave us scratching our heads, just like trying to understand why clowns always seem so happy. It's a mystery!
a) To find the x-component of the electron's velocity (vx), we need to use the uncertainty principle. According to the uncertainty principle, there is a trade-off between the precision in measuring the position and momentum of a particle. The formula for the minimum uncertainty in simultaneous measurements of position and momentum is given by:
Δx * Δ(𝑚𝑣) ≥ ℏ/2
Where Δx is the uncertainty in position, Δ(𝑚𝑣) is the uncertainty in momentum, and ℏ (h-bar) is the reduced Planck's constant (ℏ = h/2π).
In this case, the uncertainty in position (Δx) is given as 0.20 mm. We are asked to find the minimum percentage uncertainty in simultaneous measurements of vx, which is Δ(𝑚𝑣)/𝑣𝑥 expressed as a percentage. Let's assume this minimum percentage uncertainty is denoted as Δ(𝑚𝑣)/𝑣𝑥_min.
Thus, the uncertainty principle equation can be written as:
0.20 mm * (Δ(𝑚𝑣)/𝑣𝑥_min) ≥ ℏ/2
Now, we know that Δ(𝑚𝑣)/𝑣𝑥_min is given as 1.0% or 0.01. Substituting this value into the equation, and rearranging to solve for 𝑣𝑥_min:
0.20 mm * 0.01 ≥ ℏ/2
Now, we need to convert the unit of Δx from mm to meters since ℏ is given in SI units (meters):
0.20 mm = 0.20 * 10^(-3) m
0.20 * 10^(-3) m * 0.01 ≥ ℏ/2
Multiplying both sides by 2:
0.20 * 10^(-3) m * 0.01 * 2 ≥ ℏ
0.004 * 10^(-3) m ≥ ℏ
Now, we have an expression for ℏ. To find the value of ℏ, we can use the known value of the reduced Planck's constant, which is approximately ℏ = 6.62607015 × 10^(-34) J·s.
0.004 * 10^(-3) m ≥ 6.62607015 × 10^(-34) J·s
Now, we can rearrange the equation to solve for the minimum velocity (𝑣𝑥_min):
𝑣𝑥_min ≥ (6.62607015 × 10^(-34) J·s) / (0.004 × 10^(-3) m)
Calculating this expression will give you the minimum value of the x-component of the electron's velocity (𝑣𝑥_min).
b) Repeat the same process as in part (a), but now use the mass and uncertainty of a proton. The uncertainty principle equation will be the same, but the values of Δx and 𝑚 will be different. The uncertainty in position (Δx) will still be 0.20 mm (converted to meters), but the proton's mass (𝑚) needs to be considered. The proton's mass is approximately 1.673 × 10^(-27) kg. Follow the steps outlined in part (a), substituting the values for the proton, to find the minimum value of the x-component of the proton's velocity (𝑣𝑥_min).