User:Patrick0Moran/Heisenberg matrix


 * N.B. This section is getting picked up by Google for some reason, but it is only a draft in progress.
 * See User:Patrick0Moran/Rewrite_QM for rewrite of materials pertaining intro to quantum mechanics, esp. Heisenberg matrix mechanics

Equations in sequence
$$\lambda\ = B\left(\frac{m^2}{m^2 - 2^2}\right)$$
 * $$ \frac{1}{\lambda} = R\left ( \frac{1}{2^2} - \frac{1}{n^2} \right )$$
 * $$ \frac{1}{\lambda} = R\left ( \frac{1}{m^2} - \frac{1}{n^2} \right )$$

These three equations are really just more and more transparent formulations of the same thing. In the beginning, nobody understood that one term is the inverse of the square of the initial orbital of the electron and the other term is the inverse of the square of the final orbital of the electron when it makes a transition and therefore emits a photon. The equation is a mathematical model of the structure of the hydrogen atom as it pertains to frequencies/wavelengths that are emitted in various possible events.

Physicists & issues in sequence
Lord Kelvin on "two dark clouds"

Ultraviolet catastrophe

Max Planck ca 1879

Balmer 1885

Lenard photoelectric 1899

Einstein, photoelectric effect 1905

xxxBalmer 1885

RR

Bohr, bright line spectra 1913

Bohr's derivation of Balmer's formula -- now in the "one over" format

Dirac matter waves 1923

Wolfgang Pauli exclusion principle  1925

Heisenberg 1925

Born matrix mechanics

Schrödinger

Dirac

Note 1
Aitchison et al. article

symbols used:

ω(n,n-a) is the frequency involved in a transition from state n-a, to ground state n.

Classical physics uses the idea of harmonics, and they have to be integral multiples of the fundamental frequency. But they could also be conceptualized simply as other frequencies at which some physical system can vibrate.

Quantum mechanics uses the idea of set intervals between "orbitals" or between energy states that electrons can reside in. So for each atom with a characteristic set of numbers in the Balmer-related series, there are permitted energy states and therefore a defined set of frequencies at which that physical system can vibrate.

Addition of energy to the vibrating system, e.g., by accepting energy from an incoming photon, must raise the electron to the appropriate energy state, but from there is has the possibility of returning to its equilibrium state by several sequences of radiation events. Each one of these possible returns has its own frequency or series of frequencies, and it is this phenomenon that must explain what the classical theory treats as resonance. So f(n, n-a), for instance, is a "resonant" frequency. I suppose there is logically some difficulty in saying what the fundamental frequency is, but in classical physics the fundamental frequency is taken as the lowest, so perhaps it is reasonable to call f(n, n-a) fundamental frequency.

AM&S are using x to represent the position of a one-dimensional vibrating system.

In classical physics consider the following situation: n labels a "state" -- unclear of what this means in classical terms, however ω(n) is the fundamental frequency of such a "state" or physical system (n) x(n,t) is the location of the vibrating element in the system in state n at time t.

Then he gives the formula


 * $$ x(n,t) = \sum_a X_a (n) e^{i\omega(n) a t},$$

The "n" is just there to note which "state" is being computed, so as a shorthand one can write:


 * $$ x(t) = \sum_a X_a e^{i\omega a t},$$

X must indicate a series of something that relates to steps in the series a, and if ω(n) (or just ω) represents the fundamental frequency then it seems likely to me that X must represent each frequency in the series of resonant frequencies. But that can't be right because AMS call the related X(n,n-a) the amplitude.

That seems to make sense because Heisenberg is taking x(t) as a stand-in for amplitude, and a sum of amplitudes is the only thing that will result in an amplitude. But he appears to be assuming that he knows what these amplitudes are, i.e., knows their values.


 * $$ x(t) = \sum_n c_n e^{in\omega t},$$

Re-reading Heisenberg's article along with de Muynck's message it seems to me that perhaps the first two equations have to be solved first, and then everything follows from them. It seems more and more clear that the X(n,n-a)X(n-a,n-b) business has to take a series of definite values, but that the explicit pathway by which those values are achieved is only hinted at in Heisenberg's article. AM&S don't make it clear either, or at least it is not clear to me.

Dr. de Muynck says he did not start with the first two equations in the 1925 paper. Probably Heisenberg just jumped over the explicit calculation part, I guess. Strange.

Beginning with the breakthrough that let the visible bright line spectrum be calculated, it all turned out to be a matter of using difference equations, and that was because it is not what an electron is doing in an "orbital" but what energy states it transits between. 1/nsquared - 1/msquared, and so forth.

Now I see that Antonio Saraiva has an empirically derived formula for intensities. It also works with orbitals and involves Rydberg's constant. He says his values are good, but when I try to map his values to the set of 6 empirical values I have found published, then the match is not very close. If, however, it turns out that all the values are still that close, then he may be onto something.

scraps