Planck's constant and quantization of energy

[redtree]

Given:
$\textbf{E}=\hbar \textbf{k}$
where $\textbf{k} = [\vec{k}_1, \vec{k}_2,\vec{k}_3, i c \omega]$
If $\textbf{k}$ can vary continuously, how does the equation imply that energy is quantized?

For example, $y = m x +b$ where $m = \hbar$ does not imply quantized $y$.
For $\textbf{E}$ to be quantized mustn't $\textbf{k}$ be quantized?

And why should $\hbar$ be considered anything other than a unit conversion?

 

[mfb]

For $\textbf{E}$ to be quantized mustn't $\textbf{k}$ be quantized?

Right. For bound states it is.

And why should $\hbar$ be considered anything other than a unit conversion?

You can work in units where it is equal to 1. Yes, it is just a unit conversion - but the fact that this conversion is possible is not trivial.

 

[Nugatory]

If k\textbf{k} can vary continuously, how does the equation imply that energy is quantized?

It doesn't. Quantization of energy appears when you solve Schrödinger's equation for bound states. The simplest example is the one-dimensional infinite square well; in the solutions to Schrödinger's equation for that potential $k$ can only take on discrete values.

 

[vanhees71]

...and please don't use the awful $\mathrm{i} c t$ convention of the SRT pseudometric. Particularly when it comes to QFT, with that you'll confuse yourself even more than the subject itself can ever do when done in the real-time formalism ;-).