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Erwin Schrödinger: 10,000,000th birthday

Sometimes the binary numbers look more impressive

128 years ago, Erwin Rudolf Josef Alexander Schrödinger was born as the only child to a botanist (and cerecloth producer) and a daughter of a chemistry professor in Vienna. His mother was half-Austrian (after Erwin's grandpa), half-English (after Erwin's grandma) and Lutheran while his dad was Catholic. Which religion did Erwin pick? He picked the Eastern religions, pantheism, and atheism. ;-)

He is primarily well-known for wave mechanics, the most popular formulation – and a recipe for misunderstanding – of quantum mechanics. I would normally add a standard full-fledged amusing biography, explain that he liked to be accompanied both by his wife and his shadow wife at all times. He studied color perception as well as the role of entropy reduction in life etc.

But I am a bit tired tonight. Moreover, this blog is in a superposition of states "already containing a respectable biography of Schrödinger" and "not yet".

He found the equations of "wave mechanics" and also established the mathematical equivalence of these differential equations with Heisenberg's original version of quantum mechanics, the "matrix mechanics". However, I think it's just wrong when tweeting Brian Greene and lots of other popularizers of science call Schrödinger "the father of quantum mechanics". He wasn't "the father" of quantum mechanics. In fact, he wasn't even "a father". One can't be a father of a theory he completely misunderstands and fights against throughout his life.

Like Einstein, he helped the birth of the quantum theory with some inspired observations but quantum mechanics or theory was built by someone else.

Even though he found very important things, he was a typical example of "inside the box" thinker.

Let me just try to demystify his contributions a little. Quantum mechanics is so wonderfully different that 90 years have not been enough for many people – even people claiming to be immensely interested – to "get it". But if Erwin Schrödinger were "the father" of the theory, everything we could say would be totally clear, classical, comprehensible, and completely wrong, too.

What did he do when he wrote down his equation? Well, since 1924, there has been a cute idea of another guy who would turn out to be another quantum mechanics denier, Prince Louis de Broglie. There have already been experiments indicating that particles and waves tended to have both properties at the same moment. Electromagnetic waves were sort of composed of particles (photons), as seen in the photoelectric and later Compton effect. Electrons were interfering on gratings, too.

Broglie found a nice formula for a wave that may be associated with a particle. Using the modern sign conventions, it was\[

\psi_{{\rm Broglie}} = \exp (-iEt/\hbar + i \vec p \cdot \vec x/\hbar).

\] Some wave – analogous to the electromagnetic wave – was said to be associated with each particle. The precise meaning of the wave and its relationship with the particle wasn't explained by de Broglie. However, the exponent in the complex exponential above has a cute property: it is Lorentz-invariant. It is nothing else than \((-i/\hbar)\) times the inner product of the four-vector \(p^\mu\) of the particle's energy-momentum; and the location in the spacetime \(x^\mu\).

So the law that this wave is sort of "associated" with the particle seems equally valid in all relativistic inertial frames. Well, we face an immediate issue with the phase. If we shift the origin of the spacetime coordinates, the overall phase of the wave seems to change. It is a quick indication that we should take the overall phase with a grain of salt.

If you understand differential equations sufficiently well – everyone should – you may easily write down the equations that \(\psi_{\rm Broglie}\) obeys. Take its 4-gradient. You will see that\[

\partial_\mu \psi_{\rm Broglie} = \frac{p_\mu}{i \hbar} \psi_{\rm Broglie}.

\] The derivative of the exponential is the same exponential times the derivative of the argument. This gradient has too many (four) components for a good law of Nature governing the wave. But if you take the box of this, you get\[

\partial^\mu \partial_\mu \psi_{\rm Broglie} = -\frac{p^\mu p_\mu}{\hbar^2} \psi_{\rm Broglie}

\] It was therefore trivial for Schrödinger to see that the electron could have a wave associated with it – and this wave could be governed by the Klein-Gordon equation. Now, it's funny that this equation by Schrödinger is called the Klein-Gordon or Klein-Fock equation, isn't it? Well, he hasn't published it because he was able to find out that the stationary solutions to this equation (equipped with the potential term) looked nothing like the hydrogen's energy eigenstates.

OK, I forgot to say that he first generalized the "de Broglie" equation with the box and added the mass term, and a general potential term \(V(\psi)\). Because this equation didn't seem immediately useful to explain the atomic spectra, especially the hydrogen whose result should be simple – and already reproduced by the old Bohr childish model – he just discarded that Ansatz. This decision to throw the equation away was something he should have been proud about. (Klein and Gordon published the equation when it was already clear that such an equation is only relevant for other particles and fields, if any.)

Another source of pride was his decision to reduce his ambitions and ignore relativity for a while. He realized that in relativity, \(E^2=p^2+m^2\) so \(E^2\) is a quadratic function of the momentum. In non-relativistic physics, we have \(E = p^2/ 2m\), so \(E\) itself, its first power, is a quadratic function of the momentum. It was trivial to write down the non-relativistic version of the Klein-Gordon equation with the potential term: just replace the second time-derivative by the first time derivative.

And that's it. His equation was born.

It sounds cool because\[

\zzav{ -\frac{\hbar^2\nabla^2}{2m} + V(x) } \psi(x,y,z,t) =i\hbar \frac{\partial}{\partial t} \psi(x,y,z,t)

\] is perhaps the equation that people most often imagine when you say the "equation of quantum mechanics". An ingenious advance. Except that a formulation of quantum mechanics had already existed for one year, the "matrix mechanics" of Werner Heisenberg and pals.

Schrödinger was able to prove the equivalence of the Schrödinger picture and the Heisenberg picture rather soon which makes it very likely that he was actually using a reverse engineering of the Heisenberg equations to converge to the right result already when he was looking for his equation. The idea that someone found something "independently", although he was one year too late and living in the same environment, always sounds suspicious to me.

I am not really speculating that Schrödinger knew the Heisenberg equation when he was working on this equation. In 1926, Schrödinger wrote:

I knew of [Heisenberg's] theory, of course, but I felt discouraged, not to say repelled, by the methods of transcendental algebra, which appeared difficult to me, and by the lack of visualizability.
At any rate, the mathematical form of these partial differential equations is a very straightforward insight. Good high school students are capable of understanding what the equations mean and many very good high school students know how to manipulate with them. There would be no 90 years of confused criticisms and arguments if the meaning of these differential equations were "obvious".

However, it was not obvious to many people: the actual meaning of the objects in the Heisenberg picture as well as the Schrödinger picture was only known to the actual fathers of quantum mechanics and their followers. And this meaning is nothing like the meaning of objects we knew in classical physics.

The wave was associated with the particle in "some way" but Schrödinger has never ever understood in what way they were related and how the values of the wave should be used to make physical predictions. And Schrödinger has just never had any correct clue. For him, the electron was objectively spread like yoghurt and the wave described the shape of this yoghurt. Needless to say, this wasn't appropriate to predict trivial things – such as the fact that the electron makes one dot on the photographic plate.

In 1926, Heisenberg rightfully wrote to Pauli:
The more I think about the physical portion of Schrödinger's theory, the more repulsive I find it... What Schrödinger writes about the visualizability of his theory 'is probably not quite right,' in other words it's crap. Šajze. Bullšit.
Amen to that. Schrödinger may have tried to offer an alternative interpretation of the objects in Heisenberg and his equations. But he has obviously never found anything that was publishable. There is nothing of the sort that could be publishable. So relatively to quantum mechanics, Schrödinger rebranded himself as a critic of the same kind as creationists are with respect to Darwin's theory of evolution.

Even though almost all readers of popular books are led to totally misunderstand the correct answer to the gedanken experiment, Schrödinger's cat (1935) became the canonical proof showing how totally deluded this guy was concerning the meaning of quantum mechanics. He correctly argued that a decaying particle that evolves into superpositions may force a measurement apparatus – and a cat whose fate depends on this apparatus – to evolve into complex linear superpositions. And that's clearly nonsense, Schrödinger claimed, so quantum mechanics is ludicrous.

Except that the superpositions are not nonsense. It is one of the most important and universal postulates of quantum mechanics, the superposition postulate, that for all pairs of allowed states \(\ket\alpha\) and \(\ket\beta\), all of their complex superpositions \(a \ket\alpha+ b \ket\beta\) with \(a,b\in\CC\) are equally allowed, too! The Hilbert space of allowed states is unavoidably a complex linear vector space. Nothing in quantum mechanics could work if this claim were not true.

The superposition state\[

a \ket{\rm alive} + b \ket{\rm dead}, \quad |a|^2+|b|^2 = 1

\] does not describe two cats, a dead one and an alive one, overlapping each other in some way. As I have argued many times, it describes a cat that is known to the observer to be either alive, with probability \(|a|^2\), or dead, with probability \(|b|^2\). The word in between the two possibilities – the translation of the plus sign – is "or", not "and".

Quantum mechanics differs from classical physics because the assumption that one of the answers (dead/alive, in this case) is "objectively" realized in between the measurement is simply impossible. Why? Because quantum mechanics allows us to measure not only the "dead/alive" qubit. It allows us – at least in principle or in simpler systems than a cat – to measure the value of operators \(L\) that don't commute with the "dead/alive" operator, i.e. operators that are expressed by non-diagonal matrices relatively to the "dead/alive" basis.

And the cat is actually always in an eigenstate of some of these operators \(L\) – so some operators (a measure zero of them) – may be predicted at 100% certainty. The character of these operators \(L\) depends not only on the absolute values of \(a,b\) but on their relative phase, too. But if you were assuming that the cat is objectively either dead; or alive before the measurement, you would always predict uncertain results for these non-diagonal operators \(L\) – even though quantum mechanics (and experiment) predict unambiguous results for the measurement of such \(L\) that possess \(\ket\psi\) as their eigenstate.

Quite generally, I think that most people, once they understand this argument, are shocked that quantum mechanics sometimes makes predictions that are more unambiguous than what is possible in classical physics. But it's the case! Quantum mechanics isn't "universally more fuzzy" than classical physics. It is different – which may sometimes mean "more unequivocal", too. The very EPR correlations and Bell's-inequality-violating correlations are examples of quantum predictions that are more predictive, more constrained, or less ambiguous than the predictions produced by a classical model would be. And a related point is that quantum mechanics allows us to construct more accurate atomic clocks or define the energy more accurately than what would be possible in any semi-realistic classical theory. Quantum mechanics can do such things – like guaranteeing the existence of precise atomic clocks – exactly because observables often have discrete spectra and there is no need to "continuously adjust" a quantity with a discrete spectrum (such as the frequency of ticking of the atomic clocks) – there are no "wrong" nearby values.

For cats, because of decoherence, it is extremely difficult to measure the observables given by non-diagonal operators (in the "dead/alive" basis). But in principle, and for small systems, also in practice, there exists an experimental procedure to measure any operator – an observable expressed by any Hermitian matrix with respect to a basis.

At the end, the partial differential equation found by Schrödinger is one of the conceptually simplest parts of quantum mechanics which wasn't hard because of the "role models" in the form of Maxwell's equations for electromagnetic waves; de Broglie wave; and Heisenberg's matrix mechanics that simply worked. The usage of this formalism in physics is the more difficult part – which was found before Schrödinger made any contributions; and which Schrödinger has never understood.

He knew how to manipulate with the differential equations at a purely mathematical level. And he also knew that the stationary solutions – energy eigenstates – were bound to be interesting and relevant to explain the spectra. He knew how to extract the energy eigenvalues. But he didn't know how to predict the results of general, especially time-dependent experiments – and he didn't know how the spectra should exactly be derived from his equation, either.

Although his findings were very important – and I am obviously often thinking in the Schrödinger picture myself – I don't think that his contributions have made him a father of quantum mechanics and I don't think that he has made some genuine radical leaps in physics.

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