User:Patrick0Moran/Aitchison article

Notes on "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details" by Ian J.R. Aitchison, David A. MacManus, and Thomas M. Snyder, 2007-04-01.
http://arxiv.org/abs/quant-ph/0404009<t>

Symbols

E: energy

e: undefined variable Probably e is the base of the natural logarithm, but then what is ε?

(h/2π)ω(n,n-α): emitted energy in the transition between n and n-α.

i: the number which, squared, yields -1.

m: undefined variable (probably it is just mass, as usual)

n: a Bohr stationary state. (But it is also used to represent a specific value of x (the position on thex axis.) (And it also represents the state [of an electron?]

P: transition probability per unit time.

P(n,n-α): quantum version of the transition probability per unit time for a transition between n and n-α.

W(n): the energies, W, of the Bohr stationary states n.

x(n,t): coordinate associated with a certain Bohr stationary state n at time t.

X: appears to be a component of x, i.e., the value which, when summed over values of α from +∞ to -∞, yields x.

X_αn: the αth component of X for some Bohr stationary state n.

X(n,n-α): the quantum analog of X_αn, i.e. the transition amplitude.

X(n,n-α)exp[iω(n)(α+γ)t: a "magical" replacement for the classical x(t).

Y: just a variable used to represent the product of some analog of x(t)². See p 4.

Y(n,n-β): the sum of the products over the range α of X(n,n-α) and X(n-α,n-β)

Greek

α: a difference in frequency measured (by amount α) from the lowest Bohr stationary state n.

ω(n): an angular frequency associated with the Bohr stationary state n.

ω(n-α): an angular frequency associated with the Bohr stationary state n-α.

ω(n,n-α): an angular frequency associated with the transition between Bohr stationary states n and n-α.

ω(n-β): a second angular frequency associated with the Bohr stationary state ω(n-β) -- Heisenberg originally wrote ω(n-α-β)

ε₀: undefined variable --- appears in denominator of (3) which has e in the numerator.

• • ε is the small epsilon symbol and represents electrical permittivity.
  • ε₀, is a constant equal to 8.854 187 817... x 10-12 F m-1. But I don't know when exactly the constant is used. Here it is on CODATA: electric constant ε₀.

(ε = 2.7183 (According to Sears, Optics) But that is written elsewhere as "e", the base of the natural log.

log_e 10 = 2.3026)

Mixed:

X_αn: The αth X associated with Bohr stationary state n. X has got to be a quantity, the sum of a series of which (for different α and some t yields x(n,t), the coordinate associated with that Bohr stationary state at that time t.

Equations

60

"Equations (56) and (60) are Heisenberg's first results, and they are in fact appropriate to the simple (unperturbed) oscillator."

(60) a⁽⁰⁾(n,n-1)= ((h/πmω₀)^1/2•(n^1/2)

Unknowns:

What does the superior (0) mean?

What does m represent?

What does the "sub 0" mean?

82

(82) a⁽²⁾(n,n-1)= ((11 β³)/(72(ω₀)⁶) · (n·n^½)

What is a⁽²⁾? It is not a², so it has to have something to do with the second harmonic or something like that, no?

What is ω₀?

I need to define β and/or work it directly into the second equation as I did for the first.

end

General question unrelated to the following two equations:

capping a constant with an umlaut: Does that mean that the constant represents a matrix.

λ: undefined variable used frequently, but not obviously any particular wavelength.

Equations 60 and 82:
(I rewrote (60) to include β directly since I am getting ready to work these equations out in a spreadsheet.)

(I've used • to indicate ordinary multiplication.)

(60) a⁽⁰⁾(n,n-1)= ((h/πmω₀)^1/2•(n^1/2)

What does m represent? Mass of the electron? Or something else?

(82) a⁽²⁾(n,n-1)= ((11 β³)/(72(ω₀)⁶) • (n·n^½)

What is a⁽²⁾? It is not a², so it has to have something to do with the second harmonic or something like that, no?

Similarly, what is a⁽⁰⁾?

What is ω₀?

What is ε₀ -- The same equation (3) has e in the numerator. ε is base of the natural logarithm in Sears, but in this article that is e. Is ε₀ Vacuum permittivity???

---

Aitchison 2

General question unrelated to the following two equations:

capping a constant with an umlaut:

In eq. (12) $\scriptstyle {\ddot {x}}=d^{2}x/dt^{2}$

(second order time derivative).

λ: undefined variable used frequently, but not obviously any particular wavelength.

λ is a small parameter used in perturbation theory in the following way: let

$\scriptstyle a^{(0)}$

be the zero-order solution, that is the solution of the problem if there is no perturbation. The exact solution of

the problem including the perturbation is written as the series

$\scriptstyle a^{(0)}+\lambda a^{(1)}+\lambda ^{2}a^{(2)}+\lambda ^{3}a^{(3)}+\ldots$

(compare eq. (25)). Hence the unperturbed solution corresponds to

$\scriptstyle \lambda =0.$

Inserting the series solution into the equation of motion yields a power series in

$\scriptstyle \lambda$.

Requiring the solution to be satisfied for arbirary value of

$\scriptstyle \lambda$ yields the recurrence relations (compare eq. (32), etc.) from which the unknown quantities

$\scriptstyle a^{(n)}$

can be calculated.

ε₀ --

If you refer to eq. (3) then ε₀ is indeed the vacuum permittivity.

Equations 60 and 82:
(I rewrote (60) to include Aitchison's "β" equation (given under (60)) directly since I am getting ready to work these equations out in a spreadsheet.)

(60) a⁽⁰⁾(n,n-1)= ((h/πmω₀)^1/2•(n^1/2)

m represents the mass of the unperturbed harmonic oscillator, having eigenfrequency ω₀ (compare eq. (19) in which m = 1 has been taken).

What is a⁽⁰⁾? 0th harmonic? Something else?

Zero order solution (see above).

What is ω₀? Another kind of reference to the harmonic structure?

See above.

(82) a⁽²⁾(n,n-1)= ((11 β³)/(72(ω₀)⁶) • (n·n^½)

What is a⁽²⁾? It is not a², so it has to have something to do with the second harmonic or something like that, no?

a⁽²⁾ is the second order contribution to the perturbed solution (see above). (WMdeMuynck)

Matrix mechanics article

According to the article,

$X_{nm}(t)=e^{2\pi i(E_{n}-E_{m})t/h}X_{nm}(0)$.

is the original form of Heisenberg's equation of motion.

Where is that equation in Heisenberg's 1925 article? It must be written in a different form.

The article gives as the rule by which the arrays should be multiplied:

$(XP)_{mn}=\sum _{k=0}^{+\infty }X_{mk}P_{kn}$

This one is easier to find. P. 266:

$C(n,n-b)=\sum _{a={-\infty }}^{+\infty }A(n,n-a)B(n-a,n-b)$