The combinatorial core
The key observation is that computing \(langle psi_0 | (a + a^dagger)^4 | psi_0 rangle\) via normal ordering is an instance of Wick’s theorem. Writing \(phi_i = a + a^dagger\) for each of the four factors, the theorem decomposes the product into normal-ordered pieces decorated by contractions:
\[ phi_1 phi_2 phi_3 phi_4 = {:} phi_1 phi_2 phi_3 phi_4 {:} + sum_{"single"} overline{phi_i phi_j} \; {:} phi_k phi_l {:} + sum_{"double"} overline{phi_i phi_j} \; overline{phi_k phi_l} \]
where the contraction is the c-number
\[ overline{phi_i phi_j} = phi_i phi_j - {:} phi_i phi_j {:} = [a, a^dagger] = 1 \]
Since \(langle psi_0 | {:} text(anything nontrivial) {:} | psi_0 rangle = 0\), only the fully contracted terms survive — the last sum. The number of ways to partition 4 objects into 2 unordered pairs is
\[ frac(4!, 2^2 cdot 2!) = 3 \]
the pairings (12)(34), (13)(24), (14)(23). Each contraction contributes 1, so \(langle psi_0 | (a + a^dagger)^4 | psi_0 rangle = 3\).
You’ll recognise \(3 = (2 cdot 2 - 1)!! = 3!!\), the double factorial that counts perfect matchings — equivalently, the moment \(mathbb{E}[X^4] = 3\) for \(X sim cal(N)(0,1)\). This is not a coincidence: Wick’s theorem is Isserlis’ theorem, the two connected by the path-integral/Gaussian-measure perspective.
The Feynman diagram translation
Now reinterpret this diagrammatically. The quartic interaction \(lambda g \, x^4\) contributes a 4-valent vertex — a point with 4 legs. Each contraction \(overline{phi_i phi_j} = 1\) is a propagator connecting two legs of the vertex back to itself.
The vacuum expectation \(langle psi_0 | (a + a^dagger)^4 | psi_0 rangle\) asks: in how many ways can we close off all 4 legs using propagators, leaving no external lines? This is the enumeration of vacuum diagrams at first order.
With 4 legs and 2 propagators, there is topologically only one diagram: the figure-eight (two loops meeting at the vertex). But it arises from 3 distinct contractions. The relationship is controlled by the symmetry factor \(S\) of the diagram:
\[ 3 = frac(4!, S) implies S = 8 \]
The factor of 8 is visible directly: the figure-eight has a \(mathbb{Z}_2 wr mathbb{Z}_2\) symmetry — you can swap the two loops (one factor of 2), and independently reverse the orientation of each loop (two more factors of 2), giving \(2 times 2 times 2 = 8\).
So the Feynman rule for this first-order energy shift is:
\[ Delta E_0 = lambda frac(omega hbar, 2) cdot frac(1, 16) cdot frac(4!, S) = lambda frac(omega hbar, 2) cdot frac(1, 16) cdot 3 = lambda frac(3, 32) omega hbar \]
The broader picture
This is quantum mechanics — QFT in \((0+1)\) dimensions — so the diagrammatics is as simple as it gets: there’s a single mode, the “propagator” is just the number \(1/(2m omega)\) (before you absorb it into the vertex), and there’s no momentum to integrate over. But the structural pattern is identical to what happens in scalar \(phi^4\) field theory:
Normal ordering \(leftrightarrow\) Wick’s theorem \(leftrightarrow\) summing over contractions \(leftrightarrow\) summing over Feynman diagrams. These are four descriptions of the same combinatorial procedure.
Vacuum expectation values pick out the fully contracted terms, i.e., the diagrams with no external legs (vacuum bubbles).
Symmetry factors account for the overcounting when you pass between “labelled contractions” (of which there are \(frac((2n)!, 2^n cdot n!)\) for \(2n\) operators) and topologically distinct diagrams.
At higher orders in perturbation theory you’d get multiple vertices connected by internal propagators, and the diagram enumeration becomes genuinely useful as bookkeeping — especially once you also need matrix elements \(langle psi_n | cdots | psi_m rangle\), where external legs carry quantum numbers and the diagrams acquire external structure.
The fact that \(3 = (2n-1)!!\) also connects to the general result that the \(k\)-th order vacuum diagram count for a single \(phi^4\) vertex (with \(4k\) legs to pair off) relates to the enumeration of perfect matchings on \(4k\) elements — a problem with rich combinatorial structure (connection to the Hafnian, ribbon graphs, genus expansion if you track planarity, etc.).