Second Quantization: Exploring the Strange Results

[Hymne]

Didnt seem to be many threads about this subject although I don't find it trivial at all..

Lets start with a question:

If we now have <N_i - 1|â_i|N_i> = N_i^0.5 but let â operate on our ket it should give:
<N_i - 1||N_i - 1> = N_i^0.5 its adjoint however is the creation operator (right?) which gives if we let i work on our bra:

<N_i|N_i > = <N_i - 1||N_i - 1> = N_i^0.5

This seems strange! Because then the probabilitiy of finding N particles in state i is independent of N. Or where do I get i wrong?

 

[strangerep]

Or where do I get i wrong?

Such calculations depend on the normalization conventions one
chooses for the multiparticle states. I.e., you've probably missed a
normalization factor somewhere. What definition are you using for
the N-particle states (in terms of the creation op acting on the vacuum)?

 

[Hymne]

Hmm, I´m reading Landau and Lifgarbagez and he hasnt got to the vaccumstate get but otherwise he defines:

â_i|N_i> = (N_i)^0.5|N_i - 1>

and


â_i*|N_i> = (N_i+1)^0.5|N_i + 1>.

 

[cgk]

The goal of these definitions is (a) that no physically impossible states can be created (say, with N_i < 0) and (b) that the occupation number operators can be represented as
$$\hat n_i = \hat a_i^\dagger \hat a_i.$$
This, however, depends on the concrete definition of the normalization factors in your many-body basis determinants corresponding to strings of occupation numbers. For bosonic states those N_i occupation numbers would occur in these basis state definitions and cancel the sqrt(N_is) from the creation/destruction operators. Note that the |N_i - 1> in your formulas is itself /not/ a normalized N-1 body state, but rather just the state you get from |N_i> by reducing one of the occupation numbers (N_i) by one and otherwise keeping the prefactors of |N_i> (i.e., forming a determinant from one orbital less).

 

[strangerep]

Hymne, this sort of stuff is more enjoyable to read in latex form.
I've latexified your quote below as an example so you can get
the idea how it's done... (hint, hint).

$$a_i |N_i\rangle ~=~ \sqrt{N_i} ~ |N_i - 1 \rangle$$

and

$$a_i^* |N_i \rangle ~=~ \sqrt{N_i+1} ~ |N_i + 1\rangle ~.$$

OK,... but... then I don't see how you got the result you think you
did in the original post. (I.e., I'm still not sure exactly what point
you're missing.)