# What is the so called orthogonal operator basis ?

1. Jan 3, 2010

### twinphoton

What is the so called "orthogonal operator basis"?

What is the so called "orthogonal operator basis"?

2. Jan 3, 2010

### arunma

Re: What is the so called "orthogonal operator basis"?

I've never heard of the orthogonal operator basis, though I assume it would be a field theory thing, since in QFT we turn the wavefunctions into operators. I could be wrong.

Are you sure you're not simply talking about an orthogonal basis?

3. Jan 3, 2010

### twinphoton

Re: What is the so called "orthogonal operator basis"?

Yes , i am sure that it does exist the "orthogonal operator basis", i.e. if we have this operator basis, arbitrary operator can be decomposed into the sum of several operators from these operator basis.

But, i do not know how to define these basis.

4. Jan 4, 2010

### diazona

Re: What is the so called "orthogonal operator basis"?

I think we'd have to know more specifics.

I can sort of see how it might work, though... operators do form a vector space, at least in a given state basis, since the operators can be expressed as matrices. And I suppose it might be possible in at least some cases to decompose operators into basis elements in a way that's independent of the state basis being used. But I've never actually run into the idea of basis operators having any particular importance.

5. Jan 4, 2010

### Fredrik

Staff Emeritus
Re: What is the so called "orthogonal operator basis"?

I'm not familiar with the term, but it's not hard to find a basis for the algebra of operators, given a basis for the Hilbert space:

$$A|\psi\rangle=\sum_{i,j}|i\rangle\langle i|A|j\rangle\langle j|\psi\rangle=\bigg(\sum_{i,j}A_{ij}|i\rangle\langle j|\bigg)|\psi\rangle$$

6. Jan 5, 2010

### MaxwellsDemon

Re: What is the so called "orthogonal operator basis"?

An orthogonal operator basis would just be a basis for various states, associated with whatever operator you're talking about, which is orthogonal. Remember that a basis for a vector space is a set of vectors and any arbitrary vector in the vector space can be written as a linear combination of the vectors in the basis set. If a basis is orthogonal, it means that the inner product of any two of the basis vectors equals zero. A lot of time we use an orthonormal basis. This means that in addition to being orthogonal (inner product = 0) the basis vectors each have a magnitude equalling one. In quantum mechanics each operator has a set of basis vectors associated with it. Often quantum states are written in the position basis or the momentum basis...but you can have a spin basis and other kinds of bases if you wish. If you'd like more information on vector spaces and bases and things like that the subject dealing with those topics is called Linear Algebra.