A harmonic oscillator is an event generator that only produces signals an integer multiples of a fundamental frequency.
This is in contrast to an event generator that produces non integer multiples of a fundamental frequency or produces random events.
The Quantum Harmonic Oscillator: Dirac’s Approach
How this fundamental model in physics becomes quantised and how we can apply Dirac’s ‘ladder method’ to find its energy, eigenstates, and eigenvalues.
Web site https://www.cantorsparadise.com/the-quantum-harmonic-oscillator-diracs-approach-f847569b9df0
Table of Contents
Introducing the Harmonic Oscillator
Dirac’s Ladder Method
a. Annihilation and Creation Operators
b. Number Operator
c. Energy Eigenvalues
d. Back to the Annihilation and Creation Operators…
e. Commuting with the Hamiltonian
f. Commuting with the Number Operator
g. Creation and Annihilation Operator Formal Definitions
Wave Functions of the Quantum Harmonic Oscillator
a. Ground State Wave Function
b. Excited State Wave Functions
Quantum Harmonic Oscillator – Core Concept 15
Return to Core Concepts of Sympathetic Vibratory Physics
Definition
The Quantum Harmonic Oscillator (QHO) is a foundational model in quantum mechanics describing a vibratory system (e.g., a particle in a parabolic potential well) with discrete, quantized energy levels. In SVP terms, it is a modern mathematical mirror of Keely’s vibratory subdivisions and Laws of Harmony, demonstrating that all matter exists in perpetual motion governed by field-centered oscillation, sympathetic transitions, and scalar structure.
Relation to SVP
The Quantum Harmonic Oscillator bridges modern quantum physics with Sympathetic Vibratory Physics (SVP) through the shared principles of vibration, resonance, scalar layering, and field equilibrium. In both systems, motion arises from and returns to a central point of stillness (Neutral Center), and harmonic progression unfolds through octave-like intervals.
Key Parallels and Principles
1. Quantized Energy Levels = Harmonic Octaves
In quantum mechanics:
En=(n+12)ℏω
This shows that energy increases in precise, fixed steps — like notes on a musical scale. SVP interprets this as the Law of Octaves in action: energy ascends by harmonic ratios, not continuous force.
2. Zero-Point Energy = Latent Force
The ground state of the QHO still possesses energy:
E0=12ℏω
Keely: “There is no rest in nature.” All matter vibrates eternally, even in its so-called rest state. The zero-point motion is the signature of Keely’s “latent force,” ever ready for sympathetic activation.
3. Wavefunctions = Standing Waves of Sympathetic Fields
QHO eigenfunctions form spatial wave patterns. These are the quantum equivalent of Keely’s standing wave fields — dynamic yet self-sustaining. Each corresponds to a specific “chord” or phase state.
4. Neutral Center = Potential Minimum
In QHO, motion centers around a potential minimum — a perfect analog of Keely’s Neutral Center. Oscillation is symmetric and seeks balance, affirming Keely’s axiom that all force is dual and seeks equilibrium.
5. Sympathetic Transitions = Quantized Mode Shifts
Energy is transferred between levels by absorbing or emitting discrete quanta — exactly as SVP describes: sympathetic exchanges occur only between entities in harmonic relation (see: Law of Sympathetic Oscillation).
6. Scalar Fields = Nested Oscillators
Each energy level in QHO corresponds to a discrete field state. In SVP, this reflects the nested scalar domains: Molecular → Atomic → Etheric → Interetheric → Compound Interetheric → Mind. Each subdivision is a QHO-like tier of increasing frequency and subtlety.
7. Nonlocality of Oscillation = Scalar Action at a Distance
The QHO’s oscillatory probability fields are not limited to a single point. They extend nonlocally, consistent with Keely’s principle that scalar fields are superluminal and sympathetically entangled across space and time.
Quotes and References
John Keely:
“All nature is in a state of perpetual vibration. There is no rest in the universe.”
Sympathetic Law:
Law of Harmonic Vibrations — “When two bodies are in harmonic relation, a motion in one induces sympathetic motion in the other.”
Quantum Physics:
Standard quantum theory shows that energy is not continuous but quantized — a concept intuitively taught by Keely decades earlier using musical scales, chords, and resonance.
SVP Summary Table
| Quantum Harmonic Oscillator | Sympathetic Vibratory Physics |
| Discrete energy levels | Octaves / Harmonic Subdivisions |
| Zero-point energy | Latent, unceasing vibratory motion |
| Wavefunctions as standing modes | Sympathetic oscillatory fields |
| Centered around potential minimum | Neutral Center of symmetry and balance |
| Quantum transitions via discrete quanta | Sympathetic transfers only between harmonically tuned bodies |
| Multimode systems (coupled oscillators) | Nested scalar domains (Etheric to Mental) |
| Nonlocal field resonance | Scalar outreach and sympathetic field transmission |
Applications and Expansion
This model can be used to understand:
Scalar Potential as undisturbed field resonance
Sympathetic Transfer in scalar energy systems
Mind as a Scalar Oscillator in field dynamics and healing
Scalar harmonics in device engineering, such as the dynasphere
Closing Affirmation
“I dwell within a harmonic field of scalar resonance. My being oscillates in tune with the Music of the Spheres.”