Estimating Electron's Momentum Uncertainty in an Atom
Calculating the Uncertainty in Electron's Momentum
Suppose an electron is found somewhere in an atom of diameter 1.25 × 10⁻¹⁰ m. Estimate the uncertainty in the electron’s momentum (in one dimension).
How can the uncertainty in the electron's momentum be estimated using Heisenberg's Uncertainty Principle?
Final Answer: The uncertainty in the electron's momentum (in one dimension) can be estimated using Heisenberg's Uncertainty Principle as follows: Δp ≥ h / (4πΔx), where Δp is the uncertainty in momentum, h is the Planck constant, Δx is the uncertainty in position, and π is approximately 4.202 × 10⁻²⁵ kg m/s.
Explanation: Heisenberg's Uncertainty Principle states that it is impossible to know both the exact position and momentum of a particle simultaneously with absolute certainty. Δx represents the uncertainty in position, which in this case is the diameter of the atom, 1.25 × 10⁻¹⁰ m. Δx = 1.25 × 10⁻¹⁰ m. Now, we can use the formula: Δp ≥ h / (4πΔx). Δp ≥ (6.626 × 10⁻³⁴ Js) / (4π × 1.25 × 10⁻¹⁰ m). Δp ≥ 4.202 × 10⁻²⁵ kg m/s. So, the uncertainty in the electron's momentum (in one dimension) is approximately 4.202 × 10⁻²⁵ kg m/s.
What is the value of the uncertainty in an electron's momentum in the given scenario?
Final answer: Applying the Heisenberg Uncertainty Principle, we can find the uncertainty in an electron's momentum in an atom with the given diameter. With the formula rearranged to Δp = h/(4π*Δx), we find the uncertainty to be roughly 5.28 × 10⁻²⁵ kg*m/s.
Explanation: To estimate the uncertainty in the electron’s momentum, we can apply the Heisenberg Uncertainty Principle. The principle is often written as Δx*Δp ≥ h/4π, where Δx is the uncertainty in position (the diameter of the atom), Δp is the uncertainty in momentum (what we want to find), and h is Planck's constant equal to 6.626 × 10⁻³⁴ Js. So, to solve for Δp, we rearrange the formula to get Δp = h/(4π*Δx). Substituting the given values, we get Δp = 6.626 × 10⁻³⁴ Js / (4π × 1.25 × 10⁻¹⁰ m) = 5.28 × 10⁻²⁵ kg*m/s. Thus, the approximate uncertainty in the electron’s momentum in an atom of diameter 1.25 × 10⁻¹⁰ m is 5.28 × 10⁻²⁵ kg*m/s.