Photon Energy =$ h\nu$ as Deep Secret of Modern Physics?

In a classical wave equation the frequency in time $\nu$ scales with $\sqrt{E}$ with $E$ wave energy, or the other way around energy $E\sim \nu^2$. To see this recall that a classical wave appears as a real-valued solution $\phi (x,t)$ to the following classical wave equation (with $\phi_t$ the derivative with respect to $t$):

• $\phi_{tt}-\phi_{xx} =0$ for $0<x<\pi$ and $t>0$,                         (1)
• $\phi (0,t)=\phi(\pi ,t)$ for $t>0$,
• $\phi (x,0)$ and $\phi_{t}(x,0)$ given initial values.

A typical solution has the form 

• $\phi(x,t)=\cos(\nu t)\sin(\nu x)$ with $\nu =1,2,3,..$ as natural number, 
• with energy $E\equiv \int_0^\pi\vert\phi_{xx}\vert ^2dx\sim \nu^2$
• thus with frequency $\nu\sim \sqrt{E}$.  

On the other hand we know the convention of assigning the energy $E=h\nu$ to a photon in Standard Quantum Mechanics, thus as $E\sim \nu$, with $h$ a constant, which can be anything but is  prescribed to have a certain standard value in the SI Standard.  

So in classical wave mechanics $E\sim\nu^2$ and in quantum mechanics $E\sim\nu$, which to a student must be confusing, in particular since $E=h\nu$ is supposed to have a deep secret meaning. 

So why this difference? The reason is that the wave equation of quantum mechanics does not take the above form, but instead the following complex form with only one derivative in time:

• $i\phi_{t}-\phi_{xx}$ for $0<x<\pi$ and $t>0$,                         (2)
• $\phi (0,t)=\phi(\pi ,t)$ for $t>0$,
• $\phi (x,0)$ given initial value,

with typical solution 

• $\phi(x,t)=\exp(i\nu^2 t)\sin(\nu x)$ with $\nu =1,2,3,..$,  
• with energy $E\equiv \int_0^\pi\vert\phi_{xx}\vert ^2dx\sim \nu^2$,
• thus with frequency $\nu\sim E$.  

We understand that the complex form (2) can be reduced to real form:

• $\phi_{tt}-\phi_{xxxx}$,

to be compared with the classical $\phi_{tt}-\phi_{xx}$, which explains the switch from $E\sim \nu^2$ to $E\sim\nu$. 

We have learned that the connection $E=h\nu$ simply reflects the nature of the wave equation adopted and as such carries no deep secret per se and only represents an ad hoc division of global energy into little quanta which have no realisation in physics. 

If we connect an atom naturally described by (2) to light naturally described by (1), the we have to take the difference in chosen wave equations into account when connecting atomic energy and light energy recalling that incoming wave energy scales like $\nu^2$ resulting from $\nu$ incoming energy quanta of size $\nu$ per unit of time. 

The post points to basic aspects and does not seek to give a detailed account using 3d Maxwell equations for light and Schrödinger's eq for an atom. The idea is to decode the proclaimed deep secret of light particles/photons carrying energy quanta $h\nu$.

Notice that the macroscopic wave equation (1) describes waves which move rectilinearly in space, while (2) describes atomic waves which rather rotate on the spot while keeping charge density constant.  Schrödinger's equation thus connects to (2) in direct opposition to any concept of electrons moving around a nucleus.