Calculate the momentum uncertainty of (a) a tennis ball constrained to be in a fence enclosure of length 35 m surrounding the court and (b) an electron within the smallest diameter of a hydrogen atom.

Strategy We will use Equation (5.40) to find $\Delta p_{x}$ . The position uncertainty Δx is approximately half of the en closure.

$\Delta p_{x} \Delta x \geq \frac{\hbar}{2}$ (5.40)

Question

Calculate the momentum uncertainty of (a) a tennis ball constrained to be in a fence enclosure of length 35 m surrounding the court and (b) an electron within the smallest diameter of a hydrogen atom.

Strategy We will use Equation (5.40) to find [latex]\Delta p_{x}[/latex]. The position uncertainty Δx is approximately half of the en closure.

[latex]\Delta p_{x} \Delta x \geq \frac{\hbar}{2}[/latex] (5.40)

Accepted Answer

(a) If we insert the uncertainty of the location of the tennis ball, [latex]\Delta x=(35 m ) / 2[/latex], into Equation (5.40), we have

[latex]\Delta p_{x} \geq \frac{1}{2} \frac{\hbar}{\Delta x}=\frac{1.05 imes 10^{-34} J \cdot s }{2(35 m ) / 2}=3 imes 10^{-36} kg \cdot m / s[/latex]

We will have no problem specifying the momentum of the tennis ball!

b) The diameter of the hydrogen atom in its lowest energy state (smallest radius) is [latex]2 a_{0}[/latex]. We arbitrarily take the uncertainty Δx to be half the diameter or equal to the radius, [latex]\Delta x=a_{0}[/latex].

[latex]\Delta x=a_{0}=0.529 imes 10^{-10} m[/latex]

[latex]\begin{aligned}\Delta p_{x} \geq \frac{1}{2} \frac{\hbar}{\Delta x} &=\frac{1.05 imes 10^{-34} J \cdot s }{2\left(0.529 imes 10^{-10} m ight)} \&=1 imes 10^{-24} kg \cdot m / s\end{aligned}[/latex]

This may seem like a small momentum, but for an electron with a mass of about [latex]10^{-30}[/latex] kg, it corresponds to a speed of about [latex]10^{6}[/latex] m/s, which is not insignificant! Note that this is comparable to the speed of the electron in the first Bohr orbit [Equation (4.31)].

[latex]v_{n}=\frac{n \hbar}{m r_{n}}=\frac{n \hbar}{m n^{2} a_{0}}=\frac{1}{n} \frac{\hbar}{m a_{0}}[/latex] (4.31)

Question 5.8: Calculate the momentum uncertainty of (a) a tennis ball cons...

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