Are wave functions, such as those used in the Schroedinger equation just 'guessed' and verified, or are there other theories which tell us the mathematical description of the wave function for particular systems (i.e. if some new quantum phenomena is discovered, does the wave function need to be 'made up' from scratch and then experimentally verified or are there laws that give tight constraints on what form the wave function can take)?
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asked Dec 30, 2014 at 0:54
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3$\begingroup$ You've used the words "Schrodinger equation," but you should probably read up on it. $\endgroup$– jwimberleyCommented Dec 30, 2014 at 0:59
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2$\begingroup$ Solutions for wave functions are like solutions for problems in classical physics. There are very few systems for which we can calculate closed form solutions. The most important one is probably the hydrogen atom, which is extremely well understood. Beyond that we have approximation methods many complex systems. The most precise test for the validity of quantum mechanics comes from selection rules and energy eigenvalues. To the best of my knowledge we have not encountered systems for which the predictions failed reality and we can calculate many energy eigenvalue with good precision. $\endgroup$– CuriousOneCommented Dec 30, 2014 at 1:03
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1$\begingroup$ In a sense the wave function is a direct result of the Hamiltonian. The Hamiltonian is essentially made up to reproduce what we observe. You can push things back to the Lagrangian if you want, but that's just guessed and checked against reality. $\endgroup$– DanielSankCommented Dec 30, 2014 at 1:15
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$\begingroup$ Is the wave function not an independent input to the Schroedinger equation (does one not need to provide a wave function in order to solve the equation)? I'm jut curious as to where a specific wave function (input) comes from (is determining a wave function a pure trial and error process or is there a deeper theory involved)? $\endgroup$– Chris LaforetCommented Dec 30, 2014 at 1:30
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$\begingroup$ The wavefunction is the solution of Schrödinger's equation. It's the Hamiltonian that has to be more or less guessed, though it's usually inspired by the classical equivalent. $\endgroup$– JavierCommented Dec 30, 2014 at 13:55
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1 Answer
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Sometimes, our formalism is not the wave-function but all sort of symmetries. I refer here, as you asked, about cases in which what we test in the lab is a wave-function.
You ask : if some new quantum phenomena is discovered, does the wave function need to be 'made up' from scratch and then experimentally verified or are there laws that give tight constraints on what form the wave function can take?
The things go in all the ways:
Nothing is wrong in beginning from scratch.
But, a more systematic way when we find in our experiments some new effect, is to make a theoretical guess about forces or about potentials (if the forces can be considered conservative), that may lead to that effect. Then we write a Hamiltonian and obtain a wave-function by solving the Schrodinger equation, (or Dirac equation, etc., depending on the case). After that, we test the wave-function experimentally. So, it goes in nuclear interactions.
Though, I know a case in which the way above didn't lead to a satisfactory wave-function. The wave-function that was obtained explains some details of a phenomenon, but doesn't explain other details. So, people try first to modify the wave-function, check if it works, and after that they bother with the question which Hamiltonian, if any, produces the desired wave-functions.
About constraints, yes we have. A wave-function has to have a finite norm, s.t. we can normalize it to 1, because the wave-function intensity represents probability. A frequent case in which the norm is infinite, is wave-functions that normalize to $\delta$ Dirac, as the plane waves. But such wave-functions are idealizations, used for mathematical simplicity. There are other constraints too, e.g. the wave-functions of a collection of identical bosons or fermions, should obey symmetry laws with respect to the interchange of two particles. That indeed restricts the set of possible wave-functions. Particle physics gives us other symmetry constraints too.
