The term "sandwich" and the verb "to sandwich" appear pretty common but informally in quantum mechanics. Generally when describing some kind of inner product of the form: $$\langle L | H | L'\rangle $$

where $\langle L|$ and $|L'\rangle $ are some (quantum) vectors (the loaf), generally for the same state $L=L'$, and $H$ is some operator. The $\langle L | H | L'\rangle $ is called the sandwich, and to take the expectation value is "to sandwich" $H$.

I would have thought that the terminology appeared at least as early as Dirac notation, because the "sandwiching" would be less obvious with integrals.

For some examples, check these questions in Physics Stack Exchange: Solving for the density operator in the quantum Brownian motion master equation, Manipulations of the wavefunction are hard to physically understand and Is there any relationship between S-matrix elements and the path integral?

I know that sandwich can be a very coloquial term for other stuff in science. It is even sometimes used formally in mathematics for theorems like the squeeze theorem (AKA sandwich theorem) and the ham sandwich theorem. However I would like to find the earliest attestation of this term used in quantum mechanics.

I searched some of my books, but I could only find the term "sandwich" in R. Shankar Principles of Quantum Mechanics and DJ Griffiths' Introduction to Quantum Mechanics, which are relatively new (1990s).

I had sometimes joked with the idea that the term came from expectations values of a Hamiltonian (abbreviated as Ham) but maybe that's stretch.

The earliest paper I could find is Non-Lagrangian models of current algebra, KG Wilson, Phys. Rev., 1969.

In a nutshell, what's the earliest recorded use of sandwich in quantum mechanics?

Bonus: this terminology has even crossed language boundaries, see for example Resumen de mecánica cuántica, VR. Rochin UNAM, Quantique: éléments F. Balibar et al., Ferienkurs Quantenmechanik Sommersemester 2013 Technische Universität München.

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    $\begingroup$ The Hamiltonian sandwich :). Very tasty but hard to digest for most people... $\endgroup$ Jul 30, 2021 at 6:37
  • $\begingroup$ It seems to me that 'to sandwich' is used in both math and physics to mean 'to bound from avove and below' and your examples in quantum physics are just applications of this general usage. When this usage first appeared is still an interesting question though. $\endgroup$
    – quarague
    Aug 6, 2021 at 6:57


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