Is there a reason for why the wave function for a particle is the way it is?

Question

I have read that in quantum mechanics, predicting the position of an electron, say in the double slit experiment is impossible. It is thus then implied that there is no reason for the electron to end up in the position it does.

However, in the double slit experiment, a pattern does emerge. And we’re able to assign probabilities to the observed electron positions.

Is there a reason then, as to why, the wave function is the way it is? Given that the probability distribution is correlated with the wave function that is represented by Schrödinger’s equation, I suppose the equivalent question would be “why is this equation the way it is?”

I can’t seem to find a clear answer on this.

Answer 1

In addition to the references already provided, which also give an introduction to the mathematics behind:

• I recommend to follow the video of the Messenger lectures of Feynman about the double slit experiment. Feynman addresses the Messenger lectures to a broader auditorium.
• You can also read the content of the video as the first chapter of Feynman's classical textbook, the Feynman Lectures on Physics.

You are right: The Schroedinger equation describes the result of the double slit experiment and other experiments in quantum mechanics.

From the mathematical point of view the Schroedinger equation is a wave equation like many other wave equations, e.g. the wave equation for the spread of water waves on the surface of a pond.

The specific property of the Schroedinger equation is its physical interpretation: The Schroedinger equation deals with a function psi = psi(x,t) defined on physical space and time. Here x is the space coordinate and t the time coordinate. The function assumes values from the complex numbers. It is named the probability amplitude. Because the squared modulus of the function is a real number and can be interpreted as the probability, more precisely the probability density, to detect the particle at time t at the position x.

Like any other wave function the function psi shows the property of interference, which explains the observed pattern of intensity behind the double slit.

“Why is this equation [the Schroedinger equation} the way it is?"

I think the question has no physical answer.

Like similar questions from physics, e.g. why are Newton equations the way they are. One can bring the Schroedinger equation in a relation to classical mechanics, the correspondence principle of quantum mechanics. But that’s not an explanation, it is just a formal procedure.

Answer 2

Physics tells you what happens, and how to predict it, but it doesn't tell you why things are the way they are. We may discover a deeper truth – a way to predict the behaviour governed by this model, and other things besides – but that just shifts “why?” one level deeper.

The wavefunction is the way it is because that model matches the evidence well enough, and can be used to make concrete predictions about the behaviour of other experiments, and we haven't yet noticed a meaningful pattern in the differences between the predicted and observed results.

The interference pattern in the double slit experiment is determined by the wavelength (of the colour of the laser, for photons; the de Broglie wavelength determined by the momentum, for massive particles), the distance between the slits, the size of the slits, and the distance from the slits to the film / sensor / diffuse reflective surface – and whatever other parameters you use in your experiment. From an experimental perspective, there is nothing more than this.

Answer 3

Of course, we don't really know why any laws of physics are the way they are- that's still a mystery- but if you want to know why we arrived at something like the Schrodinger equation, read on...

At the time quantum theory came about, we knew that atoms were made of electrons and protons with equal but opposite charges. That was a problem straight away, because the oppositely charged particles should attract each other, and yet atoms were a lot bigger in size than protons and electrons, so there had to be something that was keeping the protons and electrons apart within the much bigger volume of the atom.

We also knew that atoms of different elements would only absorb or emit certain frequencies of light, so the idea was developed that the electrons were in orbit around the protons, like planets around the Sun, and they could jump between the different orbits by absorbing or emitting a photon of light with just the right energy.

The problem with that idea was that experiments had shown that accelerating electrical charges gave off electromagnetic fields, so if electrons were in orbit (which is accelerated motion), they should just radiate away all their energy and spiral into the middle of the atom. So we needed an idea for why that didn't happen, and why only specific electron orbits were allowed.

Separately, Louis De Broglie had come up with the idea that all particles were really waves, which was the complement of the idea that waves of light were really particles (photons). That gave an intuitive reason for the electron orbits. If an electron had a wavelength, its orbit could only have a length that was an integer multiple of the wavelength, since an orbit of any other length would interfere with itself.

So that gave all the essential ingredients for a model in which electrons were waves, with set of stable orbits around the positively charged nucleus, and the ability to jump to a higher orbit if they absorbed a photon of the right colour (energy).

Answer 4

Depends on which why you mean.

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Why is this particular wavefunction the way it is right now?

The wavefunction is the way it is now because of the way the wavefunction was earlier. The evolution of wavefunctions in time is purely deterministic until you come along and inject a measurement event into the system.

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What more fundamental theory is reducible to quantum mechanics for a particular truncated domain?

We do not yet have such a theory. We do have more-complete extensions of early 20th century quantum mechanics (i.e. the Standard Model), but that doesn't count. Our several candidates for a more fundamental theory are still underdeveloped and either have produced no testable hypotheses, or are technologically far beyond our capability to test.

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How can the Schrodinger equation be derived from axioms?

Answered here on Physics SE.

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What is the reason for selecting the axioms from which the Schrodinger equation is derived?

Answered here on Physics SE.

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What is the system really doing in between measurement events that the wave equation correlates to a function of measurement probabilities? What (and where, and when) even is the relevant system?

Nobody knows either answer.