Both $\hat{a}^\dagger\hat{a}$ and $\hat{a}\hat{a}^\dagger$ are Hermitian, how do we know which one represents the particle number?

up vote 9 down vote accepted

We require that the number operator have the following property:

$$\hat n |0\rangle = 0.$$

We know that

$$\hat a |0\rangle = 0$$

and we know that

$$\hat a |1\rangle = |0\rangle$$

and we know that

$$\hat a^{\dagger} |0\rangle = |1\rangle. $$

Thus, it follows that

$$\hat a \hat a^{\dagger} \ne \hat n$$

since

$$\hat a \hat a^{\dagger} |0\rangle = \hat a|1\rangle \ne 0.$$

Now, it remains to be shown that

$$\hat a^{\dagger}\hat a = \hat n. $$

Can you take it from here?

  • Thank you! You clearly solved my problem, but now I have another question: what's the difference between 0 and $|0\rangle$? (should I open a new question?) – LePtC Oct 11 '14 at 4:05
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    @LePtC, $|0\rangle$ is a state - the state with 0 energy quanta present while 0 is a number. The product of the number 0 and any state is the number 0. Since $|0\rangle$ is an eigenstate of the number operator with eigenvalue 0, it must be the case that $\hat n |0\rangle = 0 |0\rangle = 0$. – Alfred Centauri Oct 11 '14 at 16:56
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    @LePtC yes, that should be a new question, except that it's already been asked. – David Z Oct 12 '14 at 7:30
  • @AlfredCentauri you mean we can leave the ket when we have a 0 eigenvalue? when we get a zero spin for s=1 particle or zero energy for the Hamiltonian, why don't we leave the ket as well? – LePtC Oct 12 '14 at 16:13
  • @LePtC, I don't understand what you're trying to ask. A ket that is an eigenstate of some operator with eigenvalue zero has non-zero length, i.e., $\langle 0 | 0 \rangle \ne 0$. Only the $\varnothing$ (null) ket has zero length $\langle \varnothing | \varnothing\rangle = 0$ and it is not a state. – Alfred Centauri Oct 12 '14 at 16:35

Since you define e.g. in the bosonic case

$c_j^\dagger: H_N^S \rightarrow H_{N+1}^S,\quad c_j^\dagger | \ldots n_j \ldots \rangle := \sqrt{n_j+1} |\ldots n_j+1 \ldots \rangle$ $c_j: H_N^S \rightarrow H_{N-1}^S,\quad c_j | \ldots n_j \ldots \rangle := \sqrt{n_j} |\ldots n_j-1 \ldots \rangle$

it makes more sence to use $a^\dagger a$ which will give you $n_j$ (instead of $n_j+1$) as prefactor when acting on a state $| \ldots n_j \ldots \rangle $.

  • But you are using the eigen state of $\hat{a}^\dagger\hat{a}$, maybe we can redefine the eigen state $|n'_i\rangle=|n_i+1\rangle$ to avoid that problem? – LePtC Oct 10 '14 at 9:17
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    @LePtC: This would only "shift" your problem... – pawel_winzig Oct 10 '14 at 10:50

The vacuum should have particle number $0$. In some detail: we would like $\hat{N}\ |0\rangle=0$ and $\hat{N}=a^\dagger a$ is the only ordering that does that. It follows from the usual commutation relations that $\hat{N} |n\rangle =n\ |n\rangle$ which is in sync with interpretation of $|n\rangle $ as an $n$-particle state in second-quantisation.

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    You want to elaborate on this a bit? – Pranav Hosangadi Oct 10 '14 at 8:05
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    I think he means that $a^\dagger a$ kills the vacuum while $a \, a^\dagger$ doesn't. – Jold Oct 10 '14 at 23:50
  • @PranavHosangadi Explanation added. I hope that suffices. – suresh Oct 12 '14 at 7:15

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