Heisenberg’s Uncertainty Principle. Confining a wave/particle in a box.

In subatomic terms, because of wave particle duality, certain pairs of measurements such as where a particle is (x) and  where it is going (its position and momentum) cannot be precisely known. If we know one very precisely, the other cannot be known. Putting this another way, a particle has mass (hence momentum) also a wavelength given by the de Broglie expression

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When particles’ wavelengths interfere, they form a wave packet of finite size having a length which has to fit into the confining box, which can happen in a variety of ways…Here, we’re only really concerned with the smallest “wavefunction”, shown in red at the bottom. The diameter of the box is approximately half a wavelength. The rest are there just to show what’s possible. A wavefunction represents the probability of finding the wave in a particular space.

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Confining our wave in a box, where it is and its momentum are defined like this:

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This makes sense in the context of a problem. Imagine an alpha particle confined within a nucleus of gold. Given the alpha particle has a wavelength confined by a ‘box’ the size of the nucleus, whose diameter might be:

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Suppose we want to find the energy of the confined alpha particle. We use:

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The energy can be found using a different expression:

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We can find the mass of an alpha particle (2 protons and 2 neutrons. If we plug in the numbers, we get 4.3×10-15 J or 27keV, consistent with observed energies.

You might try the same calculation to find out the energy an electron would have to have to confine it inside the nucleus.

This is why we don’t get electrons inside nuclei…

 

 

 

 

 

 

 

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