An object is moving along a straight line and the uncertainty in its position is 1.1 m.
(a) Find the minimum uncertainty in the momentum of the object.
(b) Find the minimum uncertainty in the object's velocity, assuming that the object is a golf ball (mass = 0.045 kg).
(c) Find the minimum uncertainty in the object's velocity, assuming that the object is an electron.
To find the minimum uncertainty in the momentum, you can use Heisenberg's uncertainty principle, which states that the uncertainty in position multiplied by the uncertainty in momentum must be greater than or equal to Planck’s constant divided by 4π.
(a) The minimum uncertainty in momentum can be calculated using the formula:
Δp ≥ h / (4πΔx)
where Δp is the uncertainty in momentum, h is Planck's constant (approximately 6.63 × 10^-34 J·s), and Δx is the uncertainty in position.
Δp ≥ (6.63 × 10^-34 J·s) / (4π(1.1 m))
Calculating this yields:
Δp ≥ 1.5 × 10^-34 kg·m/s
Therefore, the minimum uncertainty in the momentum of the object is at least 1.5 × 10^-34 kg·m/s.
(b) To find the minimum uncertainty in the object's velocity, we need to use the equation:
Δv = Δp / m
where Δv is the uncertainty in velocity, Δp is the uncertainty in momentum, and m is the mass of the object.
Using the given mass of the golf ball (0.045 kg) and the minimum uncertainty in momentum from part (a), we can calculate the minimum uncertainty in velocity:
Δv = (1.5 × 10^-34 kg·m/s) / (0.045 kg)
Calculating this gives:
Δv ≈ 3.33 × 10^-33 m/s
Therefore, the minimum uncertainty in the golf ball's velocity is approximately 3.33 × 10^-33 m/s.
(c) For an electron, which has a much smaller mass, we can expect a higher uncertainty in its velocity. Using the same formula as in part (b), we find:
Δv = Δp / m
Assuming the mass of an electron is approximately 9.11 × 10^-31 kg and using the minimum uncertainty in momentum from part (a), we can calculate the minimum uncertainty in velocity:
Δv = (1.5 × 10^-34 kg·m/s) / (9.11 × 10^-31 kg)
Calculating this gives:
Δv ≈ 1.65 × 10^-3 m/s
Therefore, the minimum uncertainty in the electron's velocity is approximately 1.65 × 10^-3 m/s.
To find the minimum uncertainty in the momentum of the object, we can use the uncertainty principle, which states that the product of the uncertainty in position and the uncertainty in momentum must be greater than or equal to a constant (h-bar) divided by 2.
(a) The uncertainty principle is expressed as:
Δx * Δp ≥ h/2
Given the uncertainty in position (Δx) is 1.1 m, we can substitute this value into the equation and solve for the minimum uncertainty in momentum (Δp).
1.1 m * Δp ≥ h/2
To find the minimum uncertainty, we need to take the equal sign case, so we rewrite the inequality as an equation by replacing the ≥ sign with an equal sign:
1.1 m * Δp = h/2
Now, we can solve for Δp:
Δp = h/(2 * 1.1 m)
You might be wondering what h is. h is a fundamental constant called Planck's constant and is equal to 6.62607015 × 10^-34 Js.
So, substituting the value for h, we can calculate Δp:
Δp = (6.62607015 × 10^-34 Js)/(2 * 1.1 m)
Calculate this expression to find the minimum uncertainty in momentum.
(b) To find the minimum uncertainty in velocity, we can use the equation for momentum:
p = mv
We know the mass of the golf ball (m = 0.045 kg) and we have just found the minimum uncertainty in momentum (Δp) from part (a).
Substituting the values into the equation, we can solve for the minimum uncertainty in velocity:
Δp = Δ(mv) = mΔv
Δv = Δp/m
Substitute the value of Δp and m (0.045 kg) into the equation and calculate to find the minimum uncertainty in velocity.
(c) Similarly, to find the minimum uncertainty in velocity for an electron, we can follow the same process as in part (b), but with the mass of an electron (m = 9.10938356 × 10^-31 kg). Substitute this mass value into the equation and calculate to find the minimum uncertainty in velocity.
Using the Bohr model, estimate the wavelength ë of the Ká characteristic X-ray for a metal an atom of which contains Z = 50 protons.
Δx•Δp≥Һ/2
Δx•Δp=Һ/2
Δp = Һ/2• Δx = 6.63•10^-34/2•1.1 = 3•10^-34 kg•m/s,
Δp = m•Δv
Δv = Δp/m = 3•10^-34/0.045 = 6.7•10^-33 m/s,
Δv = Δp/m = 3•10^-34/9.1•10^-31 = 3.3•10^-4 m/s.