James Everitt explains Planck’s constant

How does Planck’s constant effect our everyday lives and how would things change if it had a different value?

Planck’s constant is named after the German physicist Max Planck, who devised the idea for atoms to only be able to vibrate at set frequencies (quantised values). This is often credited as the first theory of quantum mechanics. He named these set frequencies h, 2h, 3h …. so when this fundamental constant of quantum mechanics was first discovered it took the symbol h and was named after Max Planck. (Thompson, 2016)

Planck’s constant is not often something that is seen in macroscopic physics. Although it is something that students are made familiar with in A-level physics studies, the true significance is never fully demonstrated. The truly amazing property of this constant is how often it arises when studying an aspect of physics that seems to be completely unrelated to anything that Planck’s constant has been associated with before.

The value of Planck’s constant is one that has been continually calculated to greater and greater precision and the current accepted value is 6.62607004 x 10-34Js. (Britannica, 2018) This unit is derived from the most common equation that Planck’s constant is used in E = hf; photon energy is equal to the product of Planck’s constant and the photons frequency. This unit can be changed to kgm2s-1 which is often used to make it easier to define other units when calculating using h.

 

In 2018 Planck’s constant became a lot more important because the kilogram is now defined using Planck’s constant. The kilogram is an international standard that was previously based upon a platinum iridium cylinder kept in Versailles. (Kaplan, 2018) But now mass balances are becoming far more accurate and the loss of mass of the kilogram is now becoming measurable meaning that all the highly accurately balances in the world were having to be continually recalibrated because the mass of the kilogram was changing. (Anon., 2018)

A major quantum effect that heavily relates to Planck’s constant is wave particle duality. This is because the wavelength of any object with mass, its de Broglie wavelength, is defined by the ratio between Planck’s constant and its momentum. This wavelength determines when the wave or particle properties are shown. The tiny value of Planck’s means that anything larger than the very tiniest of sub-atomic particles don’t present these wave properties. This then presents the idea that without Planck’s constant being such a tiny value the Universe would so different that it would be unrecognisable. The need for particle interactions at the macro scale is paramount and the need for the microscopic to be able to interact differently is even more important; it is something that defines the very structure of the universe. Without quantum tunnelling at only the tiniest levels (this only happens when particles come within a distance comparable to their de Broglie wavelength to each other) no chemical reaction would happen as when molecules collided, they would simply undergo nuclear fusion. This would mean that the structure of the universe would be very different. Because any two elements couldn’t get too close to one another or a nuclear reaction will happen, most compounds would never form. The most basic bodies in the universe like planets and moons, made of large compounds, would never be able to exist.

When considering how the world might change if Planck’s constant were to change, there are multiple ways of approaching it. Because it is a constant devised by a number system created by humans it can also be changed by humans. In doing so it may change the magnitude of the current standard SI units; however, it would be entirely possible. This change would have no effect on the workings of the universe but would change the very basis of physics calculations.

 

So as the first scenario for change I will assume that the magnitude of all other universal constants are staying the same and that someone has simply decided to change this currently fixed value. The fundamental SI units are metres(m), seconds(s), moles(mol), amperes(A), kelvin(K), candela(cd) and kilograms(kg). The easiest change to show is to the kilogram. As Planck’s constant is the universal constant it is based upon so a change in the magnitude of Planck’s constant would lead to a change in the magnitude of the kilogram. (Kaplan, 2018) A mole is defined as “the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12” (Anon., 2018). This means that it is defined as a comparable unit to the kilogram; I have already shown how the kilogram will change, therefore also changing the magnitude of a mole. If the speed of light is to keep its current magnitude, then any calculation of it involving Planck’s constant would either change the value or mean changing its units. The two constants share both metres and seconds in their base unit form. However, the metre is based upon the second, so their magnitudes would both have to change to suit this new value of Planck’s constant. Candela are based upon a specific frequency of radiation giving a certain power output both from an object (540 x 1012 hertz and 1/683 watt per steradian) (Anon., 2018). Both Hertz and Watts are based upon seconds which I previously stated have had to change; this would lead to the magnitude of the candela having to be changed or the parameters that the candela is based upon being changed, either of which would change how the unit is defined. Amperes are based upon both the metre and the kilogram (from the newton) meaning that its magnitude will also change from the changes to the kilogram and the metre. The only unit that I believe would be unchanged is Kelvin as it is independent of the other base units and when it does combine with other units its magnitude has no effect upon anything like a change in Planck’s constant would.

A second scenario for the change would be the ratio it represents fundamentally changing. This would cause a huge reshape of the universe on both microscopic and macroscopic levels. Although the full extent of the changes to the universe caused by a change in Planck’s constant can never be known due to the extensive bank of equations available to the modern physicist, I am able to create many proportional relationships with many different aspects of the universe. So, in this next scenario it will be similar to the last in that all the current universal constant, other than Planck’s, will stay at their current value. All the current equations will still be assumed to be true but most important of all our measuring system will remain unchanged.

As the atomic radius of a hydrogen atom has the simplest equation to calculate, of all the elements, the relationship between Planck’s constant and atomic radius can be easily derived. This equation only works with the electron in its ground state and states the most probable distance between the hydrogen nucleus and the electron. (Yang, 2016) From the equation r1=(h20)/(π.me.e2) with ε0 being the permittivity of free space, me being the mass of an electron and e being fundamental charge. This equation shows that the minimum radius of an atom is proportional to the square of Planck’s constant meaning that if it were to double the radius of every atom in the universe would increase four-fold. However, there would be no change in mass; each atom in the universe would become massively less dense, specifically, 64 times less dense, assuming all atoms are spherical. Ignoring this, the volumes of all other objects in the universe would increase dramatically but this wouldn’t much change everyday life as everything around us would still appear to be the same size proportionally to us. However, the night sky might look at little strange; assuming the value of Planck’s constant was suddenly changed, then we would have to wait to be able to see all the other objects in the sky change size. It would be 8 minutes before we saw the newly enlarged Sun in our newly enlarged sky. In the night sky the moon would dominate even more than it does now as the surface area of the night sky is 16 times larger than previously but the light from the enlarged stars wouldn’t reach us for years after the event. When seeing the mesmerising view of the Milky Way in the night sky it might seem to be distorted as we would see the old and new images of different stars. It is an interesting thought experiment to imagine all the matter in the entire universe suddenly expanding.

Another property of the hydrogen atom that can be linked to Planck’s constant is the ionisation energy from a specific energy level within the shell. This relationship considers the energy level of the electron, the permittivity of free space and the mass and charge of an electron. (Yang, 2016) This leads to the ionisation energy being inversely proportional to the square of Planck’s constant. This means that the ionisation energy of any atom would be lowered dramatically by an increase in Planck’s constant. The consequences to life would be immense; ionisation of DNA would happen under exposure to visible light, cancer would become even more prominent than it is now. Simply to survive all life would have to become nocturnal and would most likely have to evolve to be able to return to the sea as it would be the only place that would protect us from the radiation. This is demonstrated by the belief that Earth’s oceans were the first home for life before ozone developed in the atmosphere to absorb UV radiation. The use of radioactive isotopes in medicine would have to be completely removed as the radiation that is currently see as very ionising would become even more dangerous and being in range of any large source would completely destroy tissue.
The change in ionisation energy would also affect the emission and absorption spectra of all the elements, although this might not affect people on a day to day basis, it would mean that atoms in the atmosphere are continually either being ionised or their electrons are being excited then relaxing. This means the atmosphere would begin to store solar energy in the electrons being in excited states from the radiation of the day. Although most of this may be emitted almost instantly the continual bombardment on the gases in the atmosphere would lead to them emitting infrared radiation throughout the night as all the electrons relax. It is impossible to know whether this would truly have an effect. I doubt it would cause any increase in average global temperature, but it would most likely just lead to much warmer nights than we are currently used to and it might take longer to warm up to maximum temperature during the day as some of the heat radiation would be absorbed by the atmosphere. That assumes that the wavelengths emitted by the Sun stay the same but there is evidence to suggest that this would change too.

The total radiated energy of any object is equal to the product of the Stefan-Boltzmann constant and the temperature raised to the fourth power of that object. To calculate this constant(σ) Planck’s constant must be used. Although σ can be considered a universal constant it is classified as a derived constant because it comes from calculation using other universal constants so in the scenario it can change. This change would be caused by its relationship with Planck’s constant which is that σ is inversely proportional to the cube of Planck’s constant. (Yang, 2016) This means that doubling Planck’s constant would mean a reduction by and eighth of the energy emitted by all objects. Currently the Sun emits mostly visible light with the average emission around 600 nanometres in wavelength. If the energy of those photons is reduced by a factor of an eighth, then the average wavelength will be around 4.5 micrometres which sits well within the infra-red spectrum. Currently the Sun is producing mainly wavelengths that are detectable by our eyes. However, with the Sun emitting only mostly infrared light the Earth would become dark to humans and the Earth would appear in a state of eternal night. Although it has been shown that the Sun currently emits radiation up the very highest frequencies; this is at very low intensities comparative to visible light meaning their may be a faint glow during what we currently perceive as day time.

One would then go on to think that life would simply have to evolve eyes to see infrared and then it wouldn’t have much more of an impact on life. However, our atmosphere is composed as a protective layer, only letting in the wavelengths that are beneficial to life or that won’t cause any harm; this is one of the reason Earth is such a good place for life. Infrared can cause odd heating effects but luckily it is mostly absorbed by carbon dioxide and water vapour in the atmosphere (NASA, 1999). This would lead to strange heating effects in the atmosphere as the majority of the Sun’s power output that hits the Earth is now being absorbed in the upper atmosphere. This would most likely cause a kind of accelerated greenhouse effect as this is partially what happens now. This would most likely lead to global average temperatures rising quickly and the Earth becoming almost entirely uninhabitable added to this will be the fact that the Earth would actually receive twice the energy from the Sun it currently does. Therefore, an increase in Planck’s constant would lead to rising global temperatures and the Earth would be likely to become an uninhabitable wasteland.

The second most important part of the solar system for to the human race has to be the Sun. It provides almost all the energy that ever reaches Earth. But how would it be affected by an increase in Planck’s constant? Planck’s constant simply represents a ratio between each of the quantities that allow fusion to happen in its core. This is the ratio between the thermal de Broglie wavelength of the fusing hydrogen nuclei, the critical fusion temperature and the core temperature of the star. (Yang, 2016) Both the critical fusion temperature and the core temperature are inversely proportional to the square of Planck’s constant, meaning that any change in Planck’s constant could never make it too hot or cold for fusion within the star. The de Broglie wavelength allows the nuclei to quantum tunnel to within the correct range of one another to undergo fusion. It is also proportional to the square of Planck’s constant in the same way as the size of the atoms, which I stated previously, this means that the nuclei are the same distance apart relatively and will still come within the correct distance to allow atoms to quantum tunnel at the same frequency as they do currently.

The radius of the Sun would also change as it is proportional to the square of Planck’s constant. This means that the surface area is proportional to Planck’s constant raised to the fourth power. The core temperature can be shown as inversely proportional to the radius of the star and as that is proportional to h2 then the core temperature has an inversely proportional relationship to h2, as mentioned in the previous paragraph. But across these enlarged dimensions the energy exportation is calculated by the luminosity. The luminosity has an inversely proportional relationship with the cube of Planck’s constant, meaning that the total energy emission of a star would decrease by a factor of eighth if Planck’s constant were to double. However, I previously claimed that the Earth would receive double the energy it currently does. If Planck’s constant where to double then the energy emitted by the Sun would be an eighth the value, it currently is. However, the surface area of the Earth would be 16 times what it is now due to the expansion of atomic radii and the product of 16 and eighth is 2. This means that twice the current energy would reach the Earth from the Sun adding to the global warming effect from the infrared radiation. (Yang, 2016)

In conclusion, a change in Planck’s would have huge effects on the universe and it is likely that it would be completely unrecognisable. Its importance is huge in modern physics, with it being used as the basis for the kilogram. This importance extends further into the quantum realm, where it is a ratio between both the de Broglie wavelength of a mass and its momentum and a photon’s energy and its frequency. This link between both things with mass and without mass makes it fundamental in the understanding of the quantum realm. I have shown how a simple change in this value would have disastrous consequences for life and that it would reshape the universe. It could be argued that it is a perfect ratio and that is one of the infinite number of factors that have made Earth such a haven for life.

Bibliography

Anon., 2018. Kilogram: Mass and Planck’s constant. [Online]
Available at: https://www.nist.gov/si-redefinition/kilogram-mass-and plancks-constant
[Accessed 16 March 2019].

Anon., 2018. SI Units – Amount of Substance. [Online]
Available at: https://www.nist.gov/pml/weights-and-measures/si-units-amount-substance
[Accessed 16 March 2019].

Anon., 2018. SI Units – Luminous Intensity. [Online]
Available at: https://www.nist.gov/pml/weights-and-measures/si-units-luminous-intensity
[Accessed 16 March 2019].

Britannica, T. E. o. E., 2018. Planck’s Constant. [Online]
Available at: https://www.britannica.com/science/Plancks-constant
[Accessed 16 March 2019].

George Gamow, R. S., 1999. The NEW World of Mr TOMPKINS. 1st ed. Cambridge: Cambridge University Press.

Gregersen, E., 2010. Fine structure. [Online]
Available at: https://www.britannica.com/science/fine-structure#ref70251
[Accessed 16 March 2019].

Kaplan, S., 2018. It’s Official: The Definition of a Kilogram Has Changed. [Online]
Available at: https://www.sciencealert.com/it-s-official-the-definition-of-a-kilogram-has-changed
[Accessed 16 March 2019].

NASA, 1999. Absorption Bands and Atmospheric Windows. [Online]
Available at: https://earthobservatory.nasa.gov/features/RemoteSensing/remote_04.php
[Accessed 17 March 2019].

Thompson, A., 2016. A Brief Explanation of Planck’s Constant and the Birth of Quantum Physics. [Online]
Available at: https://www.popularmechanics.com/science/news/a21490/what-is-plancks-constant/
[Accessed 17 03 2019].

Yang, P.-K., 2016. How does Planck’s constant influence the macroscopic world?, Hsinchu: IOP publishing.