Historical research guide · 1126 · 2016

University Physics Volume 3, §7.5 — The Quantum Harmonic Oscillator

Samuel J. Ling · Jeff Sanny · William Moebs · OpenStax

In one minute

This is the half-quantum of energy as an undergraduate is taught it, which is exactly why chapter 2 cites it — the zero-point energy is not a fringe proposal but a standard textbook result, worked in a free textbook anybody can open. The section sets the classical oscillator beside the quantum one and lists the three places they part company. First, the ground state of a quantum oscillator is not zero but half of h-bar times the angular frequency, and the textbook gives the reason: a particle sitting motionless at the bottom of the well would have both position and momentum exactly determined, which the uncertainty principle forbids, so the lowest state must sit above the floor of the well. Second, the particle can be found outside the classical turning points, where a classical oscillator has no energy to be at all. Third, the ground-state probability is largest at the centre of the well, the opposite of the classical particle, which lingers at the turning points. It also notes what stays the same: the levels are evenly spaced, and at high quantum numbers the quantum distribution converges on the classical one.

Connection

Chapter 2 has to establish that a ground state with energy in it is ordinary physics before it can build anything on top. This is the plainest available demonstration — a standard, free, widely used textbook section — and it also supplies the picture the chapter spends a paragraph correcting, since the stationary ground state is a fixed distribution rather than a particle rattling about too fast to see.

From the source

The allowed energies of a quantum harmonic oscillator are h-bar times the angular frequency multiplied by the quantum number plus one half, so the lowest state has energy equal to half of h-bar times the angular frequency and not zero — the standard textbook statement of zero-point energy.Section 7.5, Eq. 7.56; ‘The quantum oscillator differs from the classic oscillator in three ways’, first difference
The textbook's stated reason for that floor is the uncertainty principle, and it is general rather than special to this system: the nonexistence of a zero-energy state is common to all quantum-mechanical systems, because a particle sitting motionless at the bottom of the well would need its position and its momentum simultaneously exact.Section 7.5, first of the three stated differences
Unlike a particle in a box, the levels are evenly spaced: the gap between neighbouring levels is h-bar times the angular frequency, equal to Planck's constant times the frequency, so the smallest quantum that can be emitted or absorbed in a transition is one such gap — consistent with Planck's hypothesis for the energy exchanges in the blackbody problem.Section 7.5, Eq. 7.58 and the paragraph following it
The ground state leaks outside the classically allowed region. A particle in the quantum oscillator potential can be found with nonzero probability beyond the classical turning points, where a classical particle would have no energy to be, and the textbook gives the number: the probability of finding a ground-state quantum particle in the classically forbidden region is about 16 percent.Section 7.5, second of the three stated differences
The ground-state probability density is largest in the middle of the well, the reverse of the classical picture in which the particle spends most of its time moving slowly near the turning points — and the reversal undoes itself at high quantum numbers, where the quantum distribution converges on the classical one, illustrated for the twelfth excited state.Section 7.5, third of the three stated differences; Figure 7.15
The model is not a toy: applied to hydrogen chloride, whose lowest infrared line sits at 8.88 times ten to the thirteenth hertz, the same arithmetic gives a vibrational level spacing of 0.368 electronvolts and an interatomic force constant of 520 newtons per metre.Section 7.5, Example 7.11, ‘Vibrational Energies of the Hydrogen Chloride Molecule’

The way in

https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantum-harmonic-oscillator

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

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How to cite it

Samuel J. Ling, Jeff Sanny, William Moebs, OpenStax (2016) University Physics Volume 3, §7.5 — The Quantum Harmonic Oscillator https://openstax.org/books/university-physics-volume-3/pages/7-5-the-quantum-harmonic-oscillator

Where it sits in the curriculum

What the vacuum is