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Most quantum mechanics texts include a phrase such as 'any ket can be written as a sum of eigenkets of a given observable'.

I have problems with the generalities of any ket.

Does this literally mean any ket, or does it mean any ket of the same observable? It seems odd that any ket, say an eigenket of position, should be expressible as a sum of eigenkets of a different observable?

Edit: Unless of course a general state is dependent on more than one summation of eigenkets of more than one observable, but the measurement of one of those observables does not depend on the other summations.

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"kets" are nothing but vectors in a Hilbert space. Every vector can always be decomposed along a Hilbert basis. Every observable with (pure point spectrum) has a Hilbert basis of eigenvectors. Thus: Every vector ("ket") can be written as a (generally infinite) sum of eigenvectors of any independently given observable (with pure point spectrum). –  Valter Moretti Aug 21 at 9:18
    
@ValterMoretti covers the mathematical idea (as does Ignacio's answer). I tried to address the physical picture with my answer. –  Danu Aug 21 at 9:20
    
@ValterMoretti by the way, could I talk to you in chat for a minute? I'm about to start a math. phys. degree and am looking for some advice! –  Danu Aug 21 at 9:22
    
Both excellent answers by the way. To further clarify, does that mean when we actually measure an observable of a general state, then we're only interested in one dimension of this Hilbert space? Or to attempt geometric intuition: analogous to moving along a single axis in the Cartesian plane (but clearly in higher dimensions)? –  TheJsyChemist Aug 21 at 9:24
    
@Danu I am in chat... –  Valter Moretti Aug 21 at 9:25

2 Answers 2

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Think about it this way: When you measure some observable $O$, you get some measured value. If you measure many identically prepared systems, you may measure different values corresponding to the operator $O$. In the limit where you do this infinitely many times, you are able to recover exactly which values are possible, and with which probability. Basic quantum theory now tells you that the system (before you measured anything) was in a superposition of eigenstates corresponding to the (eigen)values for $O$ which you ended up measuring.

This means that the system is in the state $$|\psi\rangle =\sum_{i=1}^\infty c_i |o_i\rangle$$ i.e. it can be written as a weighted sum of eigenkets $|o_i\rangle$ of $O$. This basically just expresses that one can always measure the observable $O$, and get some value. In some cases, the sum should be replaced by an integral, or it can be a sum over finitely many eigenstates, but the idea is the same.

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I imagine a general state is expressible then as multiple sums of multiple eigenkets of different observables, however when we want to measure an observable A, say, then the contributions from observable B, C, D etc. are clearly zero. The simplified equations given in quantum texts neglect this, which make it seem as if a whole state can be formulated as a sum of eigenkets of a single observable all the time! –  TheJsyChemist Aug 21 at 9:28
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There are some subtleties regarding the compatibility of certain observables here, but I think you have essentially the right idea. When discussing angular momentum, for instance, the states are often labeled $|n,l,m\rangle$ for the energy, total angular momentum and angular momentum along the $z$-direction. –  Danu Aug 21 at 9:42
    
Of course I realise that not all observables commute, but yes thankyou for your responses. –  TheJsyChemist Aug 21 at 9:44

Look it like this, a ket is a vector of a (fancy) vector space and as such it has different components with different weights.

Now, going to a more familiar vector space of the $R^3$, you can choose for example the cartesian coordinate system to describe a vector in space, but you can also choose any other set of three vectors (provided they're not colinear) to describe such a vector. In particular you can pick any rotation of the initial cartesian system.

That procedure and possibility in real space, also translates to the Hilbert vector space where kets (and bras) "live". It's nothing else than to say you're expressing your vector in a different basis.

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