A molecule of roughly spherical shape has a mass of 6.10 x 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the...

Sunday, August 11, 2013

A molecule of roughly spherical shape has a mass of 6.10 x 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the...

Hello!

This problem is similar to the previous. The uncertainty principle states that both position and velocity of a particle cannot be measured exactly. Mathematically,

$\displaystyle\delta{p}\cdot\delta{x}\geq\frac{{\overline{{h}}}}{{2}},$

where $\displaystyle{p}$ is for impulse (momentum), which is equal to $\displaystyle{m}\cdot{v},$ mass by velocity, and $\displaystyle{x}$ is the position. Correspondingly, $\displaystyle\delta{p}$ means uncertainty of measuring momentum and $\displaystyle\delta{x}$ means uncertainty of measuring position. Obviously $\displaystyle\delta{p}={m}\cdot\delta{v}.$

$\displaystyle{\overline{{h}}}$ is the so-called reduced Planck's constant, $\displaystyle\frac{{h}}{{{2}\pi}}.$ Therefore the minimum uncertainty in the speed is

$\displaystyle\delta{v}=\frac{{h}}{{{4}\pi}}\cdot\frac{{1}}{{{m}\cdot\delta{x}}}.$

All quantities are given, so the numerical result is

$\displaystyle\delta{v}\approx\frac{{{6.626}\cdot{{10}}^{{-{34}}}}}{{{4}\cdot{3.14}}}\cdot\frac{{1}}{{{6.10}\cdot{{10}}^{{-{25}}}\cdot{0.7}\cdot{{10}}^{{-{9}}}}}=\frac{{{6.626}}}{{{4}\cdot{3.14}\cdot{6.10}\cdot{0.7}}}\approx{0.1235}{\left(\frac{{m}}{{s}}\right)}.$

I took into account that nano- means $\displaystyle{{10}}^{{-{9}}}.$

To obtain the result you want one should "forget" to divide by $\displaystyle{2}\pi,$ then it would be about 0.7759 m/s, C. But I'm almost sure about the values.

The second question, about the minimum speed, requires a separate consideration.

Ninenotes

Sunday, August 11, 2013

A molecule of roughly spherical shape has a mass of 6.10 x 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the...

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