About the physical definition of the Planck's constant

Question

In Quantum Mechanics (QM), the Planck's constant has the S.I. units of

(energy) x (time). So, how should one interpret that?!

How should I understand the physical meaning of the following relations?

$\frac E \nu = h$

as the "energy content" divided by the "fastness'' of oscillation equals h.

or in terms of the de Broglie definition

$ |\vec{p}| \times \lambda= h$

i.e. particle momentum times the ``size'' of wavelength equals h.

I know that the definitions are linked. However, for the sake of basic understanding, two questions

a) Should I understand that the size of the wavelength is $\underline{\text{fixed}}$ w.r.t the momentum of the particle and similarly its energy fixes the fastness of its oscillations? They are one and the same thing being looked at from different perspective?

b) Is there any classical version of a physical constant that has more than one definition like the Planck's constant?

Answer 1

There are no two definitions of Planck's constant. In fact, the two relations you gave above are connected. According to Special Relativity, the energy of an object is $$E^2 = (\vec{p} c)^2 + (mc^2)^2$$ But a photon has no mass, so you get $E = \vec{p}c$. And the wavelength of the photon is connected to the frequency ($\nu$). Actually, the wavelength $\lambda$ is equal to ${c}/{\nu}$, where $c$ is the speed of light (and conversely, $\nu = c / \lambda$). So equating, you get: $$h = \frac{E}{\nu} = \frac{\vec{p}c}{c/\lambda} = \vec{p}{\lambda}$$

As for a), actually the relation is $E = h \nu$ and $\lambda = h / {p}$. So the wavelength and momentum are proportional, but not fixed w.r.t. each other. In fact the De-Broglie wavelength, shows that the wavelength of a particle depends on the momentum. Same for the relation between energy and frequency. The energy of a particular photon is proportional to it's frequency, and $h$ is just the proportionality constant between them.

Quick Note:- How the Planck's constant gets the unit $(energy) * (time)$. The unit of the frequency is $1 / T$, where time is the period of the wave measured in seconds. Then you have energy, having SI units Joules. So by $\frac{E}{\nu} = h$, you get $\frac{E}{1 / T} = ET$, hence the units $(energy) * (time)$.

Answer 2

This is not unusual. Many quantities have two separate, common expressions in terms of two different sets of other quantities.

For example, the permeability of free space $\mu_0$ has units of $(\text{inductance})/(\text{length})$ and also of $(\text{force})/(\text{current})^2$.