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Say I have the following operator:

$$\hat { L } =\hbar { \sum_{ \sigma ,l,p } { l } \int_{ 0 }^{ \infty }\!{ \mathrm{d}{ k }_{ 0 }\,\hat { { { a }}}_{ \sigma ,l,p }^{ \dagger } } } \left({ k }_{ 0 }\right)\hat { { a }}_{ \sigma ,l,p } \left(k _{ 0 }\right)$$

How would I write this in matrix form?

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    $\begingroup$ An operator is not a matrix; what you can do is to write the matrix representation of the operator onto a given basis. $\endgroup$ – gented Jan 17 at 15:15
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    $\begingroup$ if you don't know this and if you are working in second quantized picture as you do now, it means that you are way ahead of yourself. I would strongly recommend you to stop whatever you are doing now, and start studying the first chapter of quantum mechanics of Sakurai as well as to solve exercises at the end of the chapter. it will take only 1 week. without doing that everything that you will encounter will be meaningless to you, it is basically impossible to progress . so just spend a week on that chapter and solve the exercises. $\endgroup$ – physshyp Jan 17 at 15:23
  • $\begingroup$ Actually you should rather study chapter 6 of Samurai which is about identical particles. Do you know about "second quantization" or what is the Fock space for bosons and fermions? $\endgroup$ – lcv Jan 17 at 16:24
  • $\begingroup$ I think he has trouble with basics of dirac formalism since he asks about how to write an operator in terms of matrix, so he first needs to get dirac formalism stuff then he may move to second quantization, which is pretty straigth forward after the getting the basics from dirac formalism etc. $\endgroup$ – physshyp Jan 17 at 20:05
  • $\begingroup$ Thank you for all the resources, and yes I am in over my head. The reason for this is that I am currently only in high school, I am writing a 'thesis' in, with help from PHD student for a school subject. I am investigating SPDC. I had an idea to experimentally verify the CSCO criterion for entanglement: arxiv.org/abs/1306.3325 (pretty unreliable looking paper, but, nonetheless). This requires me to find the CSCO's of a SPDC system. So I have trying to do this. I was trying to represent a set of operators in matrix form so I could test whether they commute. Is this the right approach? $\endgroup$ – Gabe Love Jan 18 at 11:49
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An operator is a linear function $f \in H \to Y$ where $H$ and $Y$ are vector spaces. For such a function there are representations that resemble matrices. For example, it is common in physics that $H=Y$ is a Hilbert space over the complex numbers and that is has a Schauder basis, in the sense that is has a countable set $(y_n)_{n=1}^\infty$ such that for any element $x \in H$ there is a sequence of of complex numbers $(a_n)_{n=1}^\infty$ such that $x=\sum_{n=1}^\infty a_ny_n$, the sum converging in the Hilber space norm. In this case, $f(x)=\sum_{n=1}^\infty a_nf(y_n)$ and also there is a sequence $(b_{n,m})_{m=1}^\infty$ such that $f(y_n)=\sum_{m=1}^\infty b_{n,m}y_m$ so

$f(x)=\sum_{n=1}^\infty\sum_{m=1}^\infty a_nb_{n,m}y_m$

which looks like how a matrix operates. You could represent that as a countably infinite matrix $(b_{n,m})_{n=1,m=1}^\infty$ being multiplied by other such matrices in the usual way and you can check that composition of operators is respected by this representation.

However this is but an example. Another one would consider bigger sets of "orthonormal" vectors and you would have integrals instead of summations (and extra trouble?). You might be using another "sense" of a basis and you would have to go through a similar procedure to build the representation. In general, once you have the representation and have chosen your "basis" the "matrix elements" would be given by the projections of $f(y_i)$ on $y_j$, $P_{y_j}f(y_i)$ where y stands for a "basis" element and its index is your way of identifying each of them (a counting set, for example the real numbers).

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The question is a little vague but you first need an orthonormal basis, i.e. you need a set of vectors $\vert \phi_k\rangle$ for which you can compute $\langle \phi_j\vert \hat L\vert \phi_k\rangle = c_{jk}$. Then it’s a matter of writing $$ \hat L=\sum_{j’,k’} c_{j’k’} \vert \phi_{j’}\rangle\langle \phi_{k’}\vert \tag{1} $$ so that $$ \langle \phi_j\vert \hat L\vert\phi_k\rangle = \sum_{j’,k’} c_{j’k’}\langle \phi_j\vert\phi_{j’}\rangle \langle \phi_{k’}\vert \phi_k\rangle= c_{jk} $$ follows from (1) by orthonormality $\langle \phi_{j’}\vert\phi_j\rangle=\delta_{j’j}$. The $c_{jk}$ can then immediately be inserted into a matrix at position $(j,k)$.

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Altough this is more mathematics than physics, I'll leave this comment:

You need to know a basis for the space. For eaxmple, in a 2-D space, you might choose the $\left( \begin{array}{} 1 \\ 0 \end{array} \right) ; \left( \begin{array}{} 0 \\ 1 \end{array} \right) $ basis.

Finally, you'd have to know how the operator transforms the initial-basis vectors. That means, the results of $L\left( \begin{array}{} 1 \\ 0 \end{array} \right) ;\quad L\left( \begin{array}{} 0 \\ 1 \end{array} \right) $

Then, it's basic algebra that the columns of the matrix are the the components of those results.

But

The first question is: What's the vector space? You need to know where you are in order to find a basis. Then you should check that you're working with functions, which are infinite dimensional vector spaces. You cannot build an infinite matrix!

So trying to find such matrix is absurd You just work with the differential operator.

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