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If charges and any other magnitude must be or well quantized by multiples of an integer, or simply discrete values. How could we us assign values to the quantized quantities?, beacuse If I measure with an analogic aparatus, how can we create a continuous scale if only values from the magntide are discrete

Raúl Aparicio Bustillo
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  • I am not sure what you are asking. For one thing, quantization doesn't automatically lead to "integer multiples". There is, for instance, no direct and simple proportionality constant between angular momentum and magnetic moment. It depends in very difficult to calculate ways on the details of the specific system. In addition, most quantities like position, momentum, time, energy etc.. are not quantized, at all. These can take on arbitrary values in our current theories and there is no sign in nature that we are missing something at the moment. – CuriousOne Mar 24 '16 at 10:21

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One might say that that assigning values to such quantities is really a historical accident that followed from a belief that these quantities might have been continuous. It is possible to simply treat the measurement of these quantities as the measurement of a number corresponding to the quantization.

For example, in the electrostatic case, we have the Coulomb force between charges $q_1$ and $q_2$: $$F = k \frac{q_1 q_2}{r^2}$$ However, when we enforce that charges are integer multiples of a fundamental charge, we can then write $q_1 = n_1 e$, $q_2 = n_2 e$ and therefore $$F = C \frac{n_1 n_2}{r^2}$$ Where $C=ke^2$ just a rescaling of the constant $k$. Here, $n_1$ and $n_2$ provide a perfectly satisfactory measure of charge, and you don't need to have a continuous set of values. This is similar to what happens when going between (say) SI units and Gaussian units.

Using the same argument in reverse, when charge, one could just measure these integers, or multiply these integers with a dimensionful quantity to get a real number/dimensionful "value". It is ultimately a matter of what you call charge.

On the other hand, adding to CuriousOne's comment, you can't always do something like this. For example, energy is quantized differently (or not at all) for different systems. A free particle can have any energy. And the standard particle-in-a-box problem in quantum mechanics has energy values given by: $$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$$ where $L$ is the width of the box. You can't reduce the energy to just a measurement of $n$, because the exact set of values the energy takes is dependent on $L$, and produces measureable consequences in interactions with other systems (for example, the frequency of light emitted in a transition between states).

Once again, you can always redefine energy (and for that matter, frequency) by multiplying by a constant, but it only makes sense to allow for a continuous range of values.

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