# Who solved the quantum harmonic oscillator

In continuation of this question, Who introduced the creation and annihilation operators for the harmonic oscillator?, I got curious who came up with the quantum harmonic oscillator originally.

I am aware that the simple classical harmonic potential arises from the definition of $$F = kx$$ as a central force, thus $$F = \nabla V$$ and $$V = \frac{1}{2} kx^2$$. But what was the purpose of plugging such a potential into the Schrödinger equation?

Nowadays the harmonic oscillator is the simplest way to model chemical bonds but I assume that it had a different purpose in the dawn of quantum mechanics.

• No, I rather suspect that this was chosen because a sin function is well understood and easy to work with. Plus when it starts to break down, just add another higher-power sin func to your energy equation, and viola [sic] a Fourier expansion! – Carl Witthoft Oct 16 '18 at 12:33

In matrix mechanics: Born, Max; Jordan, Pascual, Zur Quantenmechanik, Z. f. Physik 34, 858-888 (1925). ZBL51.0728.08. (27 September 1925)

In wave mechanics: Schrödinger, Erwin, Quantisierung als Eigenwertproblem. II, Annalen d. Physik (4) 79, 489-527 (1926). ZBL52.0966.01. (23 February 1926)

(“Old quantum theory” also discussed oscillators and even zero-point energy, but I guess that’s outside the scope of this question.)

• Schrödinger translation: 1928. – Francois Ziegler Oct 16 '18 at 14:44

Harmonic oscillator in one dimension was solved in the very first paper of Heisenberg where he proposed quantum mechanics. But it gives the same result as the one obtained "old quantum mechanics" of Bohr. To obtain something new, Heisenberg also considers "anharmonic oscillators" in the same paper.

Uber quantumtheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys, 33 (1925).

There is an English translation in the book: B. L. van der Waerden, Sources of quantum mechanics, North Holland Amsterdam 1967.

A translation of Bohr's paper is also included there.

• The van der Waerden book is online and has both Heisenberg’s paper (29 July 1925) and the Born–Jordan one I quoted. I discarded the former’s “solution” as not yet purely quantum mechanical because he still invokes “old quantum” Bohr–Sommerfeld arguments we wouldn’t (around equation 15, p. 268) — but it is true that he does work from the oscillator Hamiltonian. – Francois Ziegler Oct 16 '18 at 14:30