# What is a proportionality constant? (Planck's constant)

I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency.

There are 2 things that im trying to verify:

1. isn't the number that Planck's constant represents just an arbitrary result of the measurement system we use? As in, Planck's constant represents the ratio between 1 Joule and 1 Hertz? So if i assume correctly, Planck's constant is the increase in energy (in joules) that you would get from an increase in frequency of 1hz? Thus if i wanted to I could make an alternative unit for energy which had a 1 to 1 correspondence with hz, and no proportionality constant would be required?

2. So regardless of units used to measure, a fixed percentage increase in one quantity results in an equal percentage increase of the other, correct?

• I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. Planck's constant is essentially, or fundamentally, the quantum of action and the reduced Planck's constant is the quantum of angular momentum. – Alfred Centauri May 2 '14 at 1:28

Planck's constant can indeed be described as a proportionality constant, as

$$E=h\nu$$

for a photon with energy $E$, and frequency $\nu$. To some extent, the reductionist viewpoint of the constant may be likened to Boltzmann's constant; as Professor Tong states

There is no deep physical meaning to Boltzmann's constant. It is merely a conversion factor to allow us to go between temperature and energy.

We may if we wish view $h$ in a similar fashion. As such, we often work in natural units for high energy physics where $c=\hbar=1$. In other fields it is convenient to set $k_B=1$.

Regarding proportionality, it is correct to state that given $A\propto B$ implies a direct increase in either corresponds in a direct increase in the other quantity. However, they may increase or decrease by different percentages, in for example the case $A\propto B^3$.

First, any proportionality constant showing a direct proportion y = kx by definition indicates that if one increases, the other will increase by the same factor. So, yes, if frequency doubles, energy doubles as well, as shown in the equation $E = h \nu$. Also, if you carefully define your units, it is possible to make the proportionality constant equal to 1. This type of thing is done in a number of unit systems.

On the other hand, seeing that energy and frequency aren't the same thing, there is a little more to the equation, and to Planck's constant. It is true that inches = 2.54 * centimeters, which establishes another proportionality constant, but between two units of the same dimension. Planck's constant relates two different physical quantities. Similarly, in $E = m c^{2}$ we can consider $c^{2}$ as the proportionality constant between E and m, and in the natural (I think) unit system, is defined as 1, so that $E=m$. However, the $c$ is significant; it is the speed of light.

Another favorite example of mine is the ideal gas law. $PV=nRT$ is usually used in chemistry classes, but many physicists favor the other form, which is $PV=Nk_{B}T$. The $R$ is a proportionality constant to fit P, V, n, and T in whatever units they are given. In a physical sense, $R = N_{A} k_{B}$, where $k_{B}$ is Boltzmann's constant, which helps us relate temperature to energy.

(Last paragraph edited to remove personal preference bias for $k_{B}$ over $R$.)

• $R$ also appears in the expressions for the heat capacity of materials, among other places. – Javier May 1 '14 at 12:18
• @JavierBadia Touche. I guess any time we are working with moles, we could see it, since $R = N_{A} * k_{B}$. I'll edit to remove the bias for Boltzmann. – jdj081 May 1 '14 at 20:11

One should understand that planck's constant is 'real', but the size in any given system, such as $15.724*10^{-33}$ ft pdl s/cycle, is more an artefact of the particular measurement system in use, rather than its true nature.

e and h are the two quantum relations used to establish Bohr's atom. The modern atom descends from Bohr's model, by using debrolgie waves in Schrödinger's equations, do not necessarily say bohr's atom was 'wrong'.

Regarding using h as a unit of measure - this is already done!. Have a look into the "energy conversion tables" in the NIST CODATA information, and you will already find energy measured in Hertz being converted into foot-pdls at the rate of 1 Hz=15.724E-33 ft pdl.

Similarly there is a scale for energy in volts, originally called 'equivalent volts', but now called 'electron volts'. But because the current physics can't accept that a given quantity can have multiple dimensional analysis, it is supposed that eV is different somehow to lbf in nature (ie unit * natural constant = unit2).