Understanding Orthogonality: QM Lectures by Dr. Physics A

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john t
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I need to understand orthogonality. I am monitoring QM lectures by Dr. Physics A, and he said all basis states of a state are orthogonal. I can understand that for the topics like polarization or spin, where Cartesian coordinates obtain with reference to measurements in one of 3 perpendicular-to-each-other axes, but then he gets into the linear position operator. Every point on the line is a basis state, and their number is infinite. I understand that if a particle is at one point it is definitely not at another, so that seems to fullfill a criterion for orthogonallity, but how can they be orthogonal (perpendicular) to each other? I cannot picture it.

John Thompson
 
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The Cartesian coordinates in normal three-dimensional space are completely unrelated to the orthogonality of basis states, because that's not the "space" being spanned by these basis states. This is true even in the case of spin; the two "orthogonal" states are spin-up and spin-down, and they correspond to measurements along the same axis but 180 degrees apart, not the 90 degrees that geometric orthogonality requires.

Instead, the states are vectors in a type of abstract vector space called a Hilbert space (wikipedia and wolfram mathworld both have good definitions of "vector space" and "Hilbert space"). Two vectors are orthogonal if their inner product is zero.

In the case of the position operator, the inner product of the two vectors corresponding to the two states "the particle has a 100% probability of being found at position ##x_1##" and "the particle has a 100% probability of being found at position ##x_2##" is zero unless ##x_1=x_2##, so they're orthogonal vectors in the Hilbert space of "all possible states of the position of the particle".

But some notes and warnings:
0) If you are serious about understanding QM, you will want to quit with the videos and spend some quality time with a decent first-year QM textbook. There are some recommendations in our "books" section.
1) You will need a working understanding of linear algebra. If you can look at the wikipedia and wolfram mathworld explanations of a vector space and think "It's not that complicated. Why are they making it look so hard?" you're there.
2) The position operator brings along some additional mathematical complexities that many/most introductory treatments gloss over and that I have completely ignored in the answer above. It takes some additional machinery (the "rigged Hilbert space") to fit the basis vectors of the position operators into the formalism properly. Don't worry about this until you have to.
 
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Vectors and orthogonality, like many concepts in mathematics, get generalised way beyond the initial, intuitive definition.

The simplest vectors are 2-3D real vectors and orthogonality is easily visualised. Functions, however, also have the properties of vectors, including orthogonality, but this is more abstract and algebraic. You have to trust the maths and in many cases make do without an obvious visualisation.
 
Thank you both for taking the time to clarify this for me. Nugatory's explanation led me to Wikipedia. Although I am familiars with linear algebra (matrix manipulation, properties of vector space, etc.) I am afraid Hilbert space requires more math education than I have. I will just recognize that in QM inner product being zero implies orthogonaliy and vice versa.

jct