Second Quantization: Creation & Annihilation Operators

[Niles]

Hi all

I am reading about second quantization. The kinetic energy operator T we write as

[tex]
\hat T = \sum\limits_{i,j} {\left\langle i \right|T\left| j \right\rangle } \,c_i^\dag c_j^{}.
[/tex]

Now, the creation and annihilation operators really seem to be analogous (in some sense) to the ket and the bra in first quantization, since they tell us which matrix element we are talking about.

What is the reason for this? I understand that we have the new states in Fock space (the occupation number states), but my book never illuminates why the creation and annihilation operators designate the matrix elements just like the outer product in first quantization does.

 

[Physics Monkey]

Hi Niles,

The creation operator [tex] c_i^+ [/tex] adds one particle to the system in the single particle state labeled by i. If [tex] |i\rangle [/tex] denotes the state where the system has one particle in single particle state i, then you know that [tex] |i\rangle = c_i^+ |\text{vac} \rangle [/tex]. This means you can write [tex] c^+_i = |i \rangle \langle \text{vac} | + \text{...} [/tex] where ... consists of states with more than one particle. Thus in the single particle Hilbert space the creation operators and annihilation operators are literally the outer products you are more familiar with. For example, [tex] c_i^+ c_j = |i \rangle \langle \text{vac} | \text{vac} \rangle \langle j | + \text{...} = | i \rangle \langle j | + \text{...} [/tex].

Does this help at all?

 

[Niles]

Hi Physics_Monkey

Yes, that is a very good explanation. Although I do not quite get what you mean by: "where ... consists of states with more than one particle.". We have [itex]|i \rangle \langle \text{vac} |[/itex], which is an operator. To this operator we add multiple-particle states (i.e. vectors) - is that allowed?

 

[Physics Monkey]

What I mean is that because because [tex] c_i^+ [/tex] adds one particle to any state, the expansion of [tex] c_i^+ [/tex] in terms of outer products can't stop with [tex] | i \rangle \langle \text{vac} | [/tex]. There must be other terms like [tex] |\text{2 particles} \rangle \langle \text{1 particle} | [/tex]. However, each of these terms contains at least one bra or ket with more than one particle. This is what I mean by ... containing states with more than one particle.