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The relation is often stated as $E = h\nu$

Simple enough. But the frequency $\nu$ doesn't have to be a natural number or even an integer, correct? It can be any rational number or even any real number. So why is the energy $E$ quantized anyway? And if the given equation is technically faulty then how should it be written?

Qmechanic
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Peterr de Vries
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  • Related: https://physics.stackexchange.com/q/73959/2451 . https://physics.stackexchange.com/q/169209/2451 and links therein. – Qmechanic May 30 '20 at 03:31

2 Answers2

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Frequency isn't even a property of photons, it's an artifact of the frame in which they are measured, so in a world with "only integer frequencies", a simple boost would ruin it. (Planck units aside).

The point of:

$$ E = h\nu $$

is that it is the minimum energy observed in electromagnetic wave with frequency $\nu$, and that the only possible energies are:

$$ E_n = nh\nu\,\,\,\,\{n=1,2,3,\dots\}$$

So the field is quantized, and $n$ is interpreted as the number of photons in the mode.

JEB
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  • Thanks, that clear things up. But how would a mathematician interpret the equation? And have there been conflicts between physicists and mathematicians about what particular equations actually mean? I'm not a purist myself, but I know most - if not all - mathematicians are. – Peterr de Vries Jun 02 '20 at 12:55
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What the Planck-Einstein law implies is that if take any frequency $\nu$, irrespective of it being a natural number, rational number etc., then that light would come in photons, particles which can only have specific amounts of energy given by $E = nh\nu$. So basically, $n$ can only be a natural number. The energy of the photon is discrete. Not the frequency. And that is why the law holds. The frequency has no bounds or limitations, but the energy of any particular photons come in only discrete values.

PNS
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