I am wondering: can we explain the concept of orthogonality in physics for a beginner (without much math and linear algebra) by saying it simply means that the particle can not exist in two different states at the same time.
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2$\begingroup$ If the state you are orthogonal to is an eigenstate of some observable, then you can just say that it means you have no chance of finding the result corresponding to its eigenvalue in a measure of the said observable, right ? $\endgroup$– Barbaud JulienCommented Nov 1, 2018 at 7:18
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1$\begingroup$ You may find the answers to this question helpful $\endgroup$– By SymmetryCommented Nov 1, 2018 at 9:35
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2 Answers
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No. The wavefunction for a particle can be a superposition of orthogonal states. One can loosely say that it “exists” simultaneously in all those orthogonal states because a measurement of an observable can produce results corresponding to any of the observable’s orthogonal eigenstates.
A good way to understand orthogonality is what commenter Barbaud Julien said. The way I would put it is that orthogonal eigenstates of an observable correspond to different possible results of a measurement of that observable. You’ll observe one of the eigenvalues and not the others.
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can we explain the concept of orthogonality in physics for a beginner (without much math and linear algebra) by saying it simply means that the particle can not exist in two different states at the same time.
I think this mixes up two concepts. a) the need for a coordinate system to describe anything b) particular quantum mechanical states.
For example, we use orthogonal (x,y,z) coordinates because of the simplicity and symmetry of the system. We could have defined them with a specific angle between them,not of 90 degrees. We would still get specific locations correctly, but the algebra would be multiplied. Because vector algebra is simplfied by using orthogonal coordinates for recording locations there has never been a proposal to replace the orthogonal system in geometry.
In an analogous way, wherever one has a vector type algebra the systems chosen are orthogonal.
When it comes to quantum mechanics and wavefunctions, still the most practical system is assuming orthogonality for the corresponding system. I suppose one might still have wavefunction descriptions mathematically assuming fixed values for the dot product of the two wavefunctions on which all others would be modeled, but the complexity of expressions would be enormous and obscure the simplicity of the algebra.
That one can model reality with orthogonal wave functions is fortunate, and it allows the "it simply means that the particle can not exist in two different states at the same time", if one adds that a particle has quantum numbers that have to be represented correctly by the wavefunction describing it.
