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The mathematics and physics we have studied so far at university are heavily focused around the idea of orthogonal functions, orthogonality, sets of solutions, eigenvalues and eigenfunctions.

Why are we so interested in these properties? What are the conceptual aspects of them, mainly in quantum mechanics?

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    $\begingroup$ This is a great question but I think you should confine it to one area -- ie. QM. It's far too broad to include their use in other areas of physics because almost every area has some use for them. $\endgroup$ – tpg2114 Nov 6 '13 at 1:26
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    $\begingroup$ The short answer (at least as far as I know) is that if you have an orthogonal basis it's very easy to find the components of any vector: just take inner product with the basis elements. This is pretty much what everything Fourier is about. $\endgroup$ – Javier Nov 6 '13 at 1:39
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    $\begingroup$ The two words that cover @JavierBadia's nice comment completeness and uniqueness: for any given basis there is one and only one decompositions every member of the solution space. $\endgroup$ – dmckee --- ex-moderator kitten Nov 6 '13 at 2:03
  • $\begingroup$ To expand on the comments here, it is important that we have complete orthonormal basis sets from a practical standpoint because we can often use them in solving the differential equations that show up in mathematical physics. Other than the fact that the functions themselves are beautiful from a functional analysis standpoint, they are very useful in constructing solutions in different orthogonal curvilinear coordinate systems. $\endgroup$ – codeAndStuff Nov 6 '13 at 13:53
  • $\begingroup$ The eigenfunction of a Hermitian operator is orthogonal to each other. $\endgroup$ – user26143 Nov 6 '13 at 14:28
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Several reasons:

  1. Orthogonal functions arise naturally in the study of Sturm-Liouville theory which includes many classical and quantum system mathematical models;

  2. More generally, it is the class of normal operators (and an important special case self adjoint operators) which the spectral theorem most readily works and is most complete for. The eigenvectors of such operators are always orthogonal. The "Diagonalising" an operator in any linear system theory is an important step for understanding - it means we can decouple the operator's action into the sum of its action on altogether uncoupled eigenvectors. It's an important step in "untangling" a highly coupled problem. In the context of when the Hilbert space concerned is a function space, the relevant Sturm-Liouville theory, e.g. for the quantum harmonic oscillator shows that the linear space of all "practical", normalisable quantum states is spanned by discrete eigenfunctions. In other words, the Hilbert space's dimension is countably infinite, even though we are dealing with spaces of continuous functions and you might intuitively think the dimension cardinality might be $\aleph_1$, and that's just too scary to deal with!

  3. We deal often with two important conservation laws: conservation of energy and conservation of probability. These conservation laws are most readily expressed if the basis for the relevant state space is orthogonal - it means that energy, power or probability as appropriate is simply the $\mathbf{L}^2$ length of any vector. We don't have to manage cross coupling terms in our inner product space. Whether it be functions or Cartesian bases for three dimensional Eucliean space, projections and resolution into basis superpositions are always heaps easier and clearer if the basis is orthogonal. You'd be a sucker for punishment if you did an everyday geometric problem in $\mathbb{R}^3$ with a general, linearly independent but nonorthogonal basis, even though this can certainly be done. Exactly the same intellectual work minimalisation principles apply to functions spaces as much as they do to $\mathbb{R}^3$. Energy- or probability-conservative system transformations are then unitary and so on and so forth.

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    $\begingroup$ Interesting. Some of the things you've mentioned are beyond my current knowledge. Could you tell me something about the use of such functions in terms of the Hydrogen atom; angular momentum, spin...? (Currently studying this) $\endgroup$ – turnip Nov 6 '13 at 14:31
  • $\begingroup$ @PPG Sorry about that. I'm a little busy at the moment but I'll try to add an appendix applying the three ideas to the H atom in the next few days. Actually, do you know the quantum harmonic oscillator? (It's simply that I've only ever really studied the H atom with the Dirac equation so for the scalar QHO I know a few more specifics). $\endgroup$ – WetSavannaAnimal Nov 6 '13 at 21:11
  • $\begingroup$ I have only covered a very small section of the QHO, which looks at the expectation values and probabilities. I will read up on it though. I appreciate it a lot, take your time with the appendix. $\endgroup$ – turnip Nov 6 '13 at 21:20

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