Talk:Matrix mechanics/Archive 2

Archive 1

Archive 2

Uncertainty principle

I needed to correct a completely fallacious statement that the uncertainty principle was special to matrix mechanics and didn't appear in the Schroedinger picture at all. It is clear that the person who added this statement had never studied the Schroedinger picture, as the uncertainty principle arises naturally within the theory. See Schiff - Quantum Mechanics or any other textbook covering the Schroedinger picture for details. Elroch 13:00, 19 January 2006 (UTC)

Found this article clearer than anything else Ive seen on the topic to date. Good Job! Reflection 10:51, 23 September 2006 (UTC)

Beautiful article! —Preceding unsigned comment added by 128.84.161.129 (talk) 00:03, 25 February 2008 (UTC)

wording

"Heisenberg, after a collaboration with Kramers[4], began to understand that the transition probabilities were not quite classical quantities, because the only frequencies that appear in the Fourier series should be the ones that are observed in quantum jumps, not the fictional ones that come from Fourier-analyzing sharp classical orbits. He replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of the Fourier series."

It is not clear in what sense the coefficients are "fuzzed out." Isn't the problem that the Fourier series per se will always yield an infinite number of values forming a continuous line if graphed out, whereas the actual values found in nature are discontinuous? The writing makes it sound as though the graph of frequencies should not be drawn with a line between frequency values but with a band between frequency values, and perhaps a "fuzzy" band at that. I suppose "fuzzy" is o.k. if you can think of a graph that looks like the teeth of a comb as fuzzy. But what may look soft and fuzzy from afar may look sharp and jagged up close.

Is the author intending to imply that the values of an individual matrix, frequencies property to transitions among electron energy states, let's say, are themselves "fuzzy"? P0M (talk) 14:27, 24 June 2009 (UTC)

The "fuzziness" is an important intuitive thing in matrix mechanics. It's what Heisenberg means by "The electrons will no more move on orbits". The important thing is that matrix mechanics starts from classical Fourier series, and then changes the frequencies to be quantum frequencies instead of classical ones. This has the effect of "fuzzing out" the classically sharp phase-space trajectory into a phase-space "blob" of area about h. This is not completely precise, because it mixes up classical and quantum concepts. The way in which Heisenberg makes this intuition more precise is by stating the uncertainty principle, but that comes later. He understood the uncertainty business earlier, from working with the matrices. It's hard to explain this well in words: but if you play around with matrix mechanics a little, it's obvious.

For a classical periodic motion the Fourier series for the position is discrete, with frequencies which are integer multiples of the inverse-period. That's what makes it a Fourier series. It's discrete, not continuous, because the motion is periodic. If you call the m-th Bohr orbit trajectory X_m(t), then

$X_{m}(t)=\sum _{n}X_{mn}e^{2\pi int/T}\,$

And this is the proto-matrix description. But it's not a matrix yet, because the frequencies at each n are integer multiples of 2\pi/T. In order to make the continuity with matrix mechanics obvious, it's best to shift the zero point of n, so that n=0 is moved to n=m.

$X_{m}(t)=\sum _{n}X_{mn}e^{2pii(n-m)t/T}\,$

Heisenberg's insight is that you don't have integer multiples of a single frequency, because this is just a classical approximation. If you look at the X motion at level m, the actual decomposition of the motion is using the matrix elements and the quantum frequencies:

$X_{m}(t)=?\sum _{n}X_{mn}e^{i(E_{m}-E_{n})t}\,$

I put the question mark, precisely because this sum is stupid. It isn't even a real number, because the positive and negative frequencies don't match. But this was the thing that corresponds to the m-th Bohr orbit in the classical limit, where it is just the usual Fourier series. That's what Heisenberg means by "the electrons will no more move on orbits".

You don't get back all the Fourier coefficients of the orbits from the matrix. You sort of can, for large n and when m close to n, but that only reconstructs the gross motion. If you want the fine-scale motion, you look for the high Fourier coefficients, and these are far off the diagonal. This is the "fuzziness", and the amount of fuzziness is given by the extent to which E_m - E_n is not an integer multiple of 2pi/T. It's measured by the imaginary part of the dubious sum above, more or less. But that sum is not a good invariant construction in matrix mechanics--- the sum over n is not an allowed matrix operation. The right construction is to take products of operators and measure their imaginary part, that's XP-PX, and this is the usual uncertainty principle.Likebox (talk) 18:48, 15 July 2009 (UTC)

Allow me to make a few remarks on the above. There is no fuzziness in Heisenberg's approach. What he tried to calculate is the amplitude of the radiation emitted by an electron in the m-th Bohr state. In a first approximation restricting oneself to electric dipole radiation this amplitude is proportional to the position of the electron. Taking into account the empirically given fact that the radiation only contains Bohr frequencies one immediately arrives at the above expression (the one with the question mark). That this expression is not a real number is not at all stupid. Also in classical electrodynamics it is quite usual to use complex calculus from the start, finally taking the real or imaginary part of the complex solution if necessary.WMdeMuynck (talk) 13:32, 16 July 2009 (UTC)

The values in each matrix are not fuzzy, so I think we need some word other than "fuzziness" then. Graphing Heisenberg's values would result in a "comb" graph, not a fuzz brush. To put that idea in other terms, if one uses Heisenberg's procedures to fill out a matrix, the values are all mathematically precise. The indeterminacy is only revealed when one does the math for the multiplication of, e.g. the matrix for pq and the matrixfor qp and the math shows a difference of ih/2pi (if I remember correctly). Born says that when Heisenberg gave him his draft results and took off on vacation to recuperate, Born stared at the math for days, suddenly realized that he was looking at "instructions" for writing out matrices, and somehow is math was good enough that the fact that there would be a difference involving the factors i and h suddenly came into his consciousness. So what is the word needed to replace "fuzzed out"? P0M (talk) 18:49, 16 July 2009 (UTC)

As regards the relation of Fourier transform and matrix multiplication found by Born in Heisenberg's paper I reproduce here an answer I gave some time ago on User talk:WMdeMuynck (I have expanded it a bit and added some further remarks):

What Heisenberg probably had in mind, is calculating the classical Poynting vector from the expressions for the electric and magnetic fields on page 880 of his 1925 article. However, he restricted himself in the article to one-dimensional examples, so these expressions are not applicable as they stand. The only thing he took from it, is that the fields will depend on the position $\scriptstyle x(t)$ of the electron (which corresponds to the dipole approximation of the interaction between particles and field), this variable being considered as a classical quantity, restricted by quantum conditions of the Old quantum theory. He used for the harmonic oscillator the Fourier expansion

$x(t)=\sum _{n}c_{n}e^{in\omega t},$

This looks like equation (2a)of the English version, given there as

$x(n,t)=\sum _{\alpha }{\mathfrak {A}}_{\alpha }(n)e^{i\omega n(a)t}$

This equation is found on p. 264 of the English language edition. P0M (talk) 22:46, 19 July 2009 (UTC)

in which the quantum condition $\scriptstyle E_{n}=n\omega$ is taken into account. $\scriptstyle x(t)$ as defined above is a periodic function of t with period $\scriptstyle T=1/\omega$ (which probably corresponds to your comb graph?). The crucial difference between classical and quantum is that classically the frequency of the field is equal to the rotation frequency of the electron, whereas in quantum theory the Bohr rule is satisfied, equating the frequency of the field with the difference of the rotation frequencies. Not having studied Sommerfeld's theory I am not sure what this change implies as regards periodicity, but if not too many frequencies contribute I suppose that the motion will become quasi-periodic rather than periodic.

Now in the classical theory the amplitude of the field would be proportional to $\scriptstyle x(t)$ and its intensity proportional to $\scriptstyle x(t)^{2}$. Then (compare eq. (3) of Heisenberg's 1925 article)

$x(t)^{2}=\sum _{nm}c_{n}c_{m}e^{i(n+m)\omega t}.$

Equation 3 in the English version is

${\mathfrak {B}}_{\beta }(n)e^{i\omega (n)\beta t}=\sum _{\alpha }{\mathfrak {A}}_{\alpha }{\mathfrak {A}}_{\beta -\alpha }e^{i\omega (n)(\alpha +\beta -\alpha )t)}$

and I do not see anything more similar to the equation you present as (3). P0M (talk) 00:34, 20 July 2009 (UTC)

For some fixed N in this expression the contribution at frequency $\scriptstyle N\omega$ is

$\sum _{n}c_{n}c_{N-n}e^{iN\omega t}.$

According to Heisenberg this does not work properly, as it does not take into account the Bohr rule (the quantum theoretical expressions on p. 881) given here as

$\omega (n,m)+\omega (m,n')=\omega (n,n').$

These are the "combination relations" given just before the first numbered equation in the English version, viz:

$\nu (n-\beta ,n-\alpha -\beta )+\nu (n,n-\beta )=\nu (n,n-\alpha -\beta )$

This format disguised the "matrix nature" of the equations until he realized what was really going on and simplified them as what was "given here".

Sorry to be pedantic about all of these formulations, but everybody who writes on this subject seems to favor a different set of letters, and it is driving me to distraction.P0M (talk) 00:34, 20 July 2009 (UTC)

For the harmonic oscillator we have $\scriptstyle \omega (n,m)=(n-m)\omega .$ Heisenberg replaced in the Fourier series the frequencies $\scriptstyle n\omega$ by the Bohr frequencies $\scriptstyle \omega (n,m)$ corresponding to a transition between orbits n and m. Then eq. (7) is directly obtained as the component of $\scriptstyle x(t)^{2}$ at the difference frequency $\scriptstyle \omega (n,m)$:

$\sum _{m'}c_{nm'}c_{m'm}e^{i[(n-m')+(m'-m)]\omega t}=\sum _{m'}c_{nm'}c_{m'm}e^{i(n-m)\omega t}.$

It is strange that this is not equation (7) in the English version, nor is it anything like it. Or did you mean to communicate some calculated consequence between equation (7) and what you have written? I have (7) as:

${\mathfrak {B}}(n,n-\beta )e^{i\omega (n,n-\beta )t}=\sum _{\alpha }{\mathfrak {A}}(n,n-\alpha ){\mathfrak {A}}(n-\alpha ,n-\beta )e^{i\omega (n,n-\beta )t}$

Perhaps after my confusion over the equations is resolved, someone can provide a calculation that involves frequency, amplitude, momentum, or some other such predicted measure that would be used as an element of a matrix, but not the result of multiplying two or more matrices, and would itself involve "± k" where k is not anticipated experimental error. P0M (talk) 01:14, 20 July 2009 (UTC)

Here matrix multiplication is evident since the component of $\scriptstyle x(t)^{2}$ at frequency $\scriptstyle (n-m)\omega$ (in this one-dimensional example proportional to the intensity of the field at frequency $\scriptstyle (n-m)\omega$) is proportional to $\scriptstyle \sum _{m'}c_{nm'}c_{m'm}.$

Note that this is not quantum mechanics but Old quantum theory. Here no quantum mechanical transition probabilities are calculated, but just field intensities. Note also that eq. (7) is not derived, but guessed by varying the classical theory so as to comply with the Bohr rule.

I hope this will helpWMdeMuynck (talk) 23:29, 30 May 2009 (UTC)

By itself there is nothing quantum mechanical in Heisenberg's application of Fourier theory (this theory being widely applied in classical wave theories too), nor in matrix multiplication (being in a natural way implied by considering products of functions represented by Fourier series). Basically, Born just learned from Heisenberg's paper that a product of functions corresponds to a matrix multiplication of Fourier representations.

Quantum conditions were applied for the first time to matrices by Born and Jordan (Zeitschr. f. Phys. 34, 858-888 (1925)), starting from the usual conditions of the Old quantum theory. In this way for a very special problem the (canonical) commutation relation for q and p was derived, which became the basis of quantum mechanics.

Whether one wants to associate the Heisenberg inequality with fuzziness depends on one's interpretation of the mathematical formalism of quantum mechanics. Heisenberg was the first to derive an uncertainty relation (Zeitschr. f. Phys. 43,172-198 (1927)) from the commutation relation for q and p found by Born and Jordan. Heisenberg considered this uncertainty relation as a consequence of the disturbing influence of measurement, referring to measurement inaccuracies. He did not believe that this proved the non-existence of sharp trajectories, but he considered it metaphysical to assume their existence because as a consequence of the inaccuracies of the measurements it would be experimentally impossible to prove their existence. Physicists like Jordan were more rigorously denying the existence of trajectories, thus giving rise to the idea that trajectories do not exist at all and should be replaced by some fuzzy notion.

Whether one believes the one or the other depends on one's knowledge and belief in the sense and nonsense generated on this issue in the 90 odd years since quantum mechanics started. It is impossible to deal with this in a few lines (you might have a look at my web site, referred to on User:WMdeMuynck which I am currently updating). Suffice it to say here that I think that Heisenberg's empiricist ideas, inducing him to take into account the influence of measurement, should be taken very seriously, but that the founding fathers of quantum mechanics were hampered by too poor experimental material and too elementary a variant of quantum mechanics to have a sufficient basis for a reliable assessment of the meaning of the theory. For this reason it seems to me that it is impossible to arrive at an understanding of the meaning of quantum mechanics on the basis of the first few papers of the founding fathers.WMdeMuynck (talk) 00:13, 18 July 2009 (UTC)

The original question was whether the statement "[Heisenberg replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of the Fourier series, is a misleading way of describing what Heisenberg did. Above, you said, "There is no fuzziness in Heisenberg's approach." Are you now changing your position? If so, how is "fuzziness" to be defined?

I believe that the average well-informed reader would understand by a "fuzzy" frequency something like 656±k nm. with the understanding that the range is not due to experimental error. But as far as I know, the only "fuzziness" is what Heisenberg called indeterminacy, and that kind of uncertainty comes out of the consequences of matrix multiplications involving mathematically precise values. In their article on the subject of Heisenberg's 1925 paper Aitchison, MacManus, and Snyder speak of Heisenberg having "a scheme which was capable in principle of determining uniquely the relevant physical quantities (transition frequencies and amplitudes." (p. 2)P0M (talk)

If anyone knows any mathematical function that can itself provide a truly random result I am sure that you should patent and copyright it -- or just sell it to NSA or somebody with deep pockets. As far as I know, the existing algorithms for producing "random numbers" all produce numbers the determinative nature of which is difficult to ascertain. For real random numbers one is advised to watch a geiger counter or some other quantum-based producer of events. P0M (talk) 01:57, 20 July 2009 (UTC)

Misunderstandings

The above comments have a few misunderstandings, which is completely understandable because the original literature is so opaque. To make the literature readable, a few notation modernizations are essential:

• radian measure for all frequencies, so that there are no stupid 2pi factors to keep straight.
• hbar is 1, so that the radian frequency and energy are equal, and both denoted by the letter "E".
• Instead of using the frequency combination formula nu(a,b) + nu(b,c) = nu(a,c), use the equivalent more intuitive fact that the energy emitted in a trasition from state n to state m is the difference in energy of state n and state m: E(n,m) = E_n - E_m.

Connes, for example, uses the same conventions in his explanation of Heisenberg in "Noncommutative geometry".

I'm puzzled. Why bother to insist on discussions of energy in preference to discussions of frequency above, and then speak entirely in terms of frequency below? Either way is fine with me, and as long as people bother to tell me that they are making h = 1, that is fine too. P0M (talk) 06:13, 27 July 2009 (UTC)

It's not a puzzle: energy and frequency are synonyms when hbar is 1, you can use one word or the other interchangably. Remember that h is not 1, hbar is 1, h is 2pi. This is a human convention for frequency measure, but it is a good convention, and it is universal.

Notation aside, the main misunderstanding is that the Heisenberg matrices are not just the same as the classical Fourier series. The Fourier series sort of corresponds to a single row of the matrix (or column, I'll get back to that) at position m, A(m+n,m), as n varies from -m to m (but imagine that m is large so that this might as well be -infinity to infinity).

"Row of the matrix at position m"? I'm guess you mean that there is some row labeled something like v and m is the column label for the value you want us to look at? P0M (talk) 06:13, 27 July 2009 (UTC)

The matrix is constructed from the Fourier series of the m-th Bohr orbit. Let's call the position of the electron in the m-th Bohr orbit X_m(t). Call the n-th Fourier coefficient of this position X_m(n). Then the semi-classical matrix for X would be "something like" X(m,n)= X_m(m+n).

This construction is only the crudest possible semi-classical approximation, and it is not even qualitatively correct. It is wrong, because it doesn't construct a hermitian matrix--- X(m,n) is not the complex conjugate of X(n,m). This construction produces rows which are symmetric around the diagonal point: X(m+n,m) = X*(m-n,m). That's not hermiticity. That's something crazy and not covariant. But these wrong type of matrices are what Kramers was using to calculate transition probabilities in the Bohr theory.

Notice that in these "matrices", The indices n and m are not symmetrical, like they are in a real full-fledged matrix. The index m in X(m,m+n) is counting the bohr-orbit, while the index n is the n-th Fourier coefficient, from a zero point shifted by m.

But Heisenberg (or maybe Heisenberg and Kramers, not sure) now realizes that the correct way to fill out the matrix is by using X(n,m) as complex conjugate to X(m,n), because he has an intuition that the frequencies of the classical Fourier series are just an approximation to the correct motion. To make that work, you need to put the m-th Bohr orbit into the matrix in a cockeyed sort of way. You put it in by making the point X(m+n,m) be the complex conjugate of X(m,m+n). That's hermiticity--- you reflect across the diagonal. Now you have a real matrix--- the rows and the columns are symmetrical.Likebox (talk) 22:33, 27 July 2009 (UTC)

The classical Fourier series has the property that A(m+n,m) = A*(m-n,m), which is the statement that the positive frequencies are the complex conjugates of the negative frequencies. This is exactly the translation of the reality of the Fourier series to matrix language, and it is absolutely not Heisenberg's condition. Heisenberg's condition is that A(m+n,m)=A*(m,m+n).

The difference is because classically, the Fourier frequencies are multiples of 1/T, so that the negative frequencies are exactly negative of the positive frequencies A(-n) and A(n) have opposite frequencies. In quantum mechanics, the frequencies are the differences of energy levels, so that the opposite frequencies are at A(n,m) with frequency E_n - E_m and A(m,n) with frequency E_m - E_n. This means that if you try to find too high fourier coefficients in quantum mechanics, you fail.

The product of Fourier series is by convolution, and the product of matrices is by a form of row-column convolution. This is not exactly the same, for the same reason that the matrices are different than Fourier series. You can think of the matrix as like a Fourier series which has been "bent" around the diagonal in the following way. To reconstruct the Fourier series, start at the very left, go right until you hit the diagonal, then go up from there. This type of thing is very weird, but it's because when you take the classical limit, the rows of the matrix should be thought of as coming closer and closer together until they become a continuous line, while the columns stay discrete. At the same time, the diagonal of the matrix is shifted by horizontal translation so that it becomes vertical, so that the lopsided T-bend of matrix entries becomes a horizontal line (these are my personal visualizations--- the mathematics is unambiguous, but it is best if each person develops their own intuition).

The point is that the Fourier series cannot be reconstructed beyond the first few terms. The high order coefficients of the Fourier series have no quantum analog. A motion can be reconstructed vaguely from the first few coefficients of the Fourier series, but the precise position is not known. That means that the position (and momentum) of the particle has been fuzzed out.Likebox (talk) 16:28, 26 July 2009 (UTC)

I have difficulty following your logic. There seem to me to be at least two major problems, probably just in the manner of presentation:

You mention that the "misunderstanding is that the Heisenberg matrices are not just the same as the classical Fourier series." I take it that this is roughly equivalent to saying:Heisenberg matrices are just the same as the classical Fourier series.

Sorry, I was using imprecise language. Heisenberg's matrices are not the same as the Fourier series, but become the fourier series near the diagonal in the classical limit.Likebox (talk) 22:33, 27 July 2009 (UTC)

Yet you next say:

The classical Fourier series has the property that A(m+n,m) = A*(m-n,m), [which] is absolutely not Heisenberg's condition. Heisenberg's condition is that A(m+n,m)=A*(m,m+n)."

Things that have characteristics that are "absolutely not" characteristics possessed by some other things are not the same things. Their Venn diagrams in the universe of characteristics would not be the same.

Yes, what I meant to say is that the key point is that the frequencies are opposite after lower-diagonal upper-diagonal reflection of the whole matrix, when looking at points related by hermitian conjugation, not after the horizontal left-right reflection of each row around the diagonal entry, they way the classical Fourier series works.Likebox (talk) 22:33, 27 July 2009 (UTC)

You speak of "negative frequencies [that] are exactly negative of the positive frequencies." Frequency is generally calculated as the number of maxima that pass a given point in a standard length of time. Both of these factors are ordinarily considered to be positive. Perhaps you are thinking of a wave of a certain frequency as "positive" and another wave of the same absolute frequency but out of phase with the first as a "negative"? Regardless, for Heisenberg there is a clear interpretation, particularly if we talk about the energies involved. In the transition of an electron from some higher energy state to some lower energy state, energy is lost to the system in the form of a photon with the appropriate amount of energy. In the transition of an electron from some lower energy state to some higher energy state, energy is absorbed by the system in the form of an incoming photon carrying the appropriate energy. The absolute values of the energies involved will be equal, but their signs will be opposite. So in that sense I guess an absorbed photon could be said to have a "negative" frequency.

You say, "In quantum mechanics... the opposite frequencies are at A(n,m) with frequency E_n - E_m and A(m,n) with frequency E_m - E_n." That statement seems to me entirely correct if only one interprets "negative frequencies" as frequencies of light that is absorbed and "positive frequencies" as frequencies of light that is being radiated. The numbers in the Heisenberg picture indicate something that is happening "between" energy levels, and these energy levels have a natural (if oversimplified) analog in the old Bohr planetary orbits. Their absolute values are equal, so it would make sense to me to echo what you say of classic Fourier frequencies, i.e., "that the negative frequencies are exactly negative of the positive frequencies," i.e. |f(n,n-a)| = |f(n-a,n)|.

As the electron achieves higher and higher energy levels, the differences between energy levels become less and less. I think that is why you want to consider a set of values that includes not just a few near and including the ground level, but instruct us to "imagine that m is large so that this might as well be -infinity to infinity." Such a set of matrix elements will include ones such that "the rows of the matrix should be thought of as coming closer and closer together until they become a continuous line." The result is that calculating a result with terms H(l,m) and H(m,n), at a sufficiently high m, is the practical equivalent to squaring either one of them. The classical approach would be to make a single calculation, something like H(m)². At this point, the classical values and the QM values ought to be the same.

The classical picture assumes that there is a continua of frequency values, and that for each of these frequency values one can calculate an amplitude (and therefore one can calculate an intensity). So the classical view would expect glowing hydrogen to produce a continuous spectrum. But it does not. It is therefore wrong to expect a classical fourier series to work for quantum scale events.

So far your discussion has been limited to frequencies and to energies. I do not see that you have argued that either frequencies or energies involved in transitions between energy states would include any random terms. Then in your last paragraph you announce conclusions about motion and momentum of "the particle," but it is entirely unclear whether you mean the position and momentum of an electron that has made a transition from one energy state to another, or a photon that has either been emitted or absorbed, or whether you mean to refer to some other particle.

You can think of the matrix as like a Fourier series which has been "bent" around the diagonal in the following way. To reconstruct the Fourier series, start at the very left, go right until you hit the diagonal, then go up from there. This type of thing is very weird, but it's because when you take the classical limit, the rows of the matrix should be thought of as coming closer and closer together until they become a continuous line, while the columns stay discrete. At the same time, the diagonal of the matrix is shifted by horizontal translation so that it becomes vertical, so that the lopsided T-bend of matrix entries becomes a horizontal line.

I presume that you are talking about some matrix made in the Heisenberg-Born way, no? I am further assuming that this matrix starts from energy levels of "n," i.e., the ground level, as the first column label and also the first row label. In that case the upper left corner element will have a value of zero, being the frequency or the energy (or whatever) involved in a "transition" between an energy level and itself. From that initial zero there will naturally follow a diagonal line of zeroes because each column label will be matched by a like row label, and will therefore signal a non-transition, so no gain of energy, no change of frequency, etc.

Assuming that you mean to begin at any random point below the diagonal string of zeroes, e.g., E(n-q,n), and then follow that row to the right. After about q columns I should see a 0. I then follow that column up to the top element. Good enough? Then presumably I have been taking note of all of the frequencies, or whatever is involved. Further, you assert that this series of values will be that of some Fourier series. So am I to understand, then, that a Fourier series will have a set of negative values, a center value of 0, and then a set of positive values?

One of the obvious peculiarities of the frequencies at "the far left" is that there will be very large differences between values. The first five frequencies in the Lyman series are:

• 2.46738E+15
• 2.9243E+15
• 3.08423E+15
• 3.15825E+15
• 3.19846E+15

So what happens if one decides to reverse the process and interpolate a frequency such as 2.68978E+15? That should be just filling in missing values, from the standpoint of classical physics, no? But these values are ruled out by Heisenberg's scheme. It is only at the far reaches of the matrix that "the rows of the matrix should be thought of as coming closer and closer together until they become a continuous line." When I said earlier that I was not seeing a "fuzzed out" set of values but a "tooths of a comb" situation, this is the kind of "not continuous line" situation I had in mind.

Thanks for your patience.P0M (talk) 06:13, 27 July 2009 (UTC)

To clarify--- all the Heisenberg matrices are constructed from the known classical limit, at very very large values of n, using Bohr's correspondence principle. To get a consistent set of rules for small values of n and m is the whole difficulty.

So Heisenberg's analysis is always at the "far reaches of the matrix", nowhere near the small values of n. After he finds a consistent scheme, lo-and-behold, it works for all values of n.

First negative/positive: Negative frequencies and positive frequencies are concepts in Fourier series. If you use complex Fourier series, the negative frequencies are those which oscillate in one direction in the complex plane, the positive frequencies oscillate in the opposite direction. To understand Heisenberg's reasoning it is essential to be very comfortable with complex Fourier series.

a periodic function X(t) has an expansion as X_m exp(imt) where exp(imt) is a positive frequency when m is positive, and a negative frequency when m is negative (or vice versa, but people pick a convention and stick to it). in quantum mechanics, the positive frequencies of a matrix X(m,n) are positive energy differences, or a gain in energy between two levels, and the negative frequencies X(m,n) are a drop in energy.

Pedanticaly speaking, since Heisenberg's model does not include the electromagnetic field, the atom never makes transitions at all. So it doesn't emit or absorb radiation. But if Heisenberg's matrices describe a charged particle coupled to an electromagnetic field, the matrix elements of X with positive frequency describe transitions where energy is emitted, while the negative frequency matrix elements describe absorption.

Next "fuzziness": The motion of an orbit can be understood from it's Fourier series or from its X as a function of time representation. The matrices are the Fourier series in the classical limit (but only in the classical limit--- X(m,m+n) for m large and n small). This means that the classical Fourier series of X is not 100% well defined. It is only defined for n<<m. This restriction means that if you want to figure out where the particle is to a precision that is too great, you fail, not because the Fourier coefficients are uncertain, but because the concept of Fourier coefficients stops making sense.Likebox (talk) 22:33, 27 July 2009 (UTC)

Another problem

The text currently says:

Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements were the intensities of spectral lines.

For Heisenberg's work there are all kinds of matrices, else there would be no point of talking about matrix multiplication. There can be a matrix for frequencies, for amplitudes, etc. The idea that "the matrix elements were the intensities of spectral lines" is only true for an intensity matrix. Unless one is talking about a matrix filled in with empirical values, an intensity matrix would be a matrix that is the product of other matrices. P0M (talk) 04:10, 22 July 2009 (UTC)

The classical quantity which gives the intensity of radiation for a charged particle moving around is the X operator, because that is the dipole moment. So the sentence was saying hat the matrix elements of the X operator were the intensity of radiation. So it should be enough to say "The Fourier coefficients of the X operator", although I thought it was physically obvious enough to leave to the reader.Likebox (talk) 15:59, 26 July 2009 (UTC)

Main insight

There are misunderstandings in the previous section. Heisenberg's insight is not Fourier series. It's something completely new. Its replacing the classical Fourier series with a quantum mechanical matrix, which makes the frequencies be differences E_n - E_m instead of integer multiples of a basic frequency.

Classically, the frequencies in an orbit are all integer multiples of the inverse period. So the 100th Fourier coefficient is 100 times the frequency of the 1st Fourier coefficient. In quantum mechanics, the matrix element corresponding to the 100th Fourier coefficient is X(m,m+100), and the frequency is E_(m+100) - E_m. E_(m+100)-E_m is not exactly equal to 100 times E_(m+1)-E_m, it is only approximately equal to this in the classical limit: when m is enormous and the matrix elements become continuously varying in the diagonal direction.

This is the central insight, the main thing. It has no counterpart or anticedant anywhere else. Not in wave theory, not in any classical field theory. It is a completely original idea. it comes from the correspondence principle and the Bohr/Einstein relations between energy and frequency. Please internalize this idea first before making corrections to the text.Likebox (talk) 22:47, 27 July 2009 (UTC)

I think we may still have a different idea of what Heisenberg was doing. To begin with, he was not aiming at a matrix formulation. He was trying to get at the basic relationships that, at classical scale, were successfully represented by classical physics but were clearly something else because they did not work at quantum theoretical scale. He was aware that classical physics attributed photon frequencies to hypothetical "waves" emitted by electrons revolving around an atomic nucleus at certain distances. He took the idea from his work with Kramers (where it makes pretty good sense to me on an intuitive level) that what is significant is the fact that in absorbing a photon an electron in orbit around a nucleus will change energy states from a lower "step" to some higher "step," and that in returning to its equilibrium state it is significant that it can do so in a sequence of changes of energy states so that it often will produce two photons the energy of which will total the energy gained by the absorption of one photon. So in general what Heisenberg wants to do is to characterize theoretically what will happen in a transition between some energy state and some other energy state. His general notation for indicating the frequency produced by a transition from energy state n-a to the ground state n would be f(n, n-a). So if one were to construct a grid for frequencies, the natural way would be to have the columns labeled by energies of increasing magnitude, and so too for the labels for the rows, so that the intersection of any row and any column would indicate the energy gain or loss for a transition between the two energy levels (depending on the direction of the change). (The conventions for the signs in this scheme are a bit of a challenge for me to keep straight.)

I think that is probably what you mean when you say that the frequencies are differences E_n - E_m.

The 101th column could then, by my understanding, have a heading of (m =) n - 100. The intersection of that column with the row labeled n would then be the place to note the frequency of a photon produced by falling from the higher energy state of n - 100 to the lower energy state of n. (If it bothers anybody I'm perfectly willing to change the minus signs to plus signs.)

The frequencies produced by this scheme must be in accord with the Ritz Combination Principle, and they are not of equal difference from each other. Only when the energy levels become relatively high do the differences between them become so small as to be practically insignificant, and it is only at those levels that the classical equation actually give practically equivalent results.

For comparison purposes, I would be interested to see the classical predictions for frequencies in the visual spectrum range. P0M (talk) 02:51, 25 August 2009 (UTC)

Explaining the Fuzziness

Suppose you have a particle moving in an ovoid in phase space. Then the position and momentum are given as a function of time:

$X(t)=\sum _{n}X_{n}e^{2\pi int/T}\,$

$P(t)=\sum _{n}P_{n}e^{2\pi int/T}\,$

What happens in quantum mechanics is that the coefficients X_n don't make sense. They don't exist anymore. What does exist is X(j,k), the matrix element of X. You can more or less reconstruct the values of X_n and P_n from X(n,m) by the following recipe:

1. find the value for "J" of this classical motion, the action variable divided by two pi.
2. X_n is equal to X([J],[J]+n) where [J] is the integer part of J (or J/hbar if you want to have explicit hbars), and P_n is the matrix element P([J],[J]+n).

This recipe reconstructs something that you can call X_n and P_n, but it only works for small values of n. So the notion of an orbit is "fuzzy", you don't get high Fourier coefficients. High-n Fourier coefficients describe the fine details of the motion, while low-n coefficients describe the gross overall features of the motion. So the gross motion makes sense, but the fine-scale motion does not. If you figure out the amount of fuzziness in the reconstruction, the orbit is only well defined in a fuzzy region in phase space of area about h. This is the uncertainty principle.

The reason for the fuzziness is not that the matrix elements are uncertain: they are 100% precise. The reason is that they don't have the right properties of a Fourier series at large values of n. The frequencies are not integer multiples of the fundamental frequency. This means that the classical motion is ill defined on too fine a scale.Likebox (talk) 23:04, 27 July 2009 (UTC)

I think this all is clearly expressed. I still wonder some readers won't think that the quantum theoretical account fails to live up to the well-defined classical description and is therefore in some pejorative sense inaccurate. But I guess that may be another way to state the position of the hidden variable people. Don't worry, I'm not going to try to make substantive changes in the article. Thanks.P0M (talk) 03:02, 28 July 2009 (UTC)

My worry is just that, since you had trouble with it, this central point is not clear enough in the article. Perhaps saying "fuzzy" gives people the wrong idea. I don't know how to fix. I don't mind if you make changes, but some of the comments from before suggest that not everything is crystal clear yet, so please meditate on it until it is clear, and then hopefully you can communicate the idea better than it is right now.Likebox (talk) 03:29, 28 July 2009 (UTC)

The level of the article is such that my comprehension is not what it should be judged against. I was a physics major for only one year and then I realized that my poor math background would make it impossible for me to keep up -- and over the last half century things have not improved very much. I only got into this whole thing by being frustrated with all of the popularizations that only serve to confuse readerss. The standard for whether the MM article is clear enough is whether the reader who has more than ten credit hours of calculus and also more than ten hours of physics looks at the discussion of fourier functions and the words in the article generally and gets the right idea. I have seen a diagram somewhere on Commons that may help the readers that do not read mathematics the way Mozart read musical scores. Maybe I can find it. P0M (talk) 05:20, 29 July 2009 (UTC)

I had a little time to recover the matrix for frequency values (as an example for the way that Heisenberg's calculations would work out). It is easy to see what you mean by the "bending" mentioned above (in the "quote box").

For n = 1
and m =
0001 f = 0
0002 f = -1215.02273406944
0003 f = -1025.17543187109
0004 f = -972.018187255549
0005 f = -949.236510991747
0006 f = -937.303251996423
0007 f = -930.251780771912
0008 f = -925.731606910047
0009 f = -922.657888683979
0010 f = -920.471768234422
0100 f = -911.358186370715
0101 f = -911.356390458994
1000 f = -911.267961820039
1001 f = -911.267960000232

Then going around the bend, one would find the positive values (with m constant at 1 and n varying).}}

The higher the values of m above, the closer the predicted values of frequency are to predicting a continuum of values. The classical predictions are presumably then practically indistinguishable from the quantum theoretical values at classical limits. If I were to graph the classical values and then erase the values that were inconsistent with the quantum theoretical values, I would not call the resulting graphical representation "fuzzed out." Instead, I might called it "spaced out" (except that "spaced out" already means something in the drug culture), or maybe "gapped out." If that is not too inelegant an expression, perhaps it would suffice. P0M (talk) 21:01, 24 August 2009 (UTC)

Strange gaps in this discussion page

I just noticed, in re-reading this page from the top down, that there are occasional gaps. There was a time, a few weeks ago, when the server was behaving strangely. I wonder if it is possible that things dropped out at that time.P0M (talk) 00:13, 25 August 2009 (UTC)

I just checked. It's the math expressions. They are not always rendering properly. Maybe my connection is just too slow? P0M (talk) 00:20, 25 August 2009 (UTC)

It appears to be a problem with Safari on my Macintosh. No problem with Windows. P0M (talk) 00:51, 25 August 2009 (UTC)

Hbar and h

If you are going to insert factors of hbar, use the symbol $\scriptstyle \hbar$ in place of $\scriptstyle h/2\pi$. The first symbol is the fundamental constant when your angles are measured in radians, while h is the constant for angles measured in fractions of a cycle. Everyone uses radians, not cycles.

But it is best not to insert any hbar factors, keeping everything in natural units. There is no confusion possible, and the hbars are present in early parts, while the discussion of choosing units is interspersed in the main text. If you reinsert all the hbars, this stuff becomes impossible to follow. It was written in natural units for a reason.Likebox (talk) 17:53, 17 February 2010 (UTC)

It is profoundly unhelpful to set (h/2π) to one. Part of the whole deal is to be able to explicitly track the h dependence -- where does it come from? Does it still make sense if h->0 ? Does anything cancel that shouldn't in the classical limit, effectively forcing h=0; or is the classical limit consistent with h remaining small but non-zero?

It is far clearer to keep h in the text, so all this is explicitly clear.

It also means we get to the iconic equation,

$[q,p]={\frac {ih}{2\pi }}$

again with the very important h dependence.

As for using $\scriptstyle \hbar$ in place of $\scriptstyle h/2\pi$ -- they mean exactly the same thing, so what's the big deal? The original papers used h, so for ease of comparison with the original papers, that seems a good reason not to use $\scriptstyle \hbar$. It is also quite instructive to see explicitly where the factor of 2π comes from. Jheald (talk) 19:27, 17 February 2010 (UTC)

I favour keeping the constants explicit. Professional theoretical physicists may not like them, but most readers get confused when they are absent; their presence helps dimensional analysis. As for whether h should be barred or unbarred, I say go with the modern convention, which is barred. --Michael C. Price ^talk 20:28, 17 February 2010 (UTC)

It also depends where it's coming from. When we start by saying "the area of a circle must be an integer multiple of Planck's constant, and the area of a circle of radius $\scriptstyle {\sqrt {2E}}$ is 2πE, so E = nh/2π", then it seems to me it makes most sense to go on using h and to go on using 2π. Personally I find the explicit factor $\scriptstyle {\sqrt {h/2\pi }}$ makes it just a little easier to follow where the matrices are purported to be coming from.

All in all though, I'm a lot more concerned by the issue in the next thread -- that the presentation here seems on the face of it to have only the most tenuous relationship to anything published in 1925. Jheald (talk) 20:56, 17 February 2010 (UTC)

Sources?

Where is the section "mathematical development" sourced from? I'm not saying it's wrong, but it seems to bear very little similarity to the calculations presented in any of the papers in 1925. Is there a later textbook that this development of the material can be sourced to? Jheald (talk) 19:32, 17 February 2010 (UTC)

Archive 1

Archive 2