The simple harmonic oscillator is solved by geometry on the two-ring instrument and the Fibonacci 60-ring of the companion papers, with every step checked as it is made, in lengths a student can draw. The route it replaces is the operator mechanics that Freericks and Doughty (arXiv:2305.00062, 2023) propose for teaching: trade the differential equations of wave mechanics for abstraction and run every solvable problem on four operator identities, with the oscillator as the worked case, solved three ways, the operator way resting on the canonical commutation relation, the factorization of the Hamiltonian, a second singular factorization that finds the odd states, and wavefunctions built from a translation operator and an operator form of the Rodrigues formula. This paper is the fourth way. The oscillator is a point on a ring: one setting reads its position between the centers and its momentum between the crossings. Velocity sits a quarter turn ahead and acceleration a half turn ahead, and the ring's digit laws, times seven and sum to ten, check each step. The energy is the radius squared. The commutation relation, the postulate their h-bar enters by, is the size of one cell of phase space: the ground orbit encloses h/2, one triangle, and every annulus between orbits encloses h, one diamond, so h enters the system as a unit, where m and omega do, and nowhere else. On the ring the area per step is the Casoratian of the wave, one constant at all sixty positions, with Cassini's identity as its digit shadow. The stationary states are rings themselves: by Wigner's phase-space formulation, which is exact for the oscillator, time evolution is a rigid rotation, a stationary state is a function of the radius alone, the position density is that ring projected on the center line and the momentum density the same ring projected on the crossing line, and the paper verifies that these projections are the standard densities. The wavefunctions themselves come from one slide: slide the ground blob along the center line, and the powers of the slide are the Hermite functions, with the explicit formula, the recurrence and the matrix elements falling out of two multiplied series where their paper needs a Gaussian similarity transform, nested commutators and an induction. The quarter turn is the Fourier transform, which is how momentum is measured in a trap. Their singular factorization sorts the states by the sign of the ring at its center, the parity. The spectrum E = h-bar omega (n + 1/2), counted in the equilateral triangles of the Tetryonics model, is the odd numbers: the ground state is one unsplittable triangle, each rung is one diamond, one photon, the totals are the squares, hydrogen's n squared, and the three-ring oscillator's degeneracies are the triangular numbers, one triangle read three ways. Their four identities, at the oscillator, are the addition of positions plus the area term. Every formula used is standard and cited, and every number is verified by the script in the appendix.
The simple harmonic oscillator is solved by geometry on the two-ring instrument and the Fibonacci 60-ring of the companion papers, with every step checked as it is made, in lengths a student can draw. The route it replaces is the operator mechanics that Freericks and Doughty (arXiv:2305.00062, 2023) propose for teaching: trade the differential equations of wave mechanics for abstraction and run every solvable problem on four operator identities, with the oscillator as the worked case, solved three ways, the operator way resting on the canonical commutation relation, the factorization of the Hamiltonian, a second singular factorization that finds the odd states, and wavefunctions built from a translation operator and an operator form of the Rodrigues formula. This paper is the fourth way. The oscillator is a point on a ring: one setting reads its position between the centers and its momentum between the crossings. Velocity sits a quarter turn ahead and acceleration a half turn ahead, and the ring's digit laws, times seven and sum to ten, check each step. The energy is the radius squared. The commutation relation, the postulate their h-bar enters by, is the size of one cell of phase space: the ground orbit encloses h/2, one triangle, and every annulus between orbits encloses h, one diamond, so h enters the system as a unit, where m and omega do, and nowhere else. On the ring the area per step is the Casoratian of the wave, one constant at all sixty positions, with Cassini's identity as its digit shadow. The stationary states are rings themselves: by Wigner's phase-space formulation, which is exact for the oscillator, time evolution is a rigid rotation, a stationary state is a function of the radius alone, the position density is that ring projected on the center line and the momentum density the same ring projected on the crossing line, and the paper verifies that these projections are the standard densities. The wavefunctions themselves come from one slide: slide the ground blob along the center line, and the powers of the slide are the Hermite functions, with the explicit formula, the recurrence and the matrix elements falling out of two multiplied series where their paper needs a Gaussian similarity transform, nested commutators and an induction. The quarter turn is the Fourier transform, which is how momentum is measured in a trap. Their singular factorization sorts the states by the sign of the ring at its center, the parity. The spectrum E = h-bar omega (n + 1/2), counted in the equilateral triangles of the Tetryonics model, is the odd numbers: the ground state is one unsplittable triangle, each rung is one diamond, one photon, the totals are the squares, hydrogen's n squared, and the three-ring oscillator's degeneracies are the triangular numbers, one triangle read three ways. Their four identities, at the oscillator, are the addition of positions plus the area term. Every formula used is standard and cited, and every number is verified by the script in the appendix.
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
A qubit is one setting of the two sliding rings. The rotation gate Rz(θ) has entries cos(θ/2) and sin(θ/2), and the instrument's law is d = 2cos(θ/2) between the centers and e = 2sin(θ/2) between the crossings at crossing separation θ: the gate matrix is read off the setting, the Bloch amplitudes are the two half-distances, and Born's rule is the two shares of one unit. Composing rotation gates is adding positions, certified by the ring's addition law. The spinor's sign, −1 after a 360° rotation and +1 after 720°, is the ring's half turn, digits summing to ten, and its lap. The Pauli phases i, −1, −i are the quarter-turn orbit 7, 9, 3, and the phase exponent of any Pauli product is carried by the digits, verified on two hundred random products. The hidden symmetry of the Jordan–Wigner transformation that Davis and Freericks (arXiv:2512.24589) use to halve measurement circuits is the ring turning, its two angles are the quarter turn and the eighth turn, and its identities are the fold law averaged round the ring, whose coefficients 3/8 and 1/8 are exact averages over the sixty positions. Their equation 24 is verified on random fixed-number states of four qubits. From this one picture the paper places the live checksum where it can live in a quantum computer: on the classical phase registers that drive virtual-Z gates and merge rotations, on the classical Pauli frame, and on measured expectation values through the symmetry identities, which flag every number-changing error, while the law of readings of the companion paper catches the number-conserving ones. And it states where it cannot live, on the amplitudes themselves, by the no-cloning theorem. The four operator-mechanics papers of Freericks and colleagues, the operator Rodrigues formula, the momentum measurement, the free expansion of a Gaussian and the squeezed-state wavefunction, are then worked side by side as examples of the same picture: one slide, one quarter turn, one shear, one rotating ellipse. Every number is verified by the script in the appendix.
The following text develops a new method of solving the eigenvalue problem of the quantum mechanical harmonic oscillator. Thereby are introduced new notions, like that of a reduced basis or that of a huge algebra. Furthermore are presented formulas for iterated commutation relations. Commutation Relation and Hamilton Function Oscillatory properties of a diatomic molecule, a carbon-monoxide molecule for example, can be modelled by a linear elastic oscillator. The observables for describing the oscillatory states of the oscillator are taken to be the dis- �� location from the equilibrium state and the relative momentum of the two masses.
Every tool of quantum computing is one of three moves on two sliding rings marked with the sixty last digits of the Fibonacci numbers. A setting is a value read off one placement of the rings. A fold is a product turned into a sum by the half-angle law. A turn is a shift of position that carries a digit with it. Every gate is a setting, its entries cos(θ/2) and sin(θ/2) the half-distances of the rings at crossing separation θ, and the Hadamard, CZ and CNOT are quarter and half turns. Born's rule is the fold, amplitudes read at position k and expectation values at 2k. Every gradient is a quarter turn, because the parameter-shift rule is the ring's derivative law, its two shifts the ×7 and ×3 laws, with four eighth turns where a Givens gate needs them. A complete variational run is worked on a two-qubit number-conserving Hamiltonian. The energy is the ring shifted ten positions, the ground state the vesica setting at position 20, the optimizer three readings and a gnomon with nothing to tune, its convergence one line. Noise is the radius of the rings, and the square inside the ring, the stabilizer octahedron, says when the noise has cost a magic state. Phase estimation, Shor's classical end, Grover's search, Trotter chains, QAOA, Bell tests, teleportation, error mitigation, benchmarking and the [[4,2,2]] and [[5,1,3]] codes each reduce to the same three moves, forty tools on one page, and because every unitary is a finite product of rotation gates, the list is open and the three moves are closed. A checksum rides every classical number of the loop, the digits under the crossings on each setting, the addition law on each sum, the quarter-turn law on each gradient, the symmetry identity ⟨XX⟩ = ⟨YY⟩ on the measured data with its alarm threshold read off the setting, every one-, two- and three-bit upset of a 32-bit phase word and every burst of up to fifteen bits caught by three rings in lockstep, every corruption of any kind caught by the ladder, and the decoder table of the perfect code built on the four quarter turns of the net, any four simultaneous faults in an entry restored. The guard stands on every classical register of the loop, and the code space stands on the qubits. Thirty-one jobs on three IBM machines close the paper. A clean run on a 156-qubit machine had every compiled angle certified, the injected bit flip flagged at over a hundred standard errors, the clean checksum inside its line, the radius reading the noise, and the Bell value above 2, and one bit flipped in a phase word before submission, four ways, caught by the guard every time, turning the ground state into the top of its number sector and, from the entangler word, into the top of the whole spectrum when sent unguarded. The same words were corrupted up to all ten at once, by any change up to every bit inverted, and read both ways in four runs, unguarded and through the guard, and every setting word of a many-qubit tower was corrupted past any real error rate, caught and rebuilt in place, and read right on the machine. Every number is verified by the script in the appendix on explicit matrices and states, and the hardware readings are checked against the result files.
We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrödinger factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives.