# What exactly is Planck's constant? how did they calculate it?

What exactly is Planck's constant? I'm a pharmacy student and I've googled this question but couldn't find an answer I'd understand. It would be so great if somebody could tell me how we calculated Planck's constant in simple language so that I'd have a more accurate idea of what it is.

• I am not sure still this one May be worth seeing Commented Dec 5, 2020 at 17:27

The equations that govern quantum mechanics predict that the angular momentum (that is, spinning or orbiting) in a system can't take on any value, but instead come in lumps. The "reduced Planck constant" $$\hbar = h/2\pi$$ is the size of a lump of angular momentum. An electron orbiting a nucleus can do so with zero angular momentum, with angular momentum $$\hbar$$, with angular momentum $$2\hbar$$, and so on, but for a hypothetical value like $$1.37\hbar$$ there are no solutions to the electron's equations of motion.

When we look at spin angular momentum, it turns out to be slightly more subtle: $$\hbar$$ is the size of an allowed change in angular momentum, but some systems can have a total angular momentum which is a half-integer. These systems include the proton, neutron, and electron, each of which has an intrinsic spin $$\hbar/2$$. An electron's spin can "flip" along your preferred measurement axis, from $$-\hbar/2$$ to $$+\hbar/2$$ or vice-versa, but you can never remove the intrinsic spin from an electron. Light also carries angular momentum: any process that emits or absorbs a photon must involve an angular momentum change of $$\hbar$$ (or a larger integer multiple of $$\hbar$$).

The fact that angular momentum comes in lumps is responsible for, among other consequences, the shape of the periodic table.

As of 2019, the International System of Units (SI) has been modified so that the Planck constant is defined as an exact number with nine significant figures; before that change, measurements of $$\hbar$$ were traced back to the International Prototype Kilogram. My absolute favorite historical measurement of $$\hbar$$ is Richard Beth, Mechanical detection and measurement of the angular momentum of Light, Physical Review 50 115 (1936), where bright circularly-polarized light was used to drive a macroscopic torsion pendulum.

Chronological histories of Planck's constant start with Planck's work on the blackbody problem, which he solved by assuming that energy can be added or removed from the electromagnetic field only in multiples of $$hf$$. However, the quantization of angular momentum has always seemed more fundamental to me, because all angular momentum quanta are integer multiples of $$\hbar$$ in every known interaction. It seems to me like energy exchanges come in lumps because angular momentum exchanges come in lumps, even though the discoveries happened in the opposite order.

• Really nice answer. Commented Dec 5, 2020 at 17:39

The original concept of atoms relates to the idea of dividing things, until getting to objects so small that no longer can be divided.

But by things our intuition thinks of inert objects.

Now think of a pendulum for example. It is not an inert stuff, and our intuition is that it loses energy continuously until it stops.

The Planck constant is related to the notion that there is "atoms" (or quanta) of energy. The pendulum loses energy by bits of $$E = \hbar \omega$$, where $$\hbar$$ is the Planck constant, and $$\omega$$ is the frequency of the oscillation.

The Planck constant symbolized the difference between the ancient Greek notion of atoms as smallest possible objects, and the modern notion of quanta as the minimum energy for a given frequency.

May be you have already heard, that light (or every electromagnetic wave) is made up of tiny energy packets, the so-called photons.

In physical experiments (for example of the photoelectric effect) it was found that the energy $$E$$ of a photon is proportional to the frequency $$f$$ of the light. $$E=hf \tag{1}$$ The constant $$h$$ in this equation is called Planck's constant, and the equation is called the Planck-Einstein relation.

Examples:

• Red light has a frequency of $$f=4.6\cdot 10^{14}\text{ Hz}$$.
The photons of red light each have an energy of $$E=3.0\cdot 10^{-19}\text{ J}$$.
• Blue light has a frequency of $$f=6.7\cdot 10^{14}\text{ Hz}$$.
The photons of blue light each have an energy of $$E=4.4\cdot 10^{-19}\text{ J}$$.

From measuremnts like in the examples above and with the Planck-Einstein relation (1) the value of Planck's constant was calculated to be $$h=6.6\cdot 10^{-34}\text{Js}.$$

Planck solved a long standing problem by deriving the formula for thermal radiation emitted by a black body (1900). His formula contains one free parameter to be fit to experiment, the so called Planck constant $$h$$. In 1905 Einstein showed that light of frequency $$f$$ consists of particles of energy $$hf$$. Later it was found that this energy frequency relations holds for all matter.

Planck's constant is a "resolution limit" to the Universe.

The following is a very loose explanation of how that works. There's a fair bit of understanding that quantum theory is best understood as an informational theory, and from this perspective we can consider that Planck's constant sets the maximum amount of information that is allocated to each particle. What that means is that all the attributes of that particle - such as its position, the direction and speed of its motion, and how it's rotating - have to be derived from this finite amount of information. And that makes things "fuzzy", in a sense, because of the following.

Classically, i.e. in the sense of Newtonian mechanics, if we want to talk about where a particle is located, we need to specify three real numbers - you may have seen at least in school geometry that you have the Cartesian coordinates $$(x, y, z)$$. These three numbers locate any point in space, but because they're real numbers, in their most general cases they actually require an infinite amount of information to specify, because, in effect, you can't do any better than writing down every last digit after the decimal point, which goes on forever (c.f. numbers like $$\pi$$ and $$e$$, but those admit alternative ways of representing them that are compact - as I said, this is loose; there's lots of subtle points at work here to make all this mathematically rigorous). Likewise, we can give three more numbers, similarly requiring an infinite amount of information, to describe the particle motion, i.e. where it's going (we need three because we also need the direction of motion as well as the rate of motion).

But the Universe has only a limited amount of "digits", so to speak, and it can give either some to the position, or some to the velocity, but it can't give an infinite amount to both, and that leaves one or the other unspecified below a certain scale of distance or speed. If more are needed in one, then a sacrifice must be made to the other, and this is Heisenberg's uncertainty principle - most instructively written in the form

$$H_x + H_p \ge \lg(e\pi \hbar)$$

where $$H_x$$ and $$H_p$$ are the informational entropies or, better, the degree of privation of information, in the position and momentum ("velocity" was used above to make things even simpler) of the particle. The higher $$H_x$$ or $$H_p$$ are, the less information there is in either parameter. The right hand contains $$\hbar$$, the (reduced) Planck constant. Note the direction of the inequality, and note that logarithms always go up or down when their input values go up or down. If $$\hbar$$ were smaller, then the logarithm on the right would also be lower, and the information privations could be less, meaning more information. Conversely, if $$\hbar$$ were larger, then the logarithm would rise, and there'd be a greater minimum privation of information, that is, even less information.

The effect of this limitation on information is that below a certain point of scale, particles cease to be "located" any more because the information just isn't there to place them at a particular point in space - and the scale at which that happens sets the scale of things like atoms. It's as though the Universe just says "the electron is inside the atom, which has such and such a size", and not "the electron is located at this particular point". Thus, $$\hbar$$ also sets the size of atoms. A larger $$\hbar$$ would mean larger atoms - and this can also be seen reflected in the formula for one measure of the size of a hydrogen atom:

$$a_0 = \frac{4\pi \epsilon_0 \hbar^2}{m_e e^2}$$

If $$\hbar$$ were twice as large, say, the atom would be 4 times as broad, because the amount of information in the combined position and momentum would have shrunk by the order of 4 bits (note that the occurrence of two 4s here is a coincidence: bits are logarithmic, not linear), making the electron less able to locate itself, and it less able to place its speed within a given range, so that larger speeds would be consistent with the limited information available, taking it farther from the atomic nucleus.