Spectral decomposition PDF

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Quantum 5...

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Concepts Completeness (closure) relation A sum of projection operators is a sum of outer products, and thus can effectively “generate” elaborate operators. For example, let’s take a 3-dimensional vector space and construct a sum of projectors using three orthogonal states, 1, 2, and 3 in this vector space. These are labels for the states, not numbers themselves. They could be ψ1, ψ 2 and ψ3, or 𝑥,𝑦 and 𝑧… but we’ll stick with numbers for now. 0 |𝑠〉〈𝑠|

= |1〉〈1| + |2〉〈2| + |3〉〈3|

!

This sum of three projectors thus spans all possible basis states (or dimensions) in the given 3-dimensional vector space. If a sum of projector has enough terms to map all dimensions, it can be said to be complete, and effectively becomes the identity operator in that vector space. The identity operator, 𝐼6, is one that doesn’t change the function/state that it’s applied to – the matrix/vector equivalent of multiplying by 1. In maths terms, you can write this as: 𝐼6|𝜓〉 = 0|𝑠〉〈𝑠|𝜓〉8

= |1〉〈1|𝜓〉 + |2〉〈2|𝜓〉 + |3〉〈3|𝜓〉 = |𝜓〉

!

Relevance… This can help us tidy up some awkward bits of notation. For instance, the following expression with two operators 𝐴6 and 𝐵& is hard to compute because it requires multiplying two operators and the order to do it becomes ambiguous. 〈𝑟|𝐴6𝐵&|𝑐〉 So we can insert the identity operator into the middle without changing anything, replace it with a sum of projectors. A truly complete set of projectors might require an infinite sum, but if the sum has just enough projectors to describe the system properly, we can speed this calculation up: 〈𝑟|𝐴6𝐼6𝐵&|𝑐〉 = 0〈𝑟|𝐴6|𝑠〉〈𝑠|𝐵& |𝑐〉 !

Things like 〈𝑟|𝐴6|𝑠〉 and 〈𝑠|𝐵& |𝑐〉 are much easier to work with and can be multiplied in any order.

Spectral Decomposition Theorem The spectral decomposition theorem says that if we know all the eigenfunctions of an operator, we can rebuild it using a set of projectors. So if we know that 1, 2 and 3 are eigenvectors/eigenfunctions of the operator, we know that: 𝑂& |1〉 = 𝑎" |1〉

& |2〉 = 𝑎# |2〉 𝑂

& , can be described as: So the operator, 𝑂

& |3〉 = 𝑎$ |3〉 𝑂

𝑂& = 0| 𝑠〉 〈𝑠| = 𝑎"|1〉〈1 | + 𝑎# |2〉〈2 | + 𝑎$ |3〉〈3| !

Where 𝑎! are the eigenvalues, these are just numbers. So we can insert the above into any of the three eigenfunction equations. 𝑂& |1〉 = 0| 𝑠〉〈𝑠|1〉 = 𝑎" |1〉〈1|1〉 + 𝑎# |2〉〈2|1〉 + 𝑎$|3〉〈3|1〉 = 𝑎"|1〉 !

Remember that 〈 𝑥 ∣ 𝑥 〉 = 1 but 〈 𝑥 ∣ 𝑦 〉 = 0. A lot of the manipulation of these formulas is all about trying to get everything to look like these bras and kets, and finding which ones equal 1 and which ones equal 0.

Idempotent operators Projectors are idempotent operators, which means that squaring (cubing, etc..) them has no effect on them. In other words: &$ = ⋯ = 𝒫 &% 𝒫& = 𝒫& # = 𝒫

This is easily verified using for example a projector on normalised state |1⟩, i.e. 𝒫&" = |1〉〈 1|: &"# = B𝒫&" C# = 𝒫 &" 𝒫 &" = |1〉〈1|1⟩〈 1| = |1〉〈1| = 𝒫&" 𝒫

&". as 〈 1|1⟩ = 1. It is straightforward to show the same applies for other powers of 𝒫...