Dirac's Quantum Mechanics

Added: 2026-05-06

[00:00:00]In the late summer of 1925, Verer Heisenberg sent Ralph Fowler a draft of his recent groundbreaking paper creating quantum mechanics. Fowler gave it to one of his students to analyze it.

[00:00:17]In a few weeks, Paul Dak uncovered the essential aspect of Heisenber's work and created his own version of the theory.

[00:00:27]In this video, I will show you how DRA went from an unknown student to one of the key figures in the development of quantum mechanics.

[00:00:39]Heisenberg has sent a copy of the proof to RH Fowler who was my supervisor at the time. Fowler passed it on to me and asked me what I thought of it.

[00:00:54]Paul Dra is normally associated with the famous Draq equation and its prediction of antimatter but this work was done after he was widely recognized in the world of quantum theory in early 1928.

[00:01:12]Draq made his first influential contribution in the autumn of 1925 when he was still a doctoral student in Cambridge. In July of 1925, Dra was on his second year when Verer Heisenberg visited Cambridge to deliver a lecture.

[00:01:32]Heisenberger came to Tumbridge, gave a lecture there. The lecture was mainly on the anomalous s effect but he did refer at the end of his lecture to the new ideas associated with his matrix mechanics which were just uh beginning to form in his mind at the time and I afraid I was too tired to follow just what Heisenberg was saying at the time. It was very late in the evening and at the end of a complicated lecture and uh sorry I didn't pay the importance to those ideas that I should have done.

[00:02:13]>> Although Dra did not grasp the importance of this visit, his supervisor Ralph Fowler asked Heisenberg to share the proofs of his new paper when available. After receiving a copy of the now historic undo paper in early September, Fowler gave it to the raq.

[00:02:34]And this was Heisenberg's first paper on quantum mechanics. I looked through it and I didn't think very much of it.

[00:02:45]I suppose I looked through it too hurriedly and I put it aside and went back to it again after about two weeks and suddenly it got reviewed to me and quite know how that this was really important work and that it was really something new which could provide a solution to the whole of the problems of quantum mechanics.

[00:03:13]Let's see what DRA found on Heisenberg's paper after he read it for a second time.

[00:03:22]This is the definition of a poson bracket between two functions U and V.

[00:03:29]If you're unfamiliar with this, I made a whole video about it in my series on classical physics for quantum mechanics.

[00:03:38]This definition satisfies a collection of identities. Here are a few of them.

[00:03:45]At the core of quantum mechanics is the so-called commutator between two matrices U and V given by this. In my video about matrix mechanics, I presented how Max Bourne and Pascal Jordan found this definition to play a fundamental role in the theory. And just like the posum bracket, the commutator satisfies many identities. From here it seems obvious that the commutator is to quantum mechanics what the quason bracket is to classical mechanics.

[00:04:23]For this reason, it is expected that in the search of quantum equations, the pasom bracket will be replaced by the commutator. However, a direct replacement will not work because plank's constant must appear somehow in the quantum equations.

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[00:06:17]Another clear indication of the close connection between commutators and plus brackets appears when we apply them to the canonical variables Q and P. This was done by Jacobi almost 60 years before Bourne and Jordan's discovery of the canonical commutation relations in quantum mechanics. Here we see how plank's constant appears in the quantum equations and we could simply identify the commutator of any two functions as the imaginary unit and the reduced blanks constant multiplying the posum bracket. Spoiler this is the correct connection between classical and quantum mechanics discovered by dra. This parallel between classical and quantum equations is how this concept is usually presented in quantum mechanics courses.

[00:07:15]In fact, some books claim that this is how Draq found it. But this is not true.

[00:07:21]Here is the problem with this narrative.

[00:07:24]Draq did not know about the canonical commutation relations because he made his discovery before Bourne and Jordan published their paper. Draq was working in isolation during September and early October 1925, whereas the Born Jordan paper appeared only in November.

[00:07:45]It was during one of these Sunday walks, either the end of September or beginning of October, I'm not quite sure which Sunday it was in 1925, that it occurred to me that there was really a great similarity between the quantity u vus, the commutator, and the press bracket, which I had been reading about previously when I had in studying the Hamiltonian the general Hamiltonian theory in Whitaker's book would it be possible to say that this quantity just has to take the place of the price of bracket it was a very exciting idea to me and the difficulty was that I didn't remember very well what the price and bracket was a big At the time that I had been studying it, I didn't know it would be very important.

[00:08:53]One of the take-home messages of this video is debunking this myth. Dra did not use the canonical commutation relations to find his famous formula.

[00:09:05]That would have been a trivial exercise.

[00:09:07]In fact, Dra did it the other way around. He only used the quantum product introduced by Heisenberg to discover this relationship and then when applied to the canonical variables he found the canonical commutation relations independently from Bourne and Jordan. I want to show you now the actual calculation that led Draq to his groundbreaking discovery. In my video decoding the famous undo paper, I presented the four key ideas introduced by Heisenberg. The racket's starting point was Heisenberg's reinterpretation of the classical position of anatomic electron by replacing the classical fura modes by transition amplitudes and the classical harmonic frequency by the frequency associated with atomic transitions between energy levels. This is going to be crucial in a moment because as I will show you Draq worked backwards from the quantum expressions to the classical ones. The other key idea was the most important. Draq realized that the most innovative aspect of Heisenberg's work was the introduction of this non-communive quantum product which he later referred to as the Heisenberg product.

[00:10:31]When I read Heisenberg's paper second time, I picked on this non-comutation as the important feature. It seemed to me that this was really the prime difference between the new mechanics and the old mechanics and that uh what one had to do was to take the old mechanics of Newton or maybe in the lrangeian or Hamiltonian form and modify it in some way so as to bring in the non-commutation >> instead of questioning in the meaning of this non-commutativity as Heisenberg did. Direct's approach was to accept this product as a fundamental way that quantum variables must be multiplied.

[00:11:27]Note that Iraq uses two indices to denote the components of the quantum objects. But contrary to Bourne and Jordan, he does not call them matrices.

[00:11:39]For dra matrices are just a particular case of these objects. Sure we can call them matrices and refer to their product as matrix multiplication. But as a good mathematician built his theory in a more general and abstract way. Months later when the works of Bourne Heisenberg and Jordan were published, DRA introduced the special name Q numbers for quantum variables and C numbers for quantities that satisfy the standard commutative product. For Dra, the matrices of matrix mechanics were just a special type of his Q numbers. He built a physical theory around these mathematical objects. For this, Dra knew that classical differential equations containing differentials of physical quantities are fundamental. He simply postulated two rules for quantum differentiation.

[00:12:43]one for the sum and one for the product of two Q numbers where V here can be anything and the multiplication is the higher product. In a long and quite tedious calculation he obtained that in general any derivative satisfying these two rules becomes a difference of Heisenber products in this form. This is what today we call the commutator between x and a where a is a q number related to b. This is how direct discovered that the commutator of q numbers is central for his theory of quantum mechanics.

[00:13:26]Then came a more important step. He decided to understand the meaning of the commutator in the classical limit. In other words, he wondered what does this commutator represent in classical mechanics. He starts section four of his paper with we shall now consider to what the expression xy minus yx corresponds on the classical theory titled the quantum conditions. This section includes the most important calculation.

[00:14:01]Dra does it in two steps, but here I want to show you all the details because this is really not obvious.

[00:14:09]As Shinger once wrote, Dra has a completely original and unique method of thinking, but he has no idea how difficult his papers are for the normal human being. After you see it, you should be able to follow Direct's calculation with ease. But it really took me a while to process all his steps.

[00:14:33]We write the nm components of the commutator between the q numbers x and y in this form where the sum over k appears from the two heisenber products.

[00:14:47]Leaving the exponential aside for a moment, we will calculate the preactor first. From the definitions introduced by Heisenberg, the index k is short notation for n minus alpha and m is short notation for k minus beta.

[00:15:07]Replacing these definitions, we get this long expression. I remind you that alpha and beta represent the number of levels below n where the electron can transition. Note that alpha and beta can be interchanged. In fact, here is the first of direct tricks. In the second term, alpha is replaced by beta.

[00:15:34]The next step is to add zero in this form. Here we are adding and subtracting the same quantity. Now these four terms can be factorized as follows. These two have the same common factor y whereas the two remaining have the same common factor x. We have not done much only added zero in a clever way and grouped common factors.

[00:16:02]Now comes the step in which drack takes these quantum objects and applies Heisenberg's reinterpretation backwards from quantum to classical expressions.

[00:16:14]The amplitudes with indices n comma n minus alpha become fa modes with index alpha and evaluated at n. And replacing n by n minus beta, we get this second identification.

[00:16:30]Of course, the same applies to the y amplitudes in which case the expression above reduces to this. For Direct's next step, I remind you that the derivative of a function of n is defined in this way. Durac was not a big fan of Bor's correspondence principle, but he used it anyway. I made a full video about this, but all you need to remember is that according to Bor's correspondence principle in the limit of large quantum numbers, the quantum quantities must approach their classical counterparts.

[00:17:08]When it comes to calculations, this idea is expressed as alpha and beta being very small compared to n. Interpreting minus beta as delta n, we get that this difference of x free modes is proportional to the derivative of x with respect to n evaluated at n minus beta.

[00:17:32]But since beta is very small compared to n, this is approximately just n. We will simply denote this by the x alpha to the n. Of course, the same applies to the Y modes and the long expression above reduces to this.

[00:17:52]Now, we will make use of the ideas introduced by Carl Schwvartshield that I presented in a previous video and write the quantum number n in terms of the action variable J and plank's constant.

[00:18:08]So, the expression on the left becomes this.

[00:18:12]With this I bring back the original expression that we want to evaluate. We have found this prefactor. The quantum exponential must also be written in its classical limit. I remind you of Heisenberg's reinterpretation which we now need in the other direction. The exponential becomes this where instead of omega t I wrote 2 pi w is the angle variable.

[00:18:42]Grouping the alpha exponential with x and the beta exponential with y, the expression above becomes this.

[00:18:52]Now comes the last trick. Notice that if we differentiate this product with respect to the angle variable w, we get 2 pi i * the factor that we need above.

[00:19:05]The same applies for the other term.

[00:19:08]This means that including the overall factor 1 / 2 pi i, we can write the following. And the final step is just symbolic. When performing the sum, we recover the furer representation of the classical x and y in each parenthesis.

[00:19:30]And this long expression reduces to this. Since the action angle variables are canonical variables, this result is I H bar times the plus bracket between X and Y.

[00:19:46]We also know that the quason bracket is invariant under canonical transformations. So we can write this for any canonical pair of coordinates Q and P. This is DAC's great result.

[00:20:00]Classical and quantum mechanics are linked via this relation which connects the commutator of Q numbers with the poson bracket of C numbers. In his paper, Dra presented his groundbreaking result in this way. We make the fundamental assumption that the difference between the heisenber products of two quantum quantities is equal to ih bar times their pson bracket expression. This is a fundamental connection between classical and quantum variables. And other than these overall factors, the pasum bracket is the classical limit of the quantum commutator.

[00:20:46]And in fact, the relationship between them is so close that we can put bracket equal to u v minus v u with a numerical coefficient i * cos h. cost h is banks constant over 2 pi.

[00:21:04]This is the start of my work on quantum mechanics. It's turned out to be an equation of general applicability and it provides a way of passing from the classical mechanics of any dynamical system when expressed in the Hamiltonian form to the corresponding theory in terms of the new mechanics of Heisenberg.

[00:21:33]For Draq, this relation allows expressing Heisenberg's theory in formal analogy with classical mechanics. This allowed him to directly calculate what today we call the canonical commutation relations. These relations had been found by Bourne and Jordan the month before, but they were not published yet.

[00:21:56]Dra made the same discovery independently, just a few weeks apart.

[00:22:02]As we have seen all the results by born Jordan and Draq derived from Heisenberg's non-commutative product there was however one important aspect missing in Heisenberg's paper a quantum equation of motion. A complete theory requires the quantum equivalent of f= m * a for any physical system. If you watched my video, the code in Heisenberg's paper, he included only a few examples, but not a general equation. This was only achieved by Bourne and Jordan using properties of matrix algebra. In my video about matrix mechanics, I showed you all the many steps to derive what became known as the Heisenber equation. But since the B Jordan paper was only published in November, this equation was still unknown. Contrary to the long calculation by Bourne and Jordan, Draq was able to find the same quantum equation of motion in a single line. One has equations of motion for any dynamical variable U.

[00:23:16]The classical theory in the hammer towing form says that du by dt equals the passion bracket of u with this capital h. This capital h called the hamiltonian is the total energy expressed in the hamiltonian variables and p.

[00:23:38]And this leads at once to the equation of motion in the new mechanics I hdu by dt = uh h - hu.

[00:23:54]We had then a general equation of motion applicable to any system in quantum mechanics.

[00:24:06]If we know the corresponding theory in classical mechanics expressed in the Hamiltonian form.

[00:24:18]This is the Heisenber equation. Although it was discovered by Bourne and Jordan and independently by Dra.

[00:24:26]Drag wrote his results and submitted his paper during the first week of November 1925 which was published on December 1st. The Royal Society has made this paper publicly available.

[00:24:42]After what I show you in this video, you should be able to read most of it. I highly encourage anybody, in particular physics students, to read this paper. I think that it shows drag at his best. A high level of ingenuity combined with brilliant calculations.

[00:25:02]Years later, Max Bourne recalled reading this paper for the first time. The name Dak was completely unknown to me. The author appeared to be a youngster, yet everything was perfect in its way and admirable.

[00:25:20]This work was Durac's breakthrough into quantum mechanics, but it was far from the last. He continued the mathematical development of his algebra of Q numbers and successfully applied it to the hydrogen atom. He almost beat Powi in demonstrating that Heisenber's seinal idea leads to the Balmer formula for spectral lines. During this time and during the last semester of his doctorate, Durac began teaching the first course on quantum mechanics at Cambridge. Among the few students who attended Dur's lectures was a young American equally fascinated by physics and poetry named J. Robert Openenheimimer.