What is Planck constant? What is the difference between $h$ and $\hslash $ bar

Dr. Shahid Ali Yousafzai

We discussed the difference between linear frequency and angular frequency. We defined linear frequency as the number of cycles which are passing through a certain point in a unit time which is one second $$ \text { Linear frequency } = \frac { \text { no. of cycles } }{ \text { time } } $$ and we defined this in units of Hertz, so how many cycles per second are there is called Hertz. $$ v = \frac { cycles }{ 1s } =Hz=Hertz $$ Angular frequency means how many rotations are completed or how many cycles are completed in one second. So I can write that $$ \omega =\frac { \text { no. of revolutions } }{ s } =\frac { rad }{ s } $$The angular frequency or the angular velocity is actually the same thing. We define this in radians per second and sometimes in SI units we ignore the word cycle or radians and we only call it per second, this create a confusion between the linear dynamics and the angular dynamics. Let's consider a circle and let's say we are having $$ \frac { 1 cycle }{ s } =v=2\pi rad=1Hz $$ In angular frequency we will call this $$ \frac { 1 rad }{ s } $$ Now one radian here is actually not $2 \pi $ but one radian, as we discussed in earlier lectures, that one radian is the angle in which the arc length becomes equal to the radius of the circle, so I can write that, $\omega $ is $2 \pi $ greater than $v$ $$ \omega =2\pi v $$ Here we can now understand the basic difference between $h$ and $\hslash $. We know $h$ is the Planck's constant and $\hslash $ is the reduced Planck's constant. From the Einstein's relation $$ E = hv $$ Here $v$ is the linear frequency. What is $h$? It is the Planck constant, this constant is giving us the equality of this equation, means energy is proportional to $v$, now the proportionality constant is actually $h$. It relates that a photon of this much frequency will possess how much energy, so it gives us the energy of that frequency photon or that frequency radiation. We can write this in angular terms as: $$ E=\hslash \omega $$ which I call the reduced Planck's constant. For the energy it doesn't matter whether it is in linear dynamics or it is in angular dynamics. Now let's understand this thing $$ h=\frac { E }{ v } $$ So, this is the ratio of the energy of this much frequency radiation where energy is in Joules and $v$ is in cycles per second, I can write $$ h=\frac { E }{ v } =\frac { J }{ \left( \frac { cycle }{ s } \right) } =\frac { Js }{ cycle } $$ And $$ \hslash =\frac { E }{ \omega } =\frac { J }{ \left( \frac { rad }{ s } \right) } =\frac { Js }{ rad } $$ In SI units, we normally neglect that $h$ is Js/cycle and $\hslash $ is actually Js/rad . Now the $2 \pi $ factor will come in that $$ \hslash =\frac { h }{ 2\pi } $$ That's why we call this one as the reduced Planck's constant, and $$ h=6.6025\times { 10 }^{ -34 }\frac { Js }{ cycle } $$