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The frequency v can be less than 1. So the energy is starting from zero, and is continuous. The black body radiation is also continuous spectrum given an arbitrary temperature T.

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The fact that $\nu$ is continuous and unbounded simply tells you that, for example, a 'free' photon(boundary conditions often enforce quantization, eg standing waves) can take up any frequency whatsoever, and so, it can have any real number for energy(as there will always be a corresponding $\nu$ for it).

Now if you consider a monochromatic beam of light, which is made up of photons, then the beam can only have energies of the form-$$E=nh\nu$$, where $n$ is the number of photons in the beam. In this sense, the energy of the beam is quantised.

Finally, this holds for any kind of electromagnetic radiation. You can think of it as being made up of a discrete number of photons, and once you have assigned a frequency $\nu$ to each of them, the energy of radiation is quantized by the number of photons. There was no restriction, however, on what $\nu$ you would assign to each photon. As energy exchange by radiation occurs via these photons, this energy is called 'quantised'.

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  • $\begingroup$ Given a specific frequency, total energy of the light must be Nhv. This is a self evident math. Please see my words answering gary p in above floor. The total emissions of all cars running at 100km/s on road are the same quantum value. But we all know car emission is continuous without discrete number. $\endgroup$ – Cang Ye Jul 8 at 12:13
  • $\begingroup$ The total weight of all 60kg body-weight people on earth MUST be 60kg multiplied by N. Is this so-caled "discovery of the new quantum world"? Can this lead the physics to the conclusion that every man's body weight is quantised? Hard to believe. $\endgroup$ – Cang Ye Jul 8 at 15:22
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    $\begingroup$ The point is that light, previously assumed to be a wave, must be simultaneously thought to be made up of particles (photons) to explain the apparent quantisation of energy despite no quantisation of frequency. This wave particle duality is at the heart of quantum mechanics, and it is certainly not trivial. $\endgroup$ – GRrocks Jul 8 at 20:58
  • $\begingroup$ GRrocks, yes it's not trivial, I agree. What I am saying is that the formula doesn't mean light energy is of quantum mature. It means light of given frequency is of quantum nature. Just the same as that total gravity of all 60kg body-weight men is quantised value, which is self evident and simplest math. A lot of physics textbook misinterpreted the formula as energy has quantum nature. $\endgroup$ – Cang Ye Jul 10 at 9:47
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You don't have that formula quite right. It's really $$E=nh\nu$$ where $n$ is an integer and $h$ is a constant. $\nu$ can take the value of any positive real number, so the black body curve is indeed continuous. But $n$ is an integer. So the energy can take only discrete values.

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  • $\begingroup$ For a given frequency v, of course, the total energy is N times hv. But this is a always correct but meaningless conclusion. Take automobile emissions for example. Assume (just assume) that vehicle emissions are proportional to vehicle speed. Obviously, the emission of a single vehicle is continuous, because the vehicle speed is continuous from 0 to 200 km/s. $\endgroup$ – Cang Ye Jul 8 at 11:29
  • $\begingroup$ There are millions of cars on the road. If we look at the total emissions of these cars at specific speeds, it's quantum. If the emission of a 100 km vehicle is w, then two 100 km vehicles must be 2w, and a million 100 km vehicles is 1000000w. The quantum nature discovered by Planck is the total emission value of specific speed cars. $\endgroup$ – Cang Ye Jul 8 at 11:35
  • $\begingroup$ The emission value of selected 100km cars must be quantum. What I said always-correct but meaningless is referring to this . $\endgroup$ – Cang Ye Jul 8 at 11:46
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It can also be explained by the fishes in a lake. The weight of fish is not quantized. There are eggs, small fishes and adult fishes in the lake. They have been growing up bit by bit. So the weight of a fish is a continuous variable. But if we look at all five kilograms of adult fishes in the lake, their total weight is of quantum distribution. If there is only one in the lake, the total weight will be 5 kilograms, if there are two in the lake, it will be 10 kilograms, if there are N in the lake, the total weight will be 5N kilograms. The quantum nature discovered by Planck is the total weight of all five kilograms fishes in the lake. This is, of course, a value with quantum distribution. This quantum property does not mean that the weight of fish is quantum value, but mean that the number of fishes is integers and the specific choice of the observer is 5 kilograms.

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  • $\begingroup$ Total energy of given frequency light is quantized, but light energy is not quantized. Total weight of all 5kg fishes is quantized,but fish weight is continuous. For anything, if we use a fixed amount as unit, the total summary of all that units must be quantized. This is a common sense, can't lead to quantum theory, and shouldn't be mistakenly interpreted. $\endgroup$ – Cang Ye Jul 10 at 10:44

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