Heisenberg again!

1. An electron is found in a spherical metallic particle having a volume of $V =10^{-6} cm$. The kinetic energy of the electron is $E_k =10 eV$. Using Heisenberg uncertainty relation compute the relative error in determining the speed of electron.

2. The position of the mass center of a ball having a mass $m =1 mg$ can be measured with an imprecision of $2 \mu m$. Does the uncertainty Heisenberg relation have some some practical importance when determining the speed of the ball?

3. Give a few numerical examples to show that in experiments with objects having a mass of about $m=1 g$ the Heisenberg principle has no importance.

1. The Heisenberg equation  $\Delta x*\Delta p_x \geq \hbar$ becomes in this case

$\Delta x*m\Delta v_x \geq \hbar$

Thus the relative error in determining the speed is

$\frac{\Delta v_x}{v_x} =\frac{\frac{\hbar}{m\Delta v_x}}{\sqrt{\frac{2E_c}{m}}}=\frac{\hbar}{\Delta x*\sqrt{2mE_c}}$

Supposing that the imprecision in finding the electron position is comparable with the radius of the particle that contains it

$\Delta x\simeq x \simeq \sqrt[3]{V}=10^{-4} m$

one finds for the minimum relative error of speed the value:

$\Delta v_x/v_x =6.2*10^{-7}$

2. From the equation $\Delta x*m\Delta v_x \geq \hbar$, the minimum imprecision in determining the speed is $\Delta v_x \geq \hbar/(m*\Delta x)$, where $\Delta x$ is the imprecision in position and $m$ is the ball mass.

By doing the math one has $\delta v_x \geq 5.27*10^{-23} m/s$.

It is obvious that in this case the uncertainty relation has no practical importance.

3.  Suppose a spherical ball having a mass $m =1 g$ travels with a speed $v = 100 m/s$ measured with an imprecision of $0.01%$ (a real life example).

The momentum of the ball is $p =m v =0.001*100=0.1 (kg*m/s)$ and the imprecision is

$\Delta p =(0.01/100)*0.1 =10^{-5} (kg*m/s)$

From the Heisenberg relation one can find the imprecision in the ball position:

$\Delta x = \hbar/\Delta p =(6.62*10^{-34})/(2 \pi *10^{-5}) \simeq 10^{-29} m$

This value is so small that can not be measured with our present capabilities, so that in this case the Heisenberg  principle does not have practical importance.