User:Patrick0Moran/Heisenberg matrix2

As a result of many interactions with his fellow researchers, by 1925 Heisenberg had arrived a two very important guiding principles: (1) The models (theories) by which he attempted to account for the frequencies and intensities of the bright line spectrum of hydrogen would only make use of things that could actually be observed. (2) The theoretical description of these phenomena should not be formulated in some arbitrary way dependent only on the whim of the physicist, but should attempt to mirror the processes that occur in nature. Heisenberg recalled, "Bohr would always say, 'First we have to understand how physics works; only when we have completely understood what it is about can we hope to represent it by mathematical schemes.' " (Heisenberg, Conversations, p. 230)

Heisenberg had only the frequencies and the intensities of the bright line spectum to use as basic evidence. The transition amplitude is computed by taking the square root of the transition intensity (i.e., the measured intensity for a spectral line associated with some possible transition).

The importance of taking account of multi-step transitions of electron energy (symbolized by terms such as X(n, n-a) and X(n-a. n-b)) first achieved theoretical importance when Heisenberg helped Kramer in his successful attempt to explain light dispersion in translucent media. It was necessary to take account of the fact that incoming photons of a certain energy/frequency are absorbed and electrons undergo a change of state, essentially being "boosted" into a higher energy state, and that thereafter these electrons will return to their equilibrium state. Sometimes an electron will lose exactly the same energy it just gained, in which case it will emit a photon of the same frequency as the incoming photon. At other times an electron will make two transitions, losing part of its energy and generating a photon of corresponding energy, at each transition. (See the diagram above.) Thus, in an experiment involving many photons, part of the dispersed radiation will not be at the same frequency as the incoming radiation.

In his own work, Heisenberg discovered that a central problem would be how to multiply quantum theoretical quantities. "I gradually saw that the first thing I must know was: if I have all of the amplitudes of a coordinate x, and of another coordinate y, how can I get the amplitudes of the product xy?" (Heisenberg, Conversations, pp. 260-261) He saw a possible answer in using the general method employed in the Kramers dispersion paper.

Although Heisenberg originally did not think about it in such a technical way, the formula given above contains instructions for the diagonal matrix that represents the product of the multiplication of a pair of matrices. Shortly after Bohr saw Heisenberg's complete set of calculations he realized that matrix mathematics would be the natural way to express the above equation in visual form. It says, in effect, that the intensity of radiation in a transition from n to n-b is equal to the sum of the products found by multiplying the amplitude of a transition from n to n-a by the amplitude of a transition from to n-a to n-b, and so on.

The formula directs us to compute the sum of the series:
X(n,n-a)X(n-a,n-b)
X(n-a,n-b)X(n-b,n-c)
X(n-b,n-c)X(n-c,n-d)
etc.

(X just represents any quantum mechanical variable such as amplitude, frequency, etc., that one may be interested in. No special significance lies in the fact that Heisenberg capitalized it.)

Matrices

1

The matrix formulation of the key expression in Heisenberg's new quantum mechanics is as follows:

Schema for a table of transition frequencies produced when electrons change states (sometimes called "orbitals"):

Electron States	n-a	n-b	n-c	....
n	f(n→n-a)	f(n→n-b)	f(n→n-c)	.....
n-a	f(n-a→n-a)	f(n-a→n-b)	f(n-a→n-c)	.....
n-b	f(n-b→n-a)	f(n-b→n-b)	f(n-b→n-c)	.....
transition....	.....	.....	.....	.....

xxxxx

Electron States	n-b	n-c	n-d	....
n-a	a(n-a→n-b)	a(n-a→n-c)	a(n-a→n-d)	.....
n-b	a(n-b→n-b)	a(n-b→n-c)	a(n-b→n-d)	.....
n-c	a(n-c→n-b)	a(n-c→n-c)	a(n-c→n-d)	.....
transition....	.....	.....	.....	.....

3

Heisenberg's "law for matrix multiplication" says that X, the product of two quantities such as a frequency and an amplitude, is found by summing over the range of values of a, the result found by multiplying the frequency in one transition by the amplitude associated with the following transition. That is, $X(n,n-b)=\sum _{a}^{}\,F(n,n-a)A(n-a,n-b)$

(Here, F represents frequency and A represents amplitude.)

What is the reason for the staggering of positions in the two matrices, or, to put it in other words, why didn't Heisenberg use X(n→b)= Σ F(n→n-a)A(n→n-a) as his definition of matrix multiplication? The explanation can be given in several different ways: (1) The formula that Heisenberg uses follows from the Ritz Combination Principle -- something that was well established as a fundamental description of the class of phenomena being described in the new formula. (2) One could never see a case in which an electron makes a transition from some initial state n to some consequent state n-a and then immediately makes the same transition. There would have to be an intervening event for the electron in question to make a transition from n-a to n so that it could once again go from n to n-a. The object of the matrix multiplication schema is to show all possible transitions that could occur in a given physical system, so it naturally excludes those transitions that would be impossible. (3) This is the formulation that, to Heisenberg's delight, gave theoretical answers that corresponded closely to the well known facts.

Moreover, doing things this way has a very important consequence. It matters which order the two matrices are put in. X(n→b)= Σ F(n→n-a)A(n-a→n-b) says that if one starts computing a product X by making the first term a frequency, then the first frequency used will be the one that is produced by a transition from n to n-a, and the first amplitude used used will be the one from n-a to n-b. But if one reverses the order of matrices (and, hence the order of measurements) the appropriate formula is X(n→b)= Σ A(n→n-a)F(n-a→n-b) so the first frequency used will be the one that is produced by the transition from n-a to n-b and the first amplitude used will be the one from n to n-a. The rather magical seeming consequence that pq is not equal to qp is then seen to be the consequence of electrons never magically repeating the same transition without anything happening between times to restore them to their initial state. On the macro scale it is a commonplace experience that no book falls off a shelf twice unless in the intervening time someone has restored it to its initial position in the bookcase. It is possible, however, to see a book first fall from shelf to table, and then to see it fall from table to floor.

The matrix for the product of the above two matrices is:

Electron States	n-b	n-c	n-d	.....
n	A	.....	.....	.....
n-a	.....	B	.....	.....
n-b	.....	.....	C	.....

Where:
A=f(n→n-a)*a(n-a→n-b)+f(n→n-b)*a(n-b→n-b)+f(n→n-c)*a(n-c→n-b)+f(n→n-d)*a(n-d→n-b)+f(n→n-e)*a(n-e→n-b).....

B=f(n-a→n-a)*a(n-a→n-c)+f(n-a→n-b)*a(n-b→n-c)+f(n-a→n-c)*a(n-c→n-c)+f(n-a→n-d)*a(n-d→n-c)+f(n-a→n-e)*a(n-e→n-c).....

C=f(n-b→n-a)*a(n-a→n-d)+f(n-b→n-b)*a(n-b→n-d)+f(n-b→n-c)*a(n-d→n-d)+f(n-b→n-d)*a(n-d→n-d)+f(n-b→n-e)*a(n-e→n-d).....

D=f(n-c→n-a)*a(n-a→n-e)+f(n-c→n-b)*a(n-b→n-e)+f(n-c→n-c)*a(n-c→n-e)+f(n-c→n-d)*a(n-d→n-e)+f(n-c→n-e)*a(n-e→n-e).....

E=f(n-d→n-a)*a(n-a→n-f)+f(n-d→n-b)*a(n-b→n-f)+f(n-d→n-c)*a(n-c→n-f)+f(n-d→n-d)*a(n-d→n-f)+f(n-d→n-e)*a(n-e→n-f).....

If the matrices were reversed, the following values would result:

A=a(n→n-a)*f(n-a→n-b)+a(n→n-b)*f(n-b→n-b)+a(n→n-c)*f(n-c→n-b)+a(n→n-d)*f(n-d→n-b)+a(n→n-e)*f(n-e→n-b).....
B=a(n-a→n-a)*f(n-a→n-c)+a(n-a→n-b)*f(n-b→n-c)+a(n-a→n-c)*f(n-c→n-c)+a(n-a→n-d)*f(n-d→n-c)+a(n-a→n-e)*f(n-e→n-c).....
C=a(n-b→n-a)*f(n-a→n-d)+a(n-b→n-b)*f(n-b→n-d)+a(n-b→n-c)*f(n-d→n-d)+a(n-b→n-d)*f(n-d→n-d)+a(n-b→n-e)*f(n-e→n-d).....
D=a(n-c→n-a)*f(n-a→n-e)+a(n-c→n-b)*f(n-b→n-e)+a(n-c→n-c)*f(n-c→n-e)+a(n-c→n-d)*f(n-d→n-e)+a(n-c→n-e)*f(n-e→n-e).....
E=a(n-d→n-a)*f(n-a→n-f)+a(n-d→n-b)*f(n-b→n-f)+a(n-d→n-c)*f(n-c→n-f)+a(n-d→n-d)*f(n-d→n-f)+a(n-d→n-e)*f(n-e→n-f).....

Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied.

Populating the matrices

The energy levels, the frequencies associated with each transition between energy levels, and the energies of the photons produced in each transition could all be determined by means already in existence. What remained to be determined was how to calculate the transition amplitudes. Heisenberg achieved that goal by extrapolating from classical equations.

end

Mystery one: "Every element has its characteristic bright-line spectrum, but how can these irregular sequences be given a theoretical treatment? They don't make any rhyme or reason!" But then: Balmer: λ_n=Å•3646•(n2/(n2-4) where (n=3,4,....) and later, in a more productive format: Rydberg and Ritz: 1/λn = RH(1/4 - 1/n2) where (n = 2, 3, 4...) Mystery two: "Light waves hit a metal surface but monochromatic light gives the same voltage on the (de facto) light meter. Stronger light gives more amperage, but the same voltage. What the hay?!?" Mystery solved: E = hf (Planks's constant) f = c/λ (nothing new here, but novices need to see how to get from frequencies to wavelengths and back. Eelectron = hc/λ Planck-Einstein relation: Ephoton = Ei - Ej = hf Mystery three: "The hydrogen spectrum is predictable, for the visible spectrum, but how about ultraviolet and infrared?" Bohr's scheme of energy states predicts many spectral lines: 1/λ = RH( 1/nf 2 - 1/ni 2 ) where (nf = 1,2,3... and ni > Nf) Mystery four: "So what is with these bright lines? Why are some brighter than the others? Bet you can't explain that!"

---

The complete theory of quantum mechanics first appeared 1925, in the work of Werner Heisenberg (1901-1976), who won the Nobel Prize in Physics in 1932 .

Heisenberg followed a long path leading up to his groundbreaking formulation of a new form of quantum mechanics that was not a futile attempt to patch classical physics. One of the crucial elements in his journey was his early work on difference equations. Also important was his very close connection with Niels Bohr and with Bohr's closest collaborator Hendrik Kramers. Kramer's 1924 work on dispersion depended on difference equations. in a way that Heisenberg would eventually employ.

The goal of quantum physics is to give a theoretical account of phenomena on the atomic scale that classical physics cannot explain. Giving a quantum-theoretical treatment to the hydrogen bright line spectrum was a key step forward. The old quantum mechanics could already give a good theoretical treatment of hydrogen’s frequencies, but it could not account for its intensities.

In 1925, Heisenberg created "a scheme which was capable in principle of determining uniquely the relevant physical quantities (transition frequencies and amplitudes)." His method, building on earlier successes, was to adapt the methods of classical physics for calculating the intensities of the harmonic frequencies that compose the radiation of radio transmitters and other such systems whose dimensions were larger than atoms and molecules, but to adapt their formulas so that they would correctly apply on the atomic scale. Transition amplitudes are mathematically related to intensities.

Unlike earlier workers in this field, Heisenberg was unwilling to base theories on the supposed existence of things such as planet-like electrons in orbits. Instead, he insisted on basing himself on what could actually be seen and measured in the laboratory. So he used the frequencies and the intensities of the hydrogen bright line spectrum as his starting point. He adapted the classical way of predicting the amplitudes of the spectrum that would be produced a macro-world device such as a radio transmitter antenna to create a way to calculate amplitudes for tiny producers of electro-magnetic radiation like hydrogen atoms. Keep in mind, however, that Heisenberg wanted to be able to calculate the intensities of the bright lines in the hydrogen spectrum. In classical physics, the intensity of something is just the square of its amplitude.

Heisenberg had learn from his teacher Bohr that it is important to make one’s theoretical model of the world as close to the actual workings of nature as is possible, so at this point he borrowed an idea that had originally been discovered when he was assisting Kramers in his work on dispersion, so he wrote:

$Y(n,n-\beta )=\sum _{\alpha }^{}\,X(n,n-\alpha )X(n-\alpha ,n-\beta )$

This equation says (in case it is intensity that is being calculated), that the intensity Y associated with the transition of electrons from the energy state called “n” to the energy state after next, called “n - β" is to be found by adding together all of the products found by multiplying the amplitude of one transition, from “n” to “n -α" by the amplitude of the neighboring transition, the one from “n - α" to “n - β", and so on.

Schematic diagram of two-step return of an electron to its equilibrium position, and the resulting production of two photons

The two terms, $X(n,n-\alpha )$, and $X(n-\alpha ,n-\beta )$, follow from the Ritz combination principle. X and Y can be regarded as represeneting things such as frequencies, transition amplitudes, etc., that Heisenberg wanted to be able to work with. The fact that they are capitalized in this equation has no special significance.

At this point Heisenberg had not studied matrix mathematics. When he showed nis new work to his teachers and colleagues they very soon helped him extend and generalize his discovery into the full range of quantum phenomena.

All of the quantum theoretical calculations come out of discoveries such as Balmer’s formula, and physical regularities such as the Rydberg constant, the speed of light, etc. The mathematics needed for deriving the actual amplitude numbers is much more complicated, but in itself it does not produce any unexpected or unhoped for consequences. The rule for multiplying quantum quantities given above, however, had a consequence that Heisenberg seems to have noticed almost immediately. If one calculates some quantity U that has as components something called R and something called S it makes a difference whether one starts with:

$U(n,n-b)=\sum _{a}^{}\,R(n,n-a)S(n-a,n-b)$

or

$U(n,n-b)=\sum _{a}^{}\,S(n,n-a)R(n-a,n-b)$

As will become clear later on when some sample matrices are shown, it actually matters very much. For instance, if one made U related to products of frequencies times amplitudes, and then made another set of computations in which U’ related to products of amplitudes times frequencies, the answer for U would be different from the answer for U’. This complication, which Heisenberg originally found troubling, led to the discovery of the Heisenberg uncertainty principle.

---

The complete theory of quantum mechanics first appeared 1925, in the work of Werner Heisenberg (1901-1976), who won the Nobel Prize in Physics in 1932 .

Heisenberg followed a long path leading up to his groundbreaking formulation of a new form of quantum mechanics that was not a futile attempt to patch classical physics. One of the crucial elements in his journey was his early work on difference equations. Also important was his very close connection with Niels Bohr and with Bohr's closest collaborator Hendrik Kramers. Kramer's 1924 work on dispersion depended on difference equations. in a way that Heisenberg would eventually employ.

The goal of quantum physics is to give a theoretical account of phenomena on the atomic scale that classical physics cannot explain. Giving a quantum-theoretical treatment to the hydrogen bright line spectrum was a key step forward. The old quantum mechanics could already give a good theoretical treatment of hydrogen’s frequencies, but it could not account for its intensities.

In 1925, Heisenberg created "a scheme which was capable in principle of determining uniquely the relevant physical quantities (transition frequencies and amplitudes)." His method, building on earlier successes, was to adapt the methods of classical physics for calculating the intensities of the harmonic frequencies that compose the radiation of radio transmitters and other such systems whose dimensions were larger than atoms and molecules, but to adapt their formulas so that they would correctly apply on the atomic scale. Transition amplitudes are mathematically related to intensities.

Unlike earlier workers in this field, Heisenberg was unwilling to base theories on the supposed existence of things such as planet-like electrons in orbits. Instead, he insisted on basing himself on what could actually be seen and measured in the laboratory. So he used the frequencies and the intensities of the hydrogen bright line spectrum as his starting point. He adapted the classical way of predicting the amplitudes of the spectrum that would be produced a macro-world device such as a radio transmitter antenna to create a way to calculate amplitudes for tiny producers of electro-magnetic radiation like hydrogen atoms. Keep in mind, however, that Heisenberg wanted to be able to calculate the intensities of the bright lines in the hydrogen spectrum. In classical physics, the intensity of something is just the square of its amplitude.

Heisenberg had learn from his teacher Bohr that it is important to make one’s theoretical model of the world as close to the actual workings of nature as is possible, so at this point he borrowed an idea that had originally been discovered when he was assisting Kramers in his work on dispersion, so he wrote:

$Y(n,n-\beta )=\sum _{\alpha }^{}\,X(n,n-\alpha )X(n-\alpha ,n-\beta )$

This equation says (in case it is intensity that is being calculated), that the intensity Y associated with the transition of electrons from the energy state called “n” to the energy state after next, called “n - β" is to be found by adding together all of the products found by multiplying the amplitude of one transition, from “n” to “n -α" by the amplitude of the neighboring transition, the one from “n - α" to “n - β", and so on.

Schematic diagram of two-step return of an electron to its equilibrium position, and the resulting production of two photons

The two terms, $X(n,n-\alpha )$, and $X(n-\alpha ,n-\beta )$, follow from the Ritz combination principle. X and Y can be regarded as represeneting things such as frequencies, transition amplitudes, etc., that Heisenberg wanted to be able to work with. The fact that they are capitalized in this equation has no special significance.

At this point Heisenberg had not studied matrix mathematics. When he showed nis new work to his teachers and colleagues they very soon helped him extend and generalize his discovery into the full range of quantum phenomena.

All of the quantum theoretical calculations come out of discoveries such as Balmer’s formula, and physical regularities such as the Rydberg constant, the speed of light, etc. The mathematics needed for deriving the actual amplitude numbers is much more complicated, but in itself it does not produce any unexpected or unhoped for consequences. The rule for multiplying quantum quantities given above, however, had a consequence that Heisenberg seems to have noticed almost immediately. If one calculates some quantity U that has as components something called R and something called S it makes a difference whether one starts with:

$U(n,n-b)=\sum _{a}^{}\,R(n,n-a)S(n-a,n-b)$

or

$U(n,n-b)=\sum _{a}^{}\,S(n,n-a)R(n-a,n-b)$

As will become clear later on when some sample matrices are shown, it actually matters very much. For instance, if one made U related to products of frequencies times amplitudes, and then made another set of computations in which U’ related to products of amplitudes times frequencies, the answer for U would be different from the answer for U’. This complication, which Heisenberg originally found troubling, led to the discovery of the Heisenberg uncertainty principle.

Pre-Matrix formulation

As a result of many interactions with his fellow researchers, by 1925 Heisenberg had arrived a two very important guiding principles: (1) The models (theories) by which he attempted to account for the frequencies and intensities of the bright line spectrum of hydrogen would only make use of things that could actually be observed. (2) The theoretical description of these phenomena should not be formulated in some arbitrary way dependent only on the whim of the physicist, but should attempt to mirror the processes that occur in nature. Heisenberg recalled, "Bohr would always say, 'First we have to understand how physics works; only when we have completely understood what it is about can we hope to represent it by mathematical schemes.' " (Heisenberg, Conversations, p. 230)

Heisenberg had only the frequencies and the intensities of the bright line spectum to use as basic evidence. The transition amplitude is computed by taking the square root of the transition intensity (i.e., the measured intensity for a spectral line associated with some possible transition).

The importance of taking account of multi-step transitions of electron energy (symbolized by terms such as X(n, n-a) and X(n-a. n-b)) first achieved theoretical importance when Heisenberg helped Kramer in his successful attempt to explain light dispersion in translucent media. It was necessary to take account of the fact that incoming photons of a certain energy/frequency are absorbed and electrons undergo a change of state, essentially being "boosted" into a higher energy state, and that thereafter these electrons will return to their equilibrium state. Sometimes an electron will lose exactly the same energy it just gained, in which case it will emit a photon of the same frequency as the incoming photon. At other times an electron will make two transitions, losing part of its energy and generating a photon of corresponding energy, at each transition. (See the diagram above.) Thus, in an experiment involving many photons, part of the dispersed radiation will not be at the same frequency as the incoming radiation.

In his own work, Heisenberg discovered that a central problem would be how to multiply quantum theoretical quantities. "I gradually saw that the first thing I must know was: if I have all of the amplitudes of a coordinate x, and of another coordinate y, how can I get the amplitudes of the product xy?" (Heisenberg, Conversations, pp. 260-261) He saw a possible answer in using the general method employed in the Kramers dispersion paper.

Although Heisenberg originally did not think about it in such a technical way, the formula given above contains instructions for the diagonal matrix that represents the product of the multiplication of a pair of matrices. Shortly after Bohr saw Heisenberg's complete set of calculations he realized that matrix mathematics would be the natural way to express the above equation in visual form. It says, in effect, that the intensity of radiation in a transition from n to n-b is equal to the sum of the products found by multiplying the amplitude of a transition from n to n-a by the amplitude of a transition from to n-a to n-b, and so on.

The formula directs us to compute the sum of the series:
X(n,n-a)X(n-a,n-b)
X(n-a,n-b)X(n-b,n-c)
X(n-b,n-c)X(n-c,n-d)
etc.

(X just represents any quantum mechanical variable such as amplitude, frequency, etc., that one may be interested in. No special significance lies in the fact that Heisenberg capitalized it.)

Matrices

The matrix formulation of the key expression in Heisenberg's new quantum mechanics is as follows:

Schema for a table of transition frequencies produced when electrons change states (sometimes called "orbitals"):

Electron States	n-a	n-b	n-c	n-d	n-e	....
n	f(n→n-a)	f(n→n-b)	f(n→n-c)	f(n→n-d)	f(n→n-e)	.....
n-a	f(n-a→n-a)	f(n-a→n-b)	f(n-a→n-c)	f(n-a→n-d)	f(n-a→n-e)	.....
n-b	f(n-b→n-a)	f(n-b→n-b)	f(n-b→n-c)	f(n-b→n-d)	f(n-b→n-e)	.....
n-c	f(n-c→n-a)	f(n-c→n-b)	f(n-c→n-c)	f(n-c→n-d)	f(n-c→n-e)	.....
n-d	f(n-d→n-a)	f(n-d→n-b)	f(n-d→n-c)	f(n-d→n-d)	f(n-d→n-e)	.....
transition....	.....	.....	.....	.....	.....	.....

Schema for a related table showing the transition amplitudes:

Electron States	n-b	n-c	n-d	n-e	n-f	....
n-a	a(n-a→n-b)	a(n-a→n-c)	a(n-a→n-d)	a(n-a→n-e)	a(n-a→n-f)	.....
n-b	a(n-b→n-b)	a(n-b→n-c)	a(n-b→n-d)	a(n-b→n-e)	a(n-b→n-f)	.....
n-c	a(n-c→n-b)	a(n-c→n-c)	a(n-c→n-d)	a(n-c→n-e)	a(n-c→n-f)	.....
n-d	a(n-d→n-b)	a(n-d→n-c)	a(n-d→n-d)	a(n-d→n-e)	a(n-d→n-f)	.....
n-e	a(n-e→n-b)	a(n-e→n-c)	a(n-e→n-d)	a(n-e→n-e)	a(n-e→n-f)	.....
transition....	.....	.....	.....	.....	.....	.....

Heisenberg's "law for matrix multiplication" says that X, the product of two quantities such as a frequency and an amplitude, is found by summing over the range of values of a, the result found by multiplying the frequency in one transition by the amplitude associated with the following transition. That is, $X(n,n-b)=\sum _{a}^{}\,F(n,n-a)A(n-a,n-b)$

(Here, F represents frequency and A represents amplitude.)

What is the reason for the staggering of positions in the two matrices, or, to put it in other words, why didn't Heisenberg use X(n→b)= Σ F(n→n-a)A(n→n-a) as his definition of matrix multiplication? The explanation can be given in several different ways: (1) The formula that Heisenberg uses follows from the Ritz Combination Principle -- something that was well established as a fundamental description of the class of phenomena being described in the new formula. (2) One could never see a case in which an electron makes a transition from some initial state n to some consequent state n-a and then immediately makes the same transition. There would have to be an intervening event for the electron in question to make a transition from n-a to n so that it could once again go from n to n-a. The object of the matrix multiplication schema is to show all possible transitions that could occur in a given physical system, so it naturally excludes those transitions that would be impossible. (3) This is the formulation that, to Heisenberg's delight, gave theoretical answers that corresponded closely to the well known facts.

Moreover, doing things this way has a very important consequence. It matters which order the two matrices are put in. X(n→b)= Σ F(n→n-a)A(n-a→n-b) says that if one starts computing a product X by making the first term a frequency, then the first frequency used will be the one that is produced by a transition from n to n-a, and the first amplitude used used will be the one from n-a to n-b. But if one reverses the order of matrices (and, hence the order of measurements) the appropriate formula is X(n→b)= Σ A(n→n-a)F(n-a→n-b) so the first frequency used will be the one that is produced by the transition from n-a to n-b and the first amplitude used will be the one from n to n-a. The rather magical seeming consequence that pq is not equal to qp is then seen to be the consequence of electrons never magically repeating the same transition without anything happening between times to restore them to their initial state. On the macro scale it is a commonplace experience that no book falls off a shelf twice unless in the intervening time someone has restored it to its initial position in the bookcase. It is possible, however, to see a book first fall from shelf to table, and then to see it fall from table to floor.

The matrix for the product of the above two matrices is:

Electron States	n-b	n-c	n-d	n-e	n-f	....
n	A	.....	.....	.....	.....	.....
n-a	.....	B	.....	.....	.....	.....
n-b	.....	.....	C	.....	.....	.....
n-c	.....	.....	.....	D	.....	.....
n-d	.....	.....	.....	.....	E	.....
...	.....	.....	.....	.....	.....	.....

Where:
A=f(n→n-a)*a(n-a→n-b)+f(n→n-b)*a(n-b→n-b)+f(n→n-c)*a(n-c→n-b)+f(n→n-d)*a(n-d→n-b)+f(n→n-e)*a(n-e→n-b).....

B=f(n-a→n-a)*a(n-a→n-c)+f(n-a→n-b)*a(n-b→n-c)+f(n-a→n-c)*a(n-c→n-c)+f(n-a→n-d)*a(n-d→n-c)+f(n-a→n-e)*a(n-e→n-c).....

C=f(n-b→n-a)*a(n-a→n-d)+f(n-b→n-b)*a(n-b→n-d)+f(n-b→n-c)*a(n-d→n-d)+f(n-b→n-d)*a(n-d→n-d)+f(n-b→n-e)*a(n-e→n-d).....

D=f(n-c→n-a)*a(n-a→n-e)+f(n-c→n-b)*a(n-b→n-e)+f(n-c→n-c)*a(n-c→n-e)+f(n-c→n-d)*a(n-d→n-e)+f(n-c→n-e)*a(n-e→n-e).....

E=f(n-d→n-a)*a(n-a→n-f)+f(n-d→n-b)*a(n-b→n-f)+f(n-d→n-c)*a(n-c→n-f)+f(n-d→n-d)*a(n-d→n-f)+f(n-d→n-e)*a(n-e→n-f).....

If the matrices were reversed, the following values would result:

A=a(n→n-a)*f(n-a→n-b)+a(n→n-b)*f(n-b→n-b)+a(n→n-c)*f(n-c→n-b)+a(n→n-d)*f(n-d→n-b)+a(n→n-e)*f(n-e→n-b).....
B=a(n-a→n-a)*f(n-a→n-c)+a(n-a→n-b)*f(n-b→n-c)+a(n-a→n-c)*f(n-c→n-c)+a(n-a→n-d)*f(n-d→n-c)+a(n-a→n-e)*f(n-e→n-c).....
C=a(n-b→n-a)*f(n-a→n-d)+a(n-b→n-b)*f(n-b→n-d)+a(n-b→n-c)*f(n-d→n-d)+a(n-b→n-d)*f(n-d→n-d)+a(n-b→n-e)*f(n-e→n-d).....
D=a(n-c→n-a)*f(n-a→n-e)+a(n-c→n-b)*f(n-b→n-e)+a(n-c→n-c)*f(n-c→n-e)+a(n-c→n-d)*f(n-d→n-e)+a(n-c→n-e)*f(n-e→n-e).....
E=a(n-d→n-a)*f(n-a→n-f)+a(n-d→n-b)*f(n-b→n-f)+a(n-d→n-c)*f(n-c→n-f)+a(n-d→n-d)*f(n-d→n-f)+a(n-d→n-e)*f(n-e→n-f).....

Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied.

Populating the matrices

The energy levels, the frequencies associated with each transition between energy levels, and the energies of the photons produced in each transition could all be determined by means already in existence. What remained to be determined was how to calculate the transition amplitudes. Heisenberg achieved that goal by extrapolating from classical equations.

What was known before Heisenberg

Balmer: λ_n=Å•3646•(n²/(n²-4) where (n=3,4,....)

Rydberg and Ritz: 1/λ_n = R_H(1/4 - 1/n²) where (n = 2, 3, 4...)

E = hf

f = c/λ

E_electron = hc/λ

Planck-Einstein relation:

E_photon = E_i - E_j = hf

Bohr's scheme of energy states predicts many spectral lines:

1/λ = R_H( 1/n_f ² - 1/n_i ² ) where (n_f = 1,2,3... and n_i > N_f)

So on this basis, one could predict the spectral lines for hydrogen, and the energy of the photons associated with each of the lines. The only thing that would not be known on this basis would be the intensities of each of the spectral lines.

Symbols in Aitchison

ω(n,n-α) = 1/h (W(n)-W(n-α)) ω is frequency W is energy Equation (1) in Aitchison

In Equation (2) in Aitchison, n as a label for an electron state ω(n) as its fundamental frequency

In Equation (3) in Aitchison, the αth harmonic of ω(n) is written as ω(n)α

In Equation (4) Aitchison writes the quantum mechanical analog of ω(n)α as ω(n,n-α)
P(n,n-α) is written for the transition probability per unit time
h-barω(n,n-α) is written for emitted energy

Ritz combination principle explains: new spectral lines determined by additive and subtractive combination of two known spectral lines.

h-bar ω_ln =E_l - E_n = E_l - E_m + E_m - E_n

α starts off being used as an ordinalizer for a harmonic. β starts off being a substitution for α + γ, also from a classical formula.

Equation (10) in Aitchison: Y(n,n-β) = Σ X(n,n-α)X(n-α,n-β) sum over all possible values of α.

Thinking out loud and addressing issues with secondary sources

Lindsay and Mongenau

Hence the theory had to be built upon the facts of spectroscopy. What, then, are the immediate data of spectroscopy? In their crudest form they are spectral lines of definite frequencies and definite intensities. Upon unraveling the spectrum all frequencies appear as transitions between energy levels the positions of which can be completely determined. The intensities, on the other hand, are simply related to the probabilities of transition between the various levels. In general, therefore, atomic spectroscopy provides us with two sets of numbers: possible energies E₁,E₂,...E_i..., and transition probabilities, T₁₁,T₁₂,...T_ij...., T_ij being the probability of transition between the ith and the jth energy state, measuring the intensity of the line of frequency (E_j-E_i)/h.

-- Lindsay and Mongenau, 454

The writing of this passage is muddy. "Upon unraveling the spectrum all frequencies appear as transitions between energy levels the positions of which can be completely determined." What the authors are trying to say is that any given bright line in the spectrum, any frequency, is the result of transitions of electrons between energy levels (the so-called "orbitals"), and that these frequencies can be determined (by the equations listed at the top of the current article).

"The intensities, on the other hand, are simply related to the probabilities of transition between the various levels." The intensities are experimental data. The transition amplitudes are inferred by Heisenberg from the intensities. In other words, he argues that there must be a probability for a transition to occur between any two levels. So Heisenberg must have been arguing from the measured intensity of each line and the energy associated with each photon in that line to calculate the photon flux. Is that number the transition amplitude? Or is the square root of that number the transition amplitude? (See: http://voh.chem.ucla.edu/vohtar/fall97/20A-2/qa.html, among many other places that indicate that it should be the square root. But Heisenberg and those who write about this course of development do not make that part clear.) (In other places in quantum mechanics, the amplitudes of the psi wave are squared to give the probabilities of certain occurrences (I'm thinking of the double-slit experiment here).)

"In general, therefore, atomic spectroscopy provides us with two sets of numbers: possible energies E₁,E₂,...E_i..., and transition probabilities, T₁₁,T₁₂,...T_ij...., T_ij being the probability of transition between the ith and the jth energy state, measuring the intensity of the line of frequency (E_j-E_i)/h.}" Again, the authors seem dead set on explaining things in reverse. The intensities are measured, and their measure gives an indication of the probabilities of transition. Or, if one wants to move from theory to observation, then the photon energies associated with transitions from pairs of electron energy states are associated in the theory with transition amplitudes, and the transition amplitudes can be used to calculate the intensities of the lines in the spectrum.

More critique of L & M

Lindsay and Mongenau speak of the "immediate data of spectroscopy," What was measurable using the tools of spectroscopy was actually the frequencies of the various lines, and their intensities. Lindsay and Mongenau call them "definite," but in fact individual measurements all involve experimental errors. Nevertheless, the results of many individual measurements had led experimenters to believe that they had very reliable numbers. But one needs to read the quoted passage closely because the energies mentioned are what they call, more precisely, energy states. These energy states refer to the energy states of electrons, not to the energy carried by photons. A electron transition from one energy state to another would produce a photon equal in energy to the difference of the two electron energy states. And from this photon's energy one could compute its frequency, and vice-versa. From the measured intensity of a line in the spectrum, and from the energy carried by any one photon at that characteristic frequency, one could calculate the number of photons that would at any one instant be involved in transiting from one electron energy state to another electron energy state, and that would yield a number related to the transition probability.

What Heisenberg had to work with in the beginning

The energy states that appear in the Heisenberg matrices had values that pertained to the electron(s). The energy that was gained by absorption of an incoming photon or carried off by a newly generated photon was equal to the difference between the energy states between which the electron involved had moved. The energy states could be calculated by the Rydberg-Ritz combination principle. The computed values could be confirmed by empirical means.

Extending

The relation between the frequency of a photon and its energy is given by the Planck/Einstein relation, E=hf. Bohr postulated that hf = E_i - E_f where E_i and E_f are the initial and final electron energy states. So at this point in history, the energy states of an electron could be computed, and the frequencies of photons involved in the changes between energy states could be calculated. What could as yet not be calculated was the intensities of each of the lines in the bright line spectrum. Heisenberg asked himself how intensities were calculated in systems that could be adequately described by classical physics. The answer was that intensity equals the square of the amplitude. So Heisenberg looked for an analogous kind of amplitude, and he found it in what he called transition amplitude. A transition amplitude is a function of the number of photons generated in a radiating system between two electron energy levels (orbitals) multiplied by the energy of each photon (which is a function of its frequency).

Putting parts together

Given the state of physics in 1925, Heisenberg had all the parts, but he did not know how to put them together. He had the energy levels of the electrons, the frequencies of photons produced by transitions between any two energy levels, the energies of those individual photons, and he could compute the empirical values for the numbers of photons moving between any two energy levels (i.e., the relevant transition intensities). How the experimental values for intensities could be derived on a quantum theoretical basis was not known.

Why multiply amplitudes?

Back to the hydrogen atom. Assuming that Heisenberg really did want to multiply the transition amplitudes, as he says he does, why would he want to do so? At least in this simple context it doesn't seem to make sense. The most likely transition to occur is from the next higher electron energy level down to the ground electron energy level. The "adjacent" transition is less likely. So the product of those two transition amplitudes would be less than the square of the larger value. Why the amplitude of one line in the spectrum should affect the intensity of an adjacent line in the spectrum is not intuitively obvious, and why transitions that result in photons of other frequencies and between other electron energy state should be involved in the intensity of any transition between two electron energy states is also not intuitively obvious.

A clue from the end of the Aitchison paper

Aitchison says that one result of this Heisenberg paper is the realization that all possible paths that would result in any photon emission must be taken into account. So perhaps a more productive way to work would be to look beyond the equations that led to this conclusion and instead work backwards from an "inventory" of all possible transitions either according to where they originate or according to where they terminate.

Classical vs. quantum visualizations

Intensity and frequency

classical intensities

quantum intensities

Using the classical model for emission of radiation by vibrating electrons, one would expect a continuum of frequencies, something like the mock-up in the first image. The empirically determined frequencies, however, show very sharp peaks (represented by bars in the second diagram) in the lower frequencies, and the continuum predictions are accurate only for the highest frequencies shown. Before Heisenberg, the magnitudes of these peaks could not be predicted.

Heisenberg had empirical measures of intensity. So he wanted to be able to calculate intensities from something that he could get by computation from other quantum theoretical equations. The starting point was to learn how to write the quantum mechanical equivalent of
I = A².
If I am not mistaken, he followed the lead of the paper he worked on with Kramers to write:

$I(n,n-b)=\sum _{a}^{}\,A(n,n-a)A(n-a,n-b)$

where I = intensity and A = amplitude.

It appears that Heisenberg had in mind that in as orbitals approach the classical limits the separation between them approaches zero, so that transitions between adjacent orbitals are associated with energies that approach equality. So A(n,n-a)A(n-a,n-b) approaches A². (Note the left-most region of the above diagram.)

If this is the case, the numerical value of a term such as A(n,n-a) cannot be numerically equal to the intensity that is measured for the transition from n to n-a. That would amount to saying that
I = I²
It seems likely that in the midst of making other analogies in an attempt to see what consequences would follow, Heisenberg simply took the square root of intensity measures for each of the bright lines and interpreted those numbers as his transition amplitudes. If that is the case, then

$I(n,n-b)=\sum _{a}^{}\,A(n,n-a)A(n-a,n-b)$

would mean that the the intensity of radiation produced in the range of the spectrum between the frequency corresponding to the transition (n, n-a) to the frequency corresponding to the transition (n-a,n-b) would equal the product of the amplitudes associated with those two transitions.

Amplitudes and intensity

Amplitudes and wave lengths

What is the physical interpretation of the squaring of amplitudes to calculate the intensity? In the case of a water wave impacting some fixed object it is clear that the amplitude (height) of the wave would need to be used to calculate the pressure in square units that would be applied by the advancing wave. It would be easier to visualize a spherical mass with a vibrating surface in contact with some mass on its surface.

The empirical amplitude values are calculated by taking the square root of the intensities, but the equation that defines the intensities would seem to be inconsistent in that there is no way to calculate the intensity of the transition from the ground level, n, to the next higher orbital, n-a. The reason is that the only formula that Heisenberg gives us says:

$I(n,n-b)=\sum _{a}^{}\,A(n,n-a)A(n-a,n-b)$

In the case of hydrogen atoms radiating photons, however, the analogy given in the section above, that of the magnitude of the force applied to a mass in contact with a vibrating spherical surface, is difficult to maintain. The intensity of a beam of light is a function of the number of photons that impinge upon the target and their frequency/energy. The transition amplitudes measure the probability that transitions occur between various energy levels in an atom. The equation indicates that the intensity of radiation depends on the transition amplitudes of two adjacent electron energy states.

In the regions of the spectrum that are indistinguishable from a classical description, the "adjacent" transition amplitudes would be virtually identical, so their product would equate to the square of either of them. But in the range where only the quantum description will work, the amplitudes are not equal.

Probably I am not taking proper account of the basic formula that Heisenberg offered that indicated multiplying variables in sequential pairs of electron states and then summing the series of multiplications. Maybe the symbolism was not clear to me that the intention was to sum the products of all transitions that terminate at the ground state, then sum the products of all transitions that terminate at the the state n-a, and so forth.

Quantum theoretical amplitudes

In order to have more than a telephone book theory, Heisenberg must be able to calculate the amplitudes on a theoretical basis, i.e., a basis that states relationships among factors like the orbitals involved in a given transition, the frequency of radiation produced in each transition, the energy associated with the radiation... and what else?

Sources in Quantum Mechanics, p. 264, gives an expression that may equate to amplitude.

Bibliography

Used directly in this paper

Ian J. R. Aitchison, David A. MacManus, and Thomas M. Snyder, "Understanding Heisenberg's 'magical' paper of July 1925: a new look at the calculational details. arXiv:quant-ph/0404009v1 1 Apr 2007.

Useful readings for students

http://www.chemteam.info/Electrons/Electrons.html

move elsewhere

<a href="http://science.jrank.org/pages/10933/Quantum-New-Quantum-Mechanics-Heisenberg-Schr-dinger-Dirac.html">Quantum - The New Quantum Mechanics Of Heisenberg, Schrödinger, And Dirac</a> http://online.itp.ucsb.edu/online/utheory03/styer/pdf/Styer.pdf

trans freq 225

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Related issues: http://philsci-archive.pitt.edu/archive/00003094/01/BuschShilladay_CU.pdf