Two weeks ago, I discussed Max Born's 1954 Nobel lecture about the statistical meaning of the wave function – and the history of quantum mechanics. Many other fathers of quantum mechanics have received their Nobel prizes.
To avoid repeating Heisenberg's photographs, let's include a different hay fever sufferer. ;-)
When it comes to the basic character of quantum mechanics, the most relevant other Nobel prize went to Werner Heisenberg in 1932. Well, he only picked the prize in December 1933 – he was chosen as a winner retroactively because no candidate compatible with Nobel's will was proposed in 1932. OK, let us already look at the lecture
On the first page, Heisenberg makes it clear why Niels Bohr was viewed as the ultimate guru of the quantum community. Heisenberg himself interpreted his own efforts and achievements as refinements of some technical problems in Bohr's old model of the atom.
On one hand, it's legitimate to "discard" this not-quite-correct model of the atom entirely. On the other hand, it has played an extremely important role historically. And people like Heisenberg who went through the era in which Bohr's model of the atom was the "cutting edge of physics" may have captured some fundamentally correct ways of thinking.
On the very same first page, we already see the basic correct philosophy that reduced the rest of his groundbreaking discovery to a sufficiently intense period of mathematical reasoning:
People could see that this constant was universally important and whenever it appeared, something was behaving very differently than what classical physics was predicting. In fact, the very visualizability of the physical system – the assumption that there exists an objective state of the physical system at every moment that may be in principle captured by a picture – was breaking down whenever \(\hbar \neq 0\) was affecting the physical predictions.
Whenever \(\hbar\neq 0\), all pictures are inevitable misleading.
They are misleading because the very assumption of classical physics that certain objects objectively possess certain well-defined physical properties breaks down. The constant \(\hbar\) directly measures how much this classical assumption breaks down in Nature.
I have just visited Heisenberg's grave and asked him to record his explanations what was important for his discoveries in the audio form once again. You may also try the 1968 audio in his native tongue or the 85-minute Heisenberg and the reality question (German).
This insight is what is needed to correctly understand all the phenomena that quantum mechanics successfully explains. It is needed instead of many things that are not needed and that are not the point at all – like hidden variables or many worlds (MWI) or sudden discontinuous objective collapses (GRW) or many worlds or classical particles and waves existing simultaneously (like in Bohm's theory). None of the flawed concepts in the previous sentence has anything to do with the new assumptions that are actually needed to correctly understand all formulae with Planck's constant \(\hbar\) in them. Heisenberg et al. were going in this direction because they wanted to explain some observed phenomena. Instead, people promoting the realist pictures are doing so because they want to defend some philosophical prejudices regardless of the empirical data that don't really play an important role in their reasoning. They want to replace fundamentally unvisualizable quantum mechanics by MWI, Bohm, GRW etc. because they think that the "spirit" of their alternatives is similar to the novelties brought by quantum mechanics, but they are more philosophically pleasing. But that shouldn't matter in physics. Quantum mechanics works to predict and explain the actual phenomena, the actual equations that may extracted from the data.
Everyone who is tempted to think that there is something right about hidden variables, MWI, GRW, or Bohm's theory is urged to read Heisenberg's lecture, or at least the first pages of it, at least 12 times.
On the second page, Heisenberg continues by saying that it was understood that one needed a "new theory" that does reduce to classical physics in the right limit – effectively in a similar way that was already sketched by Bohr's principle of correspondence – but whose character before the limit is taken is qualitatively different.
He also said that the need for the new quantum laws to be probabilistic was gradually appreciated – especially when it came to Einstein's coefficient for the emission of radiation. Because the electromagnetic energy has to be emitted in discrete packets, the calculable continuous numbers can't tell us about the "intensity" because the "intensity" is supposed to be discrete. So they have to tell us about something continuous, namely the probability that the emission takes place. Max Born's Nobel lecture conveyed the same story about the initial reasons why people began to understand that the new laws were unavoidably going to be probabilistic in character.
While there was something good about the planetary model of the atom, the non-visualizability was the key to understand the atoms. The classical electron paths had to be abandoned. Some matrix elements between stationary states behaved "like" the intensity of radiation emitted while the atom is switching from one level to another. However, they had to be something else. Later, we would appreciate those things as the probability amplitudes.
On the third page, Heisenberg made us sure that he understood that it was OK to refuse to talk about the precise position of the electron while it's happily orbiting the nucleus. To measure the position accurately, you would need to send a very high-frequency photon, and that would kick the electron and ionize the atom completely. You can't really measure what the electron is doing without totally changing the future. And that's why you are allowed to say that the "exact" position of the electron at a given moment is operationally meaningless.
It was important for Heisenberg to realize that the new description that was emerging in front of his eyes was as complete as the classical trajectory. He had all these amplitudes (including phases) and he was able to see that they would play exactly the same fundamental role as the full classical information was playing in classical physics. The previous sentence doesn't mean that the game is the same. Classical and quantum mechanics are different games. But they need some information about the physical system to be applied and Heisenberg had understood what is the kind of information that has to be inserted to the emerging theory of quantum mechanics.
In classical physics, the radiation was calculable from the knowledge of the immediate state of the atom, a point in the phase space. That was enough to compute the intensity of the radiation at each frequency. In quantum mechanics, one needs to determine both the initial and the final energy eigenstate, and what is calculable is some "amplitude" that tells us something about the intensity of light at any frequency.
Because in quantum mechanics, one needs to specify both the initial and final state before we calculate anything (there are no preferred "infinitesimally adjacent" states to the given state of the atom, so all pairs are as good as others), the quantities analogous to the "intensities" actually become a matrix – the object we know from linear algebra (that Heisenberg re-discovered). Because the intensities etc. are calculable like matrices, Heisenberg was suddenly able to realize that all observables – ordinary \(c\)-numbers in classical physics – have to be promoted to matrices.
Note that in all these early investigations, he was thinking in terms of matrices expressed relatively to a basis of energy eigenstates: that's where he was led by thinking about the atomic levels. It took some time before people learned to switch from one basis to another all the time and to realize that all of them are equally good to formulate the physical theory. Heisenberg credits Dirac and Jordan for this "transformation theory".
Heisenberg "knew" that each observable was linked to a matrix. But it would be just a table – and not a full-fledged algebraic matrix – if one operation were not taking place, namely the matrix multiplication. Heisenberg was able to figure out that the matrix multiplication actually does matter – it does replace the usual multiplication of the \(c\)-number observables in classical physics. Heisenberg mentions Kramers-Ladenburg dispersion theory as a piece of research that "suggested" that it should be done in this way.
But it isn't really too hard to "guess" that the matrix multiplication should be relevant. If one computes the transition from \(\ket i\) to \(\ket f\) through some intermediate states \(\ket m\), one should naturally sum over \(\ket m\), and we therefore deal with the sum of products \(\sum_m U_{fm}V_{mi}\) which is the matrix product. Such a matrix product appears in the product of two evolution operators, for example. But evolution operators are just "some other" functions of positions and momenta etc., so the same product should be relevant for all other functions of positions and momenta, all other observables, too.
At some point, Heisenberg had a well-defined mental (not visualizable) picture: all observables are matrices, their spectrum – as suggested by Bohr's theory etc. – is a claim about the matrices, too. In combination with the (Heisenberg) equations of motion, you have everything to define the new physics. Born, Jordan, and Dirac are actually thanked for expanding the insights to a usable calculational scheme and perhaps even for\[
p_r q_s - q_s p_r = \frac{h}{2\pi i} \delta_{rs}
\] which all of us would associate purely with Heisenberg. These commutators may also be used – along with other things – to prove that things like energy and angular momentum are conserved.
All these laws are extensively formally similar to classical physics, Heisenberg says, and all the difference in the formulation of the laws is concentrated in the nonzero commutator above.
This nonzero commutator has far-reaching consequences, however. The predictions of quantum mechanics are often very different from the classical ones. Discrete atomic spectra and transition probabilities are suddenly predicted and they agree with the experiments. It works well but there is no way to visualize, no counterpart of the Wilson photographs.
Here, on the fifth page, Heisenberg gets to Schr̦dinger. He shared his 1933 Nobel prize with Dirac and they gave lectures shortly after Heisenberg. Wave mechanics was found and shown equivalent to quantum mechanics Рby which Heisenberg means "matrix mechanics" Рand Dirac and Jordan are praised for the transformation theory. Schr̦dinger's picture, a refinement of de Broglie's wave paradigm, allowed people to calculate complicated atoms etc.
However, Heisenberg quickly gets to the defects of the straightforward visualizable interpretation. Schrödinger's "waves" live in a higher-, \(3N\)-dimensional space, and they have a statistical interpretation, so they can't quite be the same thing as the classical waves that de Broglie was envisioning.
Heisenberg spends the sixth page by explaining the Pauli exclusion principle and how it may be derived from antisymmetric wave functions Рand from anticommuting creation/annihilation operators. The main point of this discussion is that if Schr̦dinger's picture is viewed as a visualization, it cannot be the only similarly allowed visualization of the physical system. The anticommuting fields provide us with a different collection of "visualizable" eigenstates etc. Dirac's and Jordan's transformation theory is essential and it is wrong to be attached to a particular basis, Heisenberg says in different words.
On the rest of the seventh page, Heisenberg tells us that classical physics has described the objective evolution of some degrees of freedom in the spacetime. And the way how we acquired the knowledge about them was totally inconsequential. Things are very different in quantum mechanics.
\Delta p \cdot \Delta q \geq \frac{h}{4\pi}
\] Now, unlike the commutator, this inequality should be attributed purely to Heisenberg. Whenever similar classical observables are described by the quantum formalism, similar inequalities simply have to hold. They say that any observable – anything that is in principle measurable i.e. anything that has a scientific meaning in principle – inevitably involves some uncertainty that can't be small if a complementary variable was measured.
A straightforward 4-minute laser experiment which – if properly interpreted – is demonstrating the uncertainty principle. Well, it's a bit demagogic because the patterns may also be explained by completely classical Maxwell's equations. But if you believe that the light may be divided to photons...
You see that from the mid 1920s, Heisenberg's thinking was the quantum thinking. It was his default state of affairs. All the people dissatisfied with quantum mechanics and proposing the alternatives choose the classical thinking as their default one.
But people who have learned to think in the newer, more correct, quantum way would have very different expectations about the "realist interpretations" of quantum mechanics. Take Bohmian mechanics. It has some (classical) guiding wave \(\psi(x,y,z,t)\) and some particles' positions \(\vec x_i(t)\). But if these degrees of freedom were transferred to someone like Heisenberg who thinks quantum mechanically, he would still say that \(\psi(x,y,z,t)\) as well as \(\vec x_i(t)\) have to be promoted to operators or matrices, many of their commutators are nonzero, and these nonzero commutators imply that all these observables can't have well-defined values at the same moment! This is the natural and correct way of thinking. We would get just another quantum mechanical theory, one with a very contrived collection of degrees of freedom and very unnatural Heisenberg equations of motion. And one that disagrees with the empirical data. To assume that all of these observables have objectively well-defined values, even though \(\hbar\) appears all over the place, is the incorrect assumption. It's exactly as incorrect as the assumption that relativistic phenomena are explained by the superluminal transfer of the information. No, relativity says exactly the opposite: superluminal motion is forbidden. Whenever \(v/c\) fails to be negligible, the limitation on speed is important. Similarly, quantum mechanics bans the visualizability. Whenever \(\hbar\) fails to be negligible, all descriptions in terms of pictures are fundamentally wrong.
Heisenberg discusses Bohr's comments to the uncertainty principle which are just half-clear, I would say. He also emphasizes that the measurement itself has to be visualizable. It's the "phenomena" that lead to the quantum predictions that are not visualizable. The visualizability is always just approximate. It's OK if \(\hbar\) is negligible relatively to the objects' properties with the same units. But that is simply not a problem. The whole process of making quantum mechanical predictions requires us to accept and understand that the microscopic phenomena are not visualizable, but their verification and measurements has to be done by some apparatuses that must be visualizable at the same level of accuracy that we demand from such measurements. So these two aspects, visualizable and unvisualizable ones, inevitably co-exist. These formulations are more careful than Landau's, for example, because he uses the adjective "visualizable" and not "classical" for the apparatus side of his cut. This side including the apparatus is "as visualizable as it was in classical physics" but that doesn't mean that there is some incompleteness – hole – in quantum mechanics that would require us to add the whole theory of classical physics. It just means that the very act of measurement has to be as visualizable as it has always been, so certain crucial aspects of classical physics are a good approximation to quantum mechanics (which is still the only exactly true theory we may talk about).
The corpuscular and wave concepts are equally acceptable starting points for visualization; he said it above (search for Pauli).
On the tenth page, Heisenberg reminds us that the laws of quantum mechanics are statistical. Does it mean that there is some chance that the conservation laws are violated? Not at all, Heisenberg replies. They work as accurately as ever. However, it's still true that if you determine things like exact positions or exact momenta, you don't determine the exact energy, and vice versa. These observables don't commute with each other. But if you're sure that the energy at the beginning is something, the energy at the end has to be exactly the same value. The problem is that you can never exactly extract the value of the energy from the measurements of positions and momenta etc. (which you might consider necessary because the energy depends on both positions and momenta) – because those can't be done exactly at the same moment. But there are other ways to observe the energy...
The same page also worships Bohr as the source of the clearest explanations of the foundations of quantum mechanics – Heisenberg particularly means Bohr's comments about the complementarity principle. I would say that Bohr's complementarity is morally correct and Heisenberg and others had to appreciate it as the very general extension of the uncertainty principle etc. On the other hand, I would still view Bohr's complementarity as a not quite well-defined philosophical paradigm which is why the word "clearest" that Heisenberg used for Bohr's comments seems slightly exaggerated to me. It's very general, universally important, and captures the essence of quantum mechanics. But it inevitably remains incomprehensible and unclear to those who don't understand quantum mechanics in any special case.
On eleventh page, he says that the growth of crystals from liquid inevitably depends on chance and it makes no sense to look for "hidden variables" that would decide what shape of a crystal we get. Even with a known initial state of the liquid, one gets nonzero probability amplitudes for very different final shapes of the crystal.
The final page is dedicated to some speculations about the future – relatively to December 1933. It's pretty funny. First, he said that the research had to continue. In particular, the relativistic quantum theory has to be found. Dirac was going to speak momentarily. An amusing speculation that seems wrong today appears in one of the sentences. He suggested that the combination of the principles of quantum mechanics and special relativity will determine the only allowed value of the fine-structure constant \(\alpha\approx 1/137\). Well, that seems incorrect. But at some moment, \(\alpha\) looked almost as universal as \(\hbar\) or \(c\) and one could enjoy similar wishful thinking.
There is a big difference here: \(c\) and \(\hbar\) are dimensionful constants which is why we can set them equal to one in special units, and this is natural for the analysis of the relativistic and quantum phenomena, respectively. The constant \(\alpha\) is dimensionless so it cannot be set equal to one. And the lessons of relativity and quanta have already been "depleted" so there is no good reason to expect that \(\alpha\) becomes uniquely determined when relativity and quanta are combined.
Today, \(\alpha\) is a universal constant that seems "important" because it describes the strength of electromagnetism which is the most important interaction in our everyday lives. But from a deeply theoretical perspective, it is in no way "uniquely" important. It is a constant that is no more important than e.g. the masses (or mass ratios) of the elementary particles in the theories we use to describe particle physics. Just another adjustable parameter. Heisenberg was ready for – and wanted – another revolution around the corner. But the first following similar revolution only occurs if you incorporate gravity and Newton's constant \(G\) – quantum gravity – because gravity does play a more special role than electromagnetism because it's linked to the dynamics of spacetime which is more unique and "omnipresent" than electromagnetic fields. (There may be – and there are – numerous different spin-one fields, think about the gauge group of the Standard Model, but there can only be one spin-two tensor field, the metric tensor, at low energies.)
Heisenberg thought that the analysis of the wave fields wasn't quite exhausted – after all, one needs quantum field theory and it was only "getting born" in the early 1930s and it seems that the very term "quantum field theory" wasn't used in 1933 yet. (But I would place the true beginning of quantum field theory to 1926 when Pascual Jordan understood that all particles are quanta of quantum fields.)
And Heisenberg notices that the heavy particles – he means the proton etc. – don't seem to obey the rules of the Dirac equation well. The magnetic moment has a wrong value, if you need a particular example of the problem. This hadronic confusion already existed in the 1930s and grew up to the 1960s (the very birth of string theory partly occurred thanks to this mythology about Bootstrap etc., and Heisenberg himself was very important in keeping it mythological up to the 1960s) – before QCD eliminated the mysterious fog in the early 1970s. Let me copy and paste a few final sentences:
It's a great recommendation for others to think equally. A theory that is specific, effective, formally analogous to some very simple and successful older theories, but able to describe the data more well and consistently is more desirable than a theory that obeys some philosophical dogmas (of visualizability) but otherwise is very ineffective, contrived, and/or disagrees with the empirical data!
To avoid repeating Heisenberg's photographs, let's include a different hay fever sufferer. ;-)
When it comes to the basic character of quantum mechanics, the most relevant other Nobel prize went to Werner Heisenberg in 1932. Well, he only picked the prize in December 1933 – he was chosen as a winner retroactively because no candidate compatible with Nobel's will was proposed in 1932. OK, let us already look at the lecture
The development of quantum mechanics (PDF, 12 pages)because I find it much wiser than what almost all people in the "foundations of quantum mechanics" are saying today, 82 years later.
On the first page, Heisenberg makes it clear why Niels Bohr was viewed as the ultimate guru of the quantum community. Heisenberg himself interpreted his own efforts and achievements as refinements of some technical problems in Bohr's old model of the atom.
On one hand, it's legitimate to "discard" this not-quite-correct model of the atom entirely. On the other hand, it has played an extremely important role historically. And people like Heisenberg who went through the era in which Bohr's model of the atom was the "cutting edge of physics" may have captured some fundamentally correct ways of thinking.
On the very same first page, we already see the basic correct philosophy that reduced the rest of his groundbreaking discovery to a sufficiently intense period of mathematical reasoning:
...This circumstance was a fresh argument in support of the assumption that the natural phenomena in which Planck’s constant plays an important part can be understood only by largely foregoing a visual description of them. Classical physics seemed the limiting case of visualization of a fundamentally unvisualizable microphysics, the more accurately realizable the more Planck’s constant vanishes relative to the parameters of the system. This view of classical mechanics as a limiting case...I've said pretty much the same thing in the past, using slightly different words. But this was a wonderful insight that opened the whole quantum treasure. You know, since the analyses of the blackbody radiation, it was known that the constant \(\hbar\) played a certain role in many related but different phenomena.
People could see that this constant was universally important and whenever it appeared, something was behaving very differently than what classical physics was predicting. In fact, the very visualizability of the physical system – the assumption that there exists an objective state of the physical system at every moment that may be in principle captured by a picture – was breaking down whenever \(\hbar \neq 0\) was affecting the physical predictions.
Whenever \(\hbar\neq 0\), all pictures are inevitable misleading.
They are misleading because the very assumption of classical physics that certain objects objectively possess certain well-defined physical properties breaks down. The constant \(\hbar\) directly measures how much this classical assumption breaks down in Nature.
I have just visited Heisenberg's grave and asked him to record his explanations what was important for his discoveries in the audio form once again. You may also try the 1968 audio in his native tongue or the 85-minute Heisenberg and the reality question (German).
This insight is what is needed to correctly understand all the phenomena that quantum mechanics successfully explains. It is needed instead of many things that are not needed and that are not the point at all – like hidden variables or many worlds (MWI) or sudden discontinuous objective collapses (GRW) or many worlds or classical particles and waves existing simultaneously (like in Bohm's theory). None of the flawed concepts in the previous sentence has anything to do with the new assumptions that are actually needed to correctly understand all formulae with Planck's constant \(\hbar\) in them. Heisenberg et al. were going in this direction because they wanted to explain some observed phenomena. Instead, people promoting the realist pictures are doing so because they want to defend some philosophical prejudices regardless of the empirical data that don't really play an important role in their reasoning. They want to replace fundamentally unvisualizable quantum mechanics by MWI, Bohm, GRW etc. because they think that the "spirit" of their alternatives is similar to the novelties brought by quantum mechanics, but they are more philosophically pleasing. But that shouldn't matter in physics. Quantum mechanics works to predict and explain the actual phenomena, the actual equations that may extracted from the data.
Everyone who is tempted to think that there is something right about hidden variables, MWI, GRW, or Bohm's theory is urged to read Heisenberg's lecture, or at least the first pages of it, at least 12 times.
On the second page, Heisenberg continues by saying that it was understood that one needed a "new theory" that does reduce to classical physics in the right limit – effectively in a similar way that was already sketched by Bohr's principle of correspondence – but whose character before the limit is taken is qualitatively different.
He also said that the need for the new quantum laws to be probabilistic was gradually appreciated – especially when it came to Einstein's coefficient for the emission of radiation. Because the electromagnetic energy has to be emitted in discrete packets, the calculable continuous numbers can't tell us about the "intensity" because the "intensity" is supposed to be discrete. So they have to tell us about something continuous, namely the probability that the emission takes place. Max Born's Nobel lecture conveyed the same story about the initial reasons why people began to understand that the new laws were unavoidably going to be probabilistic in character.
While there was something good about the planetary model of the atom, the non-visualizability was the key to understand the atoms. The classical electron paths had to be abandoned. Some matrix elements between stationary states behaved "like" the intensity of radiation emitted while the atom is switching from one level to another. However, they had to be something else. Later, we would appreciate those things as the probability amplitudes.
On the third page, Heisenberg made us sure that he understood that it was OK to refuse to talk about the precise position of the electron while it's happily orbiting the nucleus. To measure the position accurately, you would need to send a very high-frequency photon, and that would kick the electron and ionize the atom completely. You can't really measure what the electron is doing without totally changing the future. And that's why you are allowed to say that the "exact" position of the electron at a given moment is operationally meaningless.
It was important for Heisenberg to realize that the new description that was emerging in front of his eyes was as complete as the classical trajectory. He had all these amplitudes (including phases) and he was able to see that they would play exactly the same fundamental role as the full classical information was playing in classical physics. The previous sentence doesn't mean that the game is the same. Classical and quantum mechanics are different games. But they need some information about the physical system to be applied and Heisenberg had understood what is the kind of information that has to be inserted to the emerging theory of quantum mechanics.
In classical physics, the radiation was calculable from the knowledge of the immediate state of the atom, a point in the phase space. That was enough to compute the intensity of the radiation at each frequency. In quantum mechanics, one needs to determine both the initial and the final energy eigenstate, and what is calculable is some "amplitude" that tells us something about the intensity of light at any frequency.
Because in quantum mechanics, one needs to specify both the initial and final state before we calculate anything (there are no preferred "infinitesimally adjacent" states to the given state of the atom, so all pairs are as good as others), the quantities analogous to the "intensities" actually become a matrix – the object we know from linear algebra (that Heisenberg re-discovered). Because the intensities etc. are calculable like matrices, Heisenberg was suddenly able to realize that all observables – ordinary \(c\)-numbers in classical physics – have to be promoted to matrices.
Note that in all these early investigations, he was thinking in terms of matrices expressed relatively to a basis of energy eigenstates: that's where he was led by thinking about the atomic levels. It took some time before people learned to switch from one basis to another all the time and to realize that all of them are equally good to formulate the physical theory. Heisenberg credits Dirac and Jordan for this "transformation theory".
Heisenberg "knew" that each observable was linked to a matrix. But it would be just a table – and not a full-fledged algebraic matrix – if one operation were not taking place, namely the matrix multiplication. Heisenberg was able to figure out that the matrix multiplication actually does matter – it does replace the usual multiplication of the \(c\)-number observables in classical physics. Heisenberg mentions Kramers-Ladenburg dispersion theory as a piece of research that "suggested" that it should be done in this way.
But it isn't really too hard to "guess" that the matrix multiplication should be relevant. If one computes the transition from \(\ket i\) to \(\ket f\) through some intermediate states \(\ket m\), one should naturally sum over \(\ket m\), and we therefore deal with the sum of products \(\sum_m U_{fm}V_{mi}\) which is the matrix product. Such a matrix product appears in the product of two evolution operators, for example. But evolution operators are just "some other" functions of positions and momenta etc., so the same product should be relevant for all other functions of positions and momenta, all other observables, too.
At some point, Heisenberg had a well-defined mental (not visualizable) picture: all observables are matrices, their spectrum – as suggested by Bohr's theory etc. – is a claim about the matrices, too. In combination with the (Heisenberg) equations of motion, you have everything to define the new physics. Born, Jordan, and Dirac are actually thanked for expanding the insights to a usable calculational scheme and perhaps even for\[
p_r q_s - q_s p_r = \frac{h}{2\pi i} \delta_{rs}
\] which all of us would associate purely with Heisenberg. These commutators may also be used – along with other things – to prove that things like energy and angular momentum are conserved.
All these laws are extensively formally similar to classical physics, Heisenberg says, and all the difference in the formulation of the laws is concentrated in the nonzero commutator above.
This nonzero commutator has far-reaching consequences, however. The predictions of quantum mechanics are often very different from the classical ones. Discrete atomic spectra and transition probabilities are suddenly predicted and they agree with the experiments. It works well but there is no way to visualize, no counterpart of the Wilson photographs.
Here, on the fifth page, Heisenberg gets to Schr̦dinger. He shared his 1933 Nobel prize with Dirac and they gave lectures shortly after Heisenberg. Wave mechanics was found and shown equivalent to quantum mechanics Рby which Heisenberg means "matrix mechanics" Рand Dirac and Jordan are praised for the transformation theory. Schr̦dinger's picture, a refinement of de Broglie's wave paradigm, allowed people to calculate complicated atoms etc.
However, Heisenberg quickly gets to the defects of the straightforward visualizable interpretation. Schrödinger's "waves" live in a higher-, \(3N\)-dimensional space, and they have a statistical interpretation, so they can't quite be the same thing as the classical waves that de Broglie was envisioning.
Heisenberg spends the sixth page by explaining the Pauli exclusion principle and how it may be derived from antisymmetric wave functions Рand from anticommuting creation/annihilation operators. The main point of this discussion is that if Schr̦dinger's picture is viewed as a visualization, it cannot be the only similarly allowed visualization of the physical system. The anticommuting fields provide us with a different collection of "visualizable" eigenstates etc. Dirac's and Jordan's transformation theory is essential and it is wrong to be attached to a particular basis, Heisenberg says in different words.
On the rest of the seventh page, Heisenberg tells us that classical physics has described the objective evolution of some degrees of freedom in the spacetime. And the way how we acquired the knowledge about them was totally inconsequential. Things are very different in quantum mechanics.
...The very fact that the formalism of quantum mechanics cannot be interpreted as visual description of a phenomenon occurring in space and time shows that quantum mechanics is in no way concerned with the objective determination of space-time phenomena. On the contrary, the formalism of quantum mechanics should be used in such a way that the probability for the outcome of a further experiment may be concluded from the determination of an experimental situation in an atomic system, providing that the system is subject to no perturbations other than those necessitated by performing the two experiments. The fact that the only definite known result to be ascertained after the fullest possible experimental investigation of the system is the probability for a certain outcome of a second experiment shows, however, that each observation must entail a discontinuous change in the formalism describing the atomic process and therefore also a discontinuous change in the physical phenomenon itself. Whereas in the classical theory the kind of observation has no bearing on the event, in the quantum theory the disturbance associated with each observation of the atomic phenomenon has a decisive role. Since, furthermore, the result of an observation as a rule leads only to assertions about the probability of certain results of subsequent observations, the fundamentally unverifiable part of each perturbation must, as shown by Bohr, be decisive for the non-contradictory operation of quantum mechanics. This difference between classical and atomic physics is understandable, of course, since for heavy bodies such as the planets moving around the sun the pressure of the sunlight which is reflected at their surface and which is necessary for them to be observed is negligible; for the smallest building units of matter, however, owing to their low mass, every observation has a decisive effect on their physical behaviour. ...All these basic philosophical points have been said totally clearly in 1933 – and much earlier than that. Quantum mechanics doesn't and can't describe any objective state of affairs. It predicts future measurements from past measurements. Each measurement inevitably influences the measured object and creates some new uncertainty about all the other, non-commuting observables. And this uncertainty is totally needed for the internal consistency of the whole theory. He elaborates upon that point and gets to his inequality\[
\Delta p \cdot \Delta q \geq \frac{h}{4\pi}
\] Now, unlike the commutator, this inequality should be attributed purely to Heisenberg. Whenever similar classical observables are described by the quantum formalism, similar inequalities simply have to hold. They say that any observable – anything that is in principle measurable i.e. anything that has a scientific meaning in principle – inevitably involves some uncertainty that can't be small if a complementary variable was measured.
A straightforward 4-minute laser experiment which – if properly interpreted – is demonstrating the uncertainty principle. Well, it's a bit demagogic because the patterns may also be explained by completely classical Maxwell's equations. But if you believe that the light may be divided to photons...
You see that from the mid 1920s, Heisenberg's thinking was the quantum thinking. It was his default state of affairs. All the people dissatisfied with quantum mechanics and proposing the alternatives choose the classical thinking as their default one.
But people who have learned to think in the newer, more correct, quantum way would have very different expectations about the "realist interpretations" of quantum mechanics. Take Bohmian mechanics. It has some (classical) guiding wave \(\psi(x,y,z,t)\) and some particles' positions \(\vec x_i(t)\). But if these degrees of freedom were transferred to someone like Heisenberg who thinks quantum mechanically, he would still say that \(\psi(x,y,z,t)\) as well as \(\vec x_i(t)\) have to be promoted to operators or matrices, many of their commutators are nonzero, and these nonzero commutators imply that all these observables can't have well-defined values at the same moment! This is the natural and correct way of thinking. We would get just another quantum mechanical theory, one with a very contrived collection of degrees of freedom and very unnatural Heisenberg equations of motion. And one that disagrees with the empirical data. To assume that all of these observables have objectively well-defined values, even though \(\hbar\) appears all over the place, is the incorrect assumption. It's exactly as incorrect as the assumption that relativistic phenomena are explained by the superluminal transfer of the information. No, relativity says exactly the opposite: superluminal motion is forbidden. Whenever \(v/c\) fails to be negligible, the limitation on speed is important. Similarly, quantum mechanics bans the visualizability. Whenever \(\hbar\) fails to be negligible, all descriptions in terms of pictures are fundamentally wrong.
Heisenberg discusses Bohr's comments to the uncertainty principle which are just half-clear, I would say. He also emphasizes that the measurement itself has to be visualizable. It's the "phenomena" that lead to the quantum predictions that are not visualizable. The visualizability is always just approximate. It's OK if \(\hbar\) is negligible relatively to the objects' properties with the same units. But that is simply not a problem. The whole process of making quantum mechanical predictions requires us to accept and understand that the microscopic phenomena are not visualizable, but their verification and measurements has to be done by some apparatuses that must be visualizable at the same level of accuracy that we demand from such measurements. So these two aspects, visualizable and unvisualizable ones, inevitably co-exist. These formulations are more careful than Landau's, for example, because he uses the adjective "visualizable" and not "classical" for the apparatus side of his cut. This side including the apparatus is "as visualizable as it was in classical physics" but that doesn't mean that there is some incompleteness – hole – in quantum mechanics that would require us to add the whole theory of classical physics. It just means that the very act of measurement has to be as visualizable as it has always been, so certain crucial aspects of classical physics are a good approximation to quantum mechanics (which is still the only exactly true theory we may talk about).
The corpuscular and wave concepts are equally acceptable starting points for visualization; he said it above (search for Pauli).
On the tenth page, Heisenberg reminds us that the laws of quantum mechanics are statistical. Does it mean that there is some chance that the conservation laws are violated? Not at all, Heisenberg replies. They work as accurately as ever. However, it's still true that if you determine things like exact positions or exact momenta, you don't determine the exact energy, and vice versa. These observables don't commute with each other. But if you're sure that the energy at the beginning is something, the energy at the end has to be exactly the same value. The problem is that you can never exactly extract the value of the energy from the measurements of positions and momenta etc. (which you might consider necessary because the energy depends on both positions and momenta) – because those can't be done exactly at the same moment. But there are other ways to observe the energy...
The same page also worships Bohr as the source of the clearest explanations of the foundations of quantum mechanics – Heisenberg particularly means Bohr's comments about the complementarity principle. I would say that Bohr's complementarity is morally correct and Heisenberg and others had to appreciate it as the very general extension of the uncertainty principle etc. On the other hand, I would still view Bohr's complementarity as a not quite well-defined philosophical paradigm which is why the word "clearest" that Heisenberg used for Bohr's comments seems slightly exaggerated to me. It's very general, universally important, and captures the essence of quantum mechanics. But it inevitably remains incomprehensible and unclear to those who don't understand quantum mechanics in any special case.
On eleventh page, he says that the growth of crystals from liquid inevitably depends on chance and it makes no sense to look for "hidden variables" that would decide what shape of a crystal we get. Even with a known initial state of the liquid, one gets nonzero probability amplitudes for very different final shapes of the crystal.
The final page is dedicated to some speculations about the future – relatively to December 1933. It's pretty funny. First, he said that the research had to continue. In particular, the relativistic quantum theory has to be found. Dirac was going to speak momentarily. An amusing speculation that seems wrong today appears in one of the sentences. He suggested that the combination of the principles of quantum mechanics and special relativity will determine the only allowed value of the fine-structure constant \(\alpha\approx 1/137\). Well, that seems incorrect. But at some moment, \(\alpha\) looked almost as universal as \(\hbar\) or \(c\) and one could enjoy similar wishful thinking.
There is a big difference here: \(c\) and \(\hbar\) are dimensionful constants which is why we can set them equal to one in special units, and this is natural for the analysis of the relativistic and quantum phenomena, respectively. The constant \(\alpha\) is dimensionless so it cannot be set equal to one. And the lessons of relativity and quanta have already been "depleted" so there is no good reason to expect that \(\alpha\) becomes uniquely determined when relativity and quanta are combined.
Today, \(\alpha\) is a universal constant that seems "important" because it describes the strength of electromagnetism which is the most important interaction in our everyday lives. But from a deeply theoretical perspective, it is in no way "uniquely" important. It is a constant that is no more important than e.g. the masses (or mass ratios) of the elementary particles in the theories we use to describe particle physics. Just another adjustable parameter. Heisenberg was ready for – and wanted – another revolution around the corner. But the first following similar revolution only occurs if you incorporate gravity and Newton's constant \(G\) – quantum gravity – because gravity does play a more special role than electromagnetism because it's linked to the dynamics of spacetime which is more unique and "omnipresent" than electromagnetic fields. (There may be – and there are – numerous different spin-one fields, think about the gauge group of the Standard Model, but there can only be one spin-two tensor field, the metric tensor, at low energies.)
Heisenberg thought that the analysis of the wave fields wasn't quite exhausted – after all, one needs quantum field theory and it was only "getting born" in the early 1930s and it seems that the very term "quantum field theory" wasn't used in 1933 yet. (But I would place the true beginning of quantum field theory to 1926 when Pascual Jordan understood that all particles are quanta of quantum fields.)
And Heisenberg notices that the heavy particles – he means the proton etc. – don't seem to obey the rules of the Dirac equation well. The magnetic moment has a wrong value, if you need a particular example of the problem. This hadronic confusion already existed in the 1930s and grew up to the 1960s (the very birth of string theory partly occurred thanks to this mythology about Bootstrap etc., and Heisenberg himself was very important in keeping it mythological up to the 1960s) – before QCD eliminated the mysterious fog in the early 1970s. Let me copy and paste a few final sentences:
...But however the development proceeds in detail, the path so far traced by the quantum theory indicates that an understanding of those still unclarified features of atomic physics can only be acquired by foregoing visualization and objectification to an extent greater than that customary hitherto. We have probably no reason to regret this, because the thought of the great epistemological difficulties with which the visual atom concept of earlier physics had to contend gives us the hope that the abstracter atomic physics developing at present will one day fit more harmoniously into the great edifice of Science.You may see that Heisenberg expected the trend of "deobjectification" and "devisualization" to continue and he viewed it as good news. Why? Simply because the abstracter description of the atoms that emerged from quantum mechanics made more sense – was more internally consistent and compatible with the observations – than the previous, visualizable, classical models.
It's a great recommendation for others to think equally. A theory that is specific, effective, formally analogous to some very simple and successful older theories, but able to describe the data more well and consistently is more desirable than a theory that obeys some philosophical dogmas (of visualizability) but otherwise is very ineffective, contrived, and/or disagrees with the empirical data!
Heisenberg's Nobel lecture
Reviewed by DAL
on
April 28, 2015
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Reviewed by DAL
on
April 28, 2015
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