tisdag 8 april 2014

Essence of Quantum Mechanics: Energy vs Frequency in Wave Models

In Schrödinger's Equation: Smoothed Particle Dynamics we observed that Schrödinger's equation for Hydrogen atom with one electron (normalized to unit mass and charge) reads
  • $i\bar h\dot\psi + H\psi =0$,
  • $H\psi =\frac{\bar h^2}{2}\Delta\psi +\frac{1}{\vert x\vert}\psi$,
where $\psi (x,t)$ the complex-valued wave function depending on coordinates of space $x$ and time $t$ with the dot denoting differentiation with respect to time, $H$ is the Hamiltonian operator and $\bar h$ Planck's (reduced) constant.

In terms of the real part $\phi$ and imaginary part $\chi$ of $\psi =\phi +i\chi$, Schrödinger's equation takes the system form
  1. $\bar h\dot\phi +H\chi =0$,
  2. $\bar h\dot\chi - H\phi =0$.
If $\phi_E(x)$ is an eigenfunction of the Hamiltonian satisfying $H\phi_E =E\phi_E$ with $E$ the corresponding eigenvalue, then the solution can be represented as
  • $\phi (x,t)=\cos(\omega t)\phi_E(x)$,     $\chi (x,t)=\sin(\omega t)\phi_E(x)$, 
with $\bar h\omega =E$, which expresses a periodic exchange between the two real-valued wave functions $\phi$ and $\chi$ mediated by the Hamiltonian $H$. The parallel to a harmonic oscillator (with $H$ the identity) is obvious.

We see that the effect of the time derivative term is to connect energy $E$ to (angular) frequency $\omega$ by
  • $\bar h\omega = E$, 
  • or $h\nu =E$, 
where $h=2\pi\bar h$ and $\nu =\frac{\omega}{2\pi}$ is frequency in Hertz, where $h$ acts as scale factor.

Schrödinger's equation thus sets up a connection between frequency $\nu$, which can be observed as atomic emission lines, and a model of internal atomic energy $E$ as the sum of kinetic and potential energies of eigenfunctions of the Hamiltonian with the connection $\bar h\omega =h\nu = E$. Observations of atomic emission then show to fit with energy levels of the model, which gives support to the functionality of the model. 

The basic connection $\nu \sim E$ can also be seen in Planck's radiation law (with simplified high-frequency cut-off)
  • $R(\nu ,T)=\gamma T\nu^2$ for $\frac{h\nu}{kT} < 1$,
where $R(\nu ,T)$ is normalized radiance as energy per unit time, with $\gamma =\frac{2k}{c^2}$, $T$ is temperature and $k$ is Boltzmann's constant, which gives an energy per cycle scaling with $\nu$ and a high frequency cut-off $h\nu$ scaling with atomic energy $kT$.

The connection $h\nu =E$ also occurs in the law of photoelectricity
  • $h\nu = P + K$,
where $P$ is the release energy and $K=eU$ is the kinetic energy of a released electron with $e$ the electron charge and $U$ the stopping potential. 

The atomic connection $h\nu =E$ between frequency and energy thus has both theoretical and experimental support,  but it does not say that energy is "quantized" into discrete packets of energy $h\nu$ carried by particles named photons of frequency $\nu$. 

The relation $h\nu =E$ is compatible with wave models of both emission from atoms and radiation from clusters of atoms and if so by Ockham's razor particle models have no role to play.

Atomic emission and radiation is a resonance phenomenon much like the resonance in a musical instrument, both connecting frequency to matter.

Text books state that 
  1. Blackbody radiation and the photoelectric effect cannot be explained by wave models.
  2. Hence discrete quanta and particles must exist. 
  3. Hence there is particle-wave duality.    
I give on Computational Blackbody Radiation evidence that 1 is incorrect, and therefore also 2 and 3. Without particles a lot of the mysticism of quantum mechanics can be eliminated and progress made. 

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