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John Thompson

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- Thread starter john t
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John Thompson

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Nugatory

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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.

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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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.

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