Schrodinger's Interpretation of the Wave Function?

[Dopplershift]

Schrödinger developed his famous wave equation which describes how the quantum state of a system changes over time.

But, what was Schrödinger trying to initially prove with his equation?I assume that it has to do with Debrogile's hypothesis.

I know from my classes that we use the Schrödinger equation to find the probability of an electron at some position in space. But, wasn't it Born who suggested the probability distribution density interpretation? I know that Schrödinger once believed it was the electrical-charged density, but later abandoned it.

I guess my question is this: Doesn't the probability distribution interpretation rely on the belief of quantum superposition and the Copenhagen interpretation (CHI); but as from what I understand, Schrödinger did not agree with superposition or the CHI as demonstrated by his famous (although often misunderstood) cat in the box thought experiment , so what did he believe the wave function represented?

 

[Gan_HOPE326]

I am not sure of the historical perspective on this, and I would like to follow this thread and see the answers of someone who knows more than myself, but my impression from what I remember is that Schroedinger was basically searching for a wave equation of any kind. I think the idea came forth because of the wave-particle duality displayed by light, so someone obviously would think that perhaps even regular particles could have wave-like properties in certain circumstances. They probably already carried out the electron diffraction experiment by then so knew this was right; de Broglie had formulated his hypothesis about what a particle's wavelength would be like; and the picture of electrons as waves would explain very well the energetic levels of hydrogen as well, picturing them as standing wave modes. So basically Schroedinger tried a bunch of different possible equations until he found one that fit the data. I read he also stumbled upon what came to be known as the Klein-Gordon equation, because of course he was looking for a relativistic equation, except that didn't work (Dirac would fix that later on).

As for the interpretations, we're not fully sure of that even today, and IMHO the Copenhagen is a bit of a cop-out in a sense, so I can understand why he'd find it dissatisfying. Doesn't mean he necessarily had certainties about what the wave DID mean.

 

[bhobba]

See:
https://arxiv.org/pdf/1204.0653v1.pdf

Thanks
Bill

 

[vanhees71]

As far as I know, originally Schrödinger thought that particles are just described as fields and that his wave function for the electron is just describing a spreaded charge distribution $\rho(t,\vec{x})=|\psi(t,\vec{x})|^2$, but that was very soon be verified to be wrong since detecting an electron always leaves one spot on a photon plate (or any other kind of detector), and this finally brought Born to his probability interpretation, i.e., the wave function should be normalized to 1, and then gives the probability distribution of the position of the electron as a function of time. This minimal interpretation is valid until today, and there is, imho, no need for any other "interpretation" since any other attempt to interpret the formalism leads only to confusion and doesn't solve any problem, whatever some people might think there remains today. QT is the most successful theory ever, particularly the Standard Model of elementary particles, which is more successful than the physicists at CERN wish it to be! Just these days there's another great confirmation of the basic principles of relativistic local QFT: The 1s->2s transition in antihydrogen atoms has the same frequency as the corresponding transition in usual hydrogen, proving the validity of the CPT theorem to an relative accuracy of $2 \cdot 10^{-10}$.

 

[Gan_HOPE326]

This minimal interpretation is valid until today, and there is, imho, no need for any other "interpretation" since any other attempt to interpret the formalism leads only to confusion and doesn't solve any problem, whatever some people might think there remains today..

Well, there are some areas where having a clearer view might help us understand some pending questions. For example, whether the apparent breaking of unitarity involved in the wave function collapse is real, and whether it relates in some way with the 'information paradox' concerning stuff that falls in a black hole. We DO know something ought to be wrong or approximated in there somewhere, and it's not impossible that the 'real' theory, whatever it is, might fix the weird ambiguities in our interpretation of QM.

I agree beyond a point it can become a moot question and it's certainly not worth fighting over different interpretations if they do not predict testable differences (Copenhagen vs. Many Worlds, anyone?) - that is just metaphysics. But when it comes to stuff like objective collapse it makes sense to consider them, even if only to rule them out.

 

[vanhees71]

Since there is no necessity for any "collapse", that's a pseudo-problem. The question what happens close to a black hole (or its event horizon for that matter) is an open question as far as I understand it since there is no consistent relativistic theory including gravity. That's one of the big unsolved problems of contemporary theoretical physics, but I'm pretty sure that all the kind of esoterics brought up in discussions about the interpretation of QT won't bring us closer to a solution. There's a need for "another Einstein" to get a brilliant idea how to solve it in a solid mathematical way clearly based on observed facts rather than philosophical speculation!

 

[Gan_HOPE326]

Since there is no necessity for any "collapse", that's a pseudo-problem.

Could you expand on this? Because frankly I read all sorts of things and I'm kind of confused at this point.

My impression - from my practice with regular QM (not QFT of which I barely know the basics) - is that there is no unified view of how to describe measurement. If I were to simulate the quantum evolution of a system given an Hamiltonian and a starting vector I would evolve this vector with the time-dependent Schroedinger equation, and that would be a perfectly reversible unitary evolution. However if I were to simulate a 'measurement' I ought to project the vector on a basis whose eigenvalues are possible measurements for my apparatus (for example, the values of $S_z$ for a spin measurement along $z$) and grab a value based on the probability distribution given by squaring the vector. But if I were to evolve the system further, I could not restart from the vector as it was before the measurement - I would need to restart from a pure state corresponding to the outcome of the measurement.

Now of course this is a rather artificial way of going about performing this measurement. I could, in fact, simply simulate it by coupling the system to another, bigger system representing my measurement apparatus. In that case evolution of the apparatus+system total wavefunction would be unitary. But again, if the original system was in a superposition, the total wavefunction would end up in a superposition too. I could carry on this reasoning, coupling by coupling, up to the last measurement apparatus, namely my own brain. The only thing I can imagine breaks this down is somehow something along the process of 'amplification', where the single-particle property is transformed into a macroscopic signal (for example a light turning red only if $S_z=\frac{1}{2}$). In this stage the microscopic becomes macroscopic and thus quantum effects should somehow become inconsequential. I can understand how decoherence removes off-diagonal elements from a density matrix when dealing with a number of particles, but in the end, how do we explain away the fact that single particles still behave in a probabilistic manner, and yet their successive evolution is influenced by their measurement? Is it just a 'prior knowledge' thing, where the probability of the particle being in a certain point after the measurement changes because we know where it was at the time of the measurement and that biases our expectation of which paths contribute to the particle's final position? I thought that couldn't be the case, because of how performing measurements on 'which slit' an electron takes in the double slit experiment disrupts the diffraction pattern. The measurement's effect is very real.

 

[Demystifier]

But, what was Schrödinger trying to initially prove with his equation?

At that time the main issue was to explain the discrete spectrum of atoms, which Schrödinger equation does very well even if you are somewhat agnostic about the correct interpretation of $\psi$.

 

[vanhees71]

Well, I see physics as an empirical science and theory as a mathematical quantitative description of the outcome of observations and quantitative measurements. For me, there's nothing to explain when something is measured. The experimentalist has his/her devices which defines the preparation procedure of the system which then is measured by the apparati in the lab. The outcome of such experiments can with an amazing success be described by the formalism of quantum theory.

The preparation and measurement devices are designed using classical physics, and classical physics can be understood in terms of many-body quantum theory via some coarse-graining procedure, i.e., the classicality of almost all macroscopic systems (it's usually hard to prepare macroscopic systems to show quantum phenomena like sufficient isolation from uncontrolled influences from the "environment", cooled down to very low temperature etc.) can be understood from quantum theory too. So from a physicist's point of view, for me there's nothing left to be explained concerning the formalism of QT as a physical theory.

I can understand that some people feel uneasy with the intrinsic irreducible indeterminism and thus probabilistic worldview this implies, but we can't help it. Nature behaves as she does and not as we wish!

 

[mikeyork]

i.e., the wave function should be normalized to 1, and then gives the probability distribution of the position of the electron as a function of time. This minimal interpretation is valid until today, and there is, imho, no need for any other "interpretation" since any other attempt to interpret the formalism leads only to confusion and doesn't solve any problem, whatever some people might think there remains today.

I think the minimal interpretation is that it gives the probability distribution of detecting the electron at that space-time location. This is not quite the same thing as it being there unobserved.

 

[bhobba]

Could you expand on this? Because frankly I read all sorts of things and I'm kind of confused at this point.

Collapse is not something that's in the QM formalism. You will find that formalism detailed correctly in Ballentine with no mention of collapse.

Its part of some, not all, but some interpretations. Its not part of Many Worlds for instance. Its not part of the ensemble interpretation that both myself and Vanhees hold to. In that interpretation states are synonymous with preparation procedures. If a system changes state all you have done is prepared it differently. Nothing collapsed, no mystery involved or anything to worry about. Some argue its still collapse by another name, I don't think it is, I am with Ballentine and Vanhees on that one, but it isn't really important - the mystery is gone.

Thanks
Bill

 

[Dopplershift]

So, if I understand correctly. The wave equation was Schrödinger's way of fitting the data of the wavelength of Hydrogen spectrum, but the disagreement was on what the modulus squared represented? In Born's interpretation, he proposed that it was the probability amplitude of an electron's position in space?

If Schrödinger was trying to fit the data of the hydrogen spectra, but isn't that what the Rydberg formula did? I guess I'm still confused on what Debrogile defined as matter waves, what causes the waves in matter.

Sorry for all the questions and being all over the place, as you can tell I'm confused on how all of this came together?

 

[Dopplershift]

See:
https://arxiv.org/pdf/1204.0653v1.pdf

Thanks
Bill

I have yet to study hamiltonian mechanics, but it's in the upcoming semester. I have only studied Newtonian and Lagrangian mechanics in the field of classical mechanics. I will definitely read what I can and come back to the math once I cover it in my class. Thank you!

 

[vanhees71]

The important point is that Schrödinger didn't fit the hydrogen spectrum which was long known before quantum theory and brougth to a phenomenological formula by Balmer in the 19th century. Then in 1913 Bohr invented is atomic model, which shortly thereafter was mathematically completed by Sommerfeld. Fortunately the Bohr-Sommerfeld model (which is very ugly by the way, because it's based on inconsistent assumptions, we should go into here) couldn't explain much more than the hydrogen spectrum. It already failed for the Helium spectrum. Even for the hydrogen the rough other specifics come out wrong. E.g., the chemists at the time were well aware that the hydrogen atom is spherically symmetric, while the Bohr-Sommerfeld model suggested a disk-like structure (with the electron running around the proton in a plane ellipse as the planets around the Sun). That's why the physicists were forced to think deeper, and after a bit of confusion concerning the "wave-particle dualism" (socalled "old quantum theory") via de Broglie's idea of a description of particles by a wave function Schrödinger came to a consistent theory by asking for the equation governing this wave function. First he tried a relativistic wave equation (which we call the Klein-Gordon today, describing particles with spin 0), but calculating the energy levels and thus the hydrogen spectrum he got the wrong fine structure. Now it was known through the work of Sommerfeld within the old Bohr-Sommerfeld model (where ironically the finestructure comes out right by just treating the motion of the electron relativistically) that the finestructure is a relativistic effect, and that brought Schrödinger to take a step back and first try the non-relativistic version of the wave equation (which know carries his name, the Schrödinger equation), and this gave the correct hydrogen spectrum without the finestructure as he expected.

At the same time (or even a bit earlier) there were two more formulations of modern quantum theory developed. One was "matrix mechanics", discovered by Heisenberg on the island of Helgoland in the summer of 1925 (where he went to escape bad atttacs of pollen allgergy) and subsequently worked out by him, Born, and Jordan. The other was "transformation theory" by Dirac, which is the most versatile formulation, laying very clearly out the mathematical structure of the theory. Another important step, one should not foget, was made by John von Neumann, who formulated Dirac's "new math" in a mathematically rigorous way in terms of Hilbert-space theory. Nowadays Dirac's hand-waving math, which is very elegant for practical calculations, was made rigorous by G'elfand et al in terms of what physicists call the "rigged-Hilbert-space formalism".