Quantum Physics I
Wave mechanics from the Schrödinger equation to the hydrogen atom.
What this course covers
- Quantum Mechanics as a Framework
- Linearity in Physical Theories
- Linear Operators and the Equation LU = 0
- Worked Example: First-Order ODE as a Linear Equation
- Linearity of classical vs. quantum mechanics
- The wave function as the dynamical variable
- The Schrödinger equation as a linear equation
- Necessity of complex numbers in quantum mechanics
- Complex numbers: definition and representation
- Euler's formula: e^{iθ} = cos θ + i sin θ
- Complex numbers as necessary in quantum mechanics
- Born's probability interpretation of the wave function
- Photons as quantum mechanical particles
- Polarizer behaviour and Malus's law
- Loss of determinism and failure of hidden variables
- Quantum states as superpositions in Dirac notation
- Quantum vs. Classical Superposition
- Mach-Zehnder Interferometer Setup
- Photon Self-Interference in Both Beams
- Measurement Postulate and State Collapse
- Superposition and quantum measurement outcomes
- Physical assumption: overall coefficient is irrelevant
- Photon polarization parameter count as validation
- Spin-½ superposition and distinguishing quantum states from classical ensembles
- Tensor Product States for Two Particles
- Entangled States and Un-factorizability
- Measurement Correlations in Entangled Spin Pairs
- Bell Inequalities and Limits of Classical Explanation
- Photon state as a two-component complex vector
- Beam splitter action and its matrix representation
- Balanced beam splitter constraint
- Deriving a valid balanced beam splitter matrix
- Mach-Zehnder interferometer matrix rule
- Unblocked interferometer: all photons reach D0
- Blocked lower path: step-by-step amplitude calculation
- Counter-intuitive result: blocking enables D1 detection
- The Elitzur-Vaidman Bomb Problem
- Defective Bomb in the Mach-Zehnder Interferometer
- Working Bomb in the Mach-Zehnder Interferometer
- Interaction-Free Measurement and Efficiency
- Photoelectric Effect: Experimental Setup and Discovery
- Photoelectric Effect: Four Key Observed Properties
- Einstein's Photon Hypothesis and Work Function
- Millikan's Confirmation and Reluctance to Accept Photons
- Units of Planck's Constant
- Constructing Lengths from Planck's Constant
- Compton Wavelength: Definition and Physical Meaning
- Numerical Value of the Electron Compton Wavelength
- Photon momentum from relativistic energy-momentum relation
- Classical Thompson scattering and its failure
- Compton scattering: photon as a colliding particle
- Compton's experimental results and the two-peak spectrum
- Wave-particle duality as a universal property
- Matter waves as probability-amplitude waves
- The de Broglie wavelength formula
- Experimental confirmation of matter waves
- De Broglie Matter Waves and the Wave Function
- Wave Number and Angular Frequency as Notation
- Galilean Transformation of Coordinates and Momentum
- De Broglie Wavelength Changes Between Frames — Matter Waves Are Strange
- Wave phase as a Galilean invariant
- Phase transformation under Galilean coordinates
- De Broglie waves are not Galilean invariant
- Wave function Ψ is not directly measurable
- De Broglie frequency postulate: E = ℏω
- Phase velocity of matter waves
- Group velocity and recovery of particle speed
- Relativistic and photon motivations for E = ℏω
- Dispersion relations and wave packets
- Principle of stationary phase
- Applying stationary phase to the wave packet integral
- Group velocity as the velocity of a wave packet
- Wave packet integral and setup
- Taylor expansion of ω(k) around k₀
- Factoring constant-phase terms outside the integral
- Group velocity as wave-packet propagation speed
- De Broglie relations and candidate matter waves
- Superposition and the probability-vanishing argument
- Complex exponential waves survive superposition test
- Conventional selection of the matter wave e^{i(Kx−ωt)}
- Matter wave for a free particle
- Momentum operator and eigenstates
- Energy operator from the dispersion relation
- The Schrödinger equation for a free particle
- Free-particle Schrödinger equation and plane-wave solutions
- Linearity and general solution via superposition
- Group velocity and physical particle velocity
- Structural properties distinguishing the Schrödinger equation
- The Full Schrödinger Equation with Potential
- Operators in Quantum Mechanics
- The Canonical Commutation Relation [X, P] = iℏ
- Commutator Notation and Its Physical Significance
- Operators, matrices, and non-commutativity
- Momentum operator in three dimensions
- Three-dimensional Schrödinger equation
- Canonical commutation relations in 3D
- Schrödinger's original interpretation of the wave function
- Born's probabilistic interpretation of Ψ
- Probability density and the Born rule in 3D
- Consistency of normalization under time evolution
- Born Rule and Probability Normalization
- Boundary Conditions on the Wave Function
- Normalizable vs. Normalized Wave Functions
- Time Evolution and Preservation of Normalization
- Probability density and normalization condition
- Deriving dN/dt using the Schrödinger equation
- Hermitian operators and conservation of probability
- Hermitian conjugate and the definition T† = T
- Deriving the time derivative of probability density
- Expressing dρ/dt as a total spatial derivative
- Probability current density J and its formula
- Units of the probability current
- Probability current density in 3D
- Conservation of total probability (N) and Hermiticity of H
- Physical interpretation of local probability conservation
- 1D probability flux between two points
- Wave packets as superpositions of plane waves
- Momentum-space profile φ(k) and its uncertainty
- Stationary phase and the peak of ψ(x)
- Intuitive relationship between δx and δk
- Reality condition for a wave packet
- Change-of-variable technique in Fourier integrals
- Fourier uniqueness: zero transform implies zero function
- Why the example Φ(k) yields a non-real ψ
- Stationary phase and wave packet localization
- Total phase excursion and the ΔK·ΔX relation
- Quantum mechanical uncertainty principle from ΔK·ΔX ≈ 1
- Step-function Φ(K) worked example: sinc wave packet
- Wave packet shape change over time
- Taylor expansion of ω(k) and termination for non-relativistic particles
- Condition for negligible wave-packet spreading
- Physical interpretation and practical applications of packet spreading
- Step 1–2: Computing the Fourier decomposition of the initial wave function
- Step 3: Time-evolving the wave packet via plane-wave solutions
- Step 4: Performing the final k-integral
- Application: Time evolution of a free Gaussian wave packet
- Momentum space vs. position space
- Fourier transform pair for wave functions
- Integral representation of the delta function
- Basic properties of the delta function
- Normalization condition in momentum space setup
- Delta function representation and Parseval's theorem
- Fourier relations rewritten in momentum language
- Parseval's theorem restated in momentum variables
- Probabilistic interpretation of φ(P)
- 3D Fourier transform pair for ψ and φ
- 3D Dirac delta function representation
- 3D Parseval's theorem
- Expectation Values in Classical Probability
- Expectation Value of Position in Quantum Mechanics
- Deriving the Momentum Expectation Value in Position Space
- General Definition of Operator Expectation Values
- Expectation Value of the Kinetic Energy Operator
- Setting up the time derivative of an expectation value
- Applying the Schrödinger equation to the time derivatives
- Using Hermiticity to introduce the commutator
- Ehrenfest-type theorem and conservation laws
- Hermitian operators and inner product notation
- Expectation values of Hermitian operators are real
- Eigenvalues of Hermitian operators are real
- Normalisation requirement for expectation values and eigenstates
- Spectral Theorem for Hermitian Operators
- Orthonormality of Eigenfunctions
- Proof of Orthogonality from Hermiticity
- Degeneracy and Organizing Degenerate Eigenfunctions
- Eigenfunction Expansion of Arbitrary States
- Calculating Expansion Coefficients via Orthonormality
- The Quantum Measurement Postulate
- Measurement as a Separate Postulate; Wave-Function Collapse
- Expectation value from superposition states
- Particle on a circle: setup and boundary conditions
- Momentum eigenstates on a circle
- Decomposition into eigenstates and measurement probabilities
- Standard deviation as a measure of uncertainty
- Mean of the square minus the square of the mean
- Quantum mechanical uncertainty of an operator
- Two equivalent rewritings of quantum uncertainty and the eigenstate condition
- Definition and equivalence of uncertainty
- Claim 2: norm-squared integral form of uncertainty
- Zero uncertainty ↔ eigenstate equivalence
- Computational application and the uncertainty principle
- Definition of a Stationary State
- Separation of Variables and the Separation Constant E
- Time-Dependent Part: G(t) Solution
- Time-Independent Schrödinger Equation and Normalization
- Expectation value of Hamiltonian in a stationary state
- Time independence of expectation values in stationary states
- Superposition of stationary states is not stationary
- Time dependence from superpositions enables quantum dynamics
- Energy eigenstates and the spectrum
- Time-independent Schrödinger equation to solve
- Admissible potentials V(x)
- Continuity conditions on ψ and ψ′
- Particle on a circle: boundary conditions
- Proof that energy eigenvalues are strictly positive
- Wave-number k and energy–momentum relations
- General solutions and outlook for quantisation of k
- Particle on a Circle: Periodicity and Quantized Momentum
- Normalization of the Energy Eigenstates
- Degeneracy and Physical Distinguishability of States
- Completeness, Fourier Theorem, and Sine/Cosine Basis
- Particle on a Circle: Setup and Energy Quantization
- Normalization of Circle Wave Functions
- Degeneracy and Momentum as Distinguishing Observable
- Orthonormality, Completeness, and Sine/Cosine Reformulation
- Infinite Square Well Setup and Boundary Conditions
- Free Schrödinger Equation Inside the Well
- Quantization of Wave Number and Allowed n Values
- Normalized Eigenstates and Energy Spectrum
- Infinite square well wave functions and quantization
- Nodes of bound-state wave functions
- Symmetry of bound states under symmetric potentials
- Non-degeneracy of one-dimensional bound states
- Finite Square Well Setup and Bound States
- Piecewise Solutions: Trigonometric Inside, Exponential Outside
- Dimensionless Variables η, ξ, and Z₀
- Z₀ as Controller of the Number of Bound States
- Boundary condition matching at well edge
- Transcendental equation in dimensionless variables
- Dimensionless parameter Z₀ and energy readout
- Graphical solution and bound-state existence
- No degeneracy of 1D bound states
- Reality of the time-independent Schrödinger equation
- Bound-state wave functions are real up to a global phase
- Practical implication: plotting real bound states
- Property: even potentials yield even/odd eigenstates
- Proof that ψ(−x) solves the same Schrödinger equation
- Constructing symmetric and antisymmetric solution combinations
- Bound-state solutions of even potentials are automatically even or odd
- Energy, Kinetic Energy, and Momentum in a Potential
- De Broglie Wavelength as a Function of Position
- Classically Forbidden Regions and Turning Points
- Slowly Varying Potential and the WKB Approximation
- Correspondence Principle for Wave Functions
- Amplitude Proportional to de Broglie Wavelength
- Sketching Wave Functions Using Local Wavelength and Amplitude
- Node Theorem for Bound States
- Local shape equation for the wave function
- Wave function shape in classically forbidden regions
- Wave function shape in classically allowed regions
- Turning points and inflection points of ψ
- Boundary conditions for bound-state wave functions
- Energy quantization via wave-function matching
- Shooting method setup and boundary conditions
- Numerical shooting: iterating energy to find eigenstates
- Delta function potential defined
- Dimensional analysis to constrain bound-state energy
- Bound-state wave function structure and the Schrödinger equation away from x = 0
- Parity and node arguments ruling out odd bound states
- Symmetric wave function for delta potential
- Integrating the Schrödinger equation across a delta function
- Derivative discontinuity condition at a delta function
- Bound-state energy of the delta-function potential
- Node Theorem Statement for Bound States
- Impossibility of Simultaneous Zero of Ψ and Ψ′
- The Screened Potential Construction
- Continuity Argument: Nodes Cannot Be Created
- Quantum harmonic oscillator Hamiltonian and its universality
- Two solution strategies for the harmonic oscillator
- Dimensional reduction to a dimensionless variable
- Dimensionless energy and the final reduced ODE
- Normalizable solutions require discrete energies
- Large-U asymptotic analysis of the equation
- Ansatz φ(u) = H(u) e^(−u²/2) and its motivation
- Differential equation for H and quantization
- Rescaling the harmonic oscillator equation
- Asymptotic ansatz for the wave function
- Power-series solution and recursion relation
- Even and odd solution families from initial conditions
- Series truncation and energy quantization
- Quantum harmonic oscillator energy levels
- Hermite polynomials and their differential equation
- Generating function and full wavefunction solutions
- Factorizing the SHO Hamiltonian
- Operator non-commutativity in factorization
- Ladder (creation and destruction) operators A and A†
- X and P expressed in terms of A and A†
- Factorized Hamiltonian in terms of ladder operators
- Energy lower bound from factorized Hamiltonian
- Ground state from first-order differential equation
- Number operator and creation of excited states
- Harmonic oscillator algebraic solution recap
- Commutators of N with A and A†
- Commutators involving powers of ladder operators
- Number operator eigenvalue interpretation
- Creating the first excited state with A†
- Normalization of φ₁ via operator algebra
- General nth excited state and its energy eigenvalue
- Proof that φₙ is correctly normalised
- Lowering and raising operator action on energy eigenstates
- Expectation values ⟨x⟩ and ⟨p⟩ vanish for energy eigenstates
- Orthonormality of harmonic oscillator eigenstates
- Expectation value of x² and position uncertainty in φ_n
- Scattering states and non-normalizability
- The step potential and energy regimes
- Wave function ansatz for the step potential
- Reflection and transmission coefficients from boundary conditions
- Scattering stationary states and their interpretation
- Special case E = V₀ (K̄ = 0)
- Probability current in scattering regions
- Current conservation as a consistency check
- Probability current for incident and reflected waves
- Reflection coefficient R from probability current
- Transmission coefficient T: current ratio, not amplitude ratio
- Conservation law: R + T = 1
- Analytic Continuation into the Barrier Region
- Reflection and Current Conservation (|B| = |A|)
- Wave Function as a Phase-Shifted Sine
- Phase Shift δ(E) and Its Energy Dependence
- Wave packet construction from energy eigenstates
- Splitting the wave packet into incident, reflected, and transmitted parts
- Stationary phase analysis of the incident wave packet
- Stationary phase analysis of reflected and transmitted packets
- Incident and reflected wave construction
- Stationary phase applied to the reflected packet
- Time delay from phase shift
- Using time delay to probe the potential
- The negative kinetic energy paradox
- Spatial extent of the forbidden-region wavefunction
- Uncertainty principle resolution of the paradox
- Physical outcome of detecting a particle in the forbidden region
- Recap: Scattering off a Step Barrier
- Finite Square Well Scattering Setup
- Reflection and Transmission Coefficients via Probability Current
- Boundary Conditions and Coefficient Determination
- Transmission coefficient formula for finite square well
- Dimensionless form of the transmission formula
- Resonant transmission condition
- Physical interpretation: infinite square well analogy and half-wave resonance
- Transmission resonances of a finite square well
- Ramsauer–Townsend experiment
- Noble gas atom modeled as a spherical potential well
- Resonant transmission as explanation of the Ramsauer–Townsend effect
- 1D scattering setup and physical motivation
- Free-particle solution with infinite wall (no potential)
- Phase shift from probability current conservation
- Full scattered wave function and phase-shift result
- Phase shift and spatial displacement of wave features
- Physical interpretation of phase shift sign
- Definition of the scattered wave
- Explicit form of the scattered wave
- Finite-range potential and scattering solutions
- Scattered wave and scattering amplitude
- Wave-packet construction for incident and reflected waves
- Time delay and normalised delay formula
- Finite potential well setup and ansatz
- Boundary conditions at x = a
- Trigonometric identity to isolate δ
- Closed-form result for the phase shift δ
- Dimensionless variables for square-well scattering
- Explicit tan δ formula and its low-energy limit
- Numerical plots of phase shift and related observables
- Levinson's theorem: phase shift and bound-state count
- Conservation of quantum states under potential change
- Statement of Levinson's theorem
- Large-box (regulator) discretization of scattering states
- Density of positive-energy states with and without potential
- Counting positive energy states lost per interval DK
- Total states lost as boundary evaluation of phase shift
- Infinite-state subtraction problem and finite loss
- Number of bound states equals total positive-energy states lost
- Time Delay and Wigner's Causality Condition
- Physical Origin of Resonances
- Wave-Number Definitions for the Three Regions
- Wave-Function Ansatz and Matching Strategy
- Phase shift formula for finite square well with barrier
- Dimensionless variable reduction of phase shift formula
- Numerical plotting of δ(u) for specific well parameters
- Resonance identified via time delay and internal amplitude
- Resonance signatures: phase shift and amplitude
- Tangent-delta model of a resonance
- Scattering amplitude near resonance: momentum form
- Breit-Wigner distribution in energy and resonance width γ
- Gamma as energy; defining timescale Tau
- Time delay at resonance and its relation to Tau
- Physical interpretation: narrow vs. broad resonances
- Unstable particles as resonances: the Higgs boson example
- Scattering amplitude and resonance condition
- Poles of AS in the complex K-plane
- Bound states as imaginary-K poles
- The complex K-plane: unified picture
- Momentum as a generator of translations
- Angular momentum as a generator of rotations
- Central potentials and the 3-D Schrödinger equation
- L² as the angular part of the Laplacian; two-body reduction
- Angular momentum operators from classical analogy
- Commutation relations of angular momentum
- Physical meaning: rotations do not commute
- Universality of the angular momentum algebra and spin
- No simultaneous eigenstates of all three angular momentum components
- L² commutes with each component of angular momentum
- LZ as a differential operator in spherical coordinates
- L² as the angular part of the Laplacian in spherical coordinates
- Simultaneous eigenstates of L_z and L²
- Quantization of L_z eigenvalues (m must be an integer)
- The associated Legendre differential equation
- Quantization of l via Legendre polynomial termination
- Angular momentum operators and commutation relations
- Quantization of L and M quantum numbers
- Associated Legendre functions and their construction
- Spherical harmonics Y_LM
- Solid angle as integration measure for spherical harmonics
- Orthonormality of spherical harmonics
- Separation of the 3-D Schrödinger equation into radial and angular parts
- Radial equation and effective 1-D potential with centrifugal term
- Effective radial potential and centrifugal barrier
- Structure of the 3D radial wavefunction and L/M dependence
- Normalization condition for the radial function U
- Boundary condition for U at R = 0
- Two-particle quantum system: variables and wave function
- Center-of-mass canonical variables
- Relative coordinate canonical variables and reduced mass
- Separation of kinetic energy into center-of-mass and relative parts
- Hydrogen atom Hamiltonian and canonical variable pairs
- Momentum operators as coordinate-specific gradients
- Separation of variables for the two-body wave function
- Reduction to a central-potential one-body problem
- Hydrogen-like atom potential energy
- Bohr radius from dimensional analysis
- Characteristic energy scale and fine structure constant
- Hierarchy of atomic length scales
- Radial Schrödinger equation for hydrogen
- Dimensionless variable substitution and energy parameter κ
- Asymptotic analysis: behavior at ρ → ∞ and ρ → 0
- Ansatz U = ρ^(L+1) W(ρ) e^(−ρ) and the resulting one-step recursion
- Recursion relation for series coefficients
- Series divergence and the truncation requirement
- Energy quantization from series termination
- Hydrogen energy levels: the 1/n² formula
- N–L lattice and energy level organization
- Counting states per principal quantum number
- Quantum numbers of hydrogen: n, L, and m
- The ρ variable expressed in terms of n
- Full wave function of the hydrogen atom
- Hydrogen atom quantum numbers and energy levels
- Energy level diagram for the hydrogen atom
- Node theorem applied to radial wavefunctions
- Effective potential and L-dependence of radial problems
- Accidental degeneracy in the hydrogen atom
- Runge-Lenz vector and hidden symmetry
- Scaling hydrogen wave functions from Z=1 to arbitrary Z
- Asymptotic derivation of the radial exponential
- Accidental degeneracy in the hydrogen atom
- Scaling hydrogen wave functions to arbitrary nuclear charge Z
- Asymptotic analysis of the radial equation
- Physical interpretation of wave function features
- Rydberg atoms and effective nuclear charge
- Virial theorem and expectation values of 1/R and R
- Radial probability distribution and N²a₀ radius
- Physical properties and observations of Rydberg atoms
- Circular vs. elliptical orbits and angular momentum L
- Effective potential and its shape
- Turning points R+ and R− and orbit geometry
- Degeneracy of energy levels and Kepler's third law analogy
- Energy level diagram for central potentials
- Effective potential and classical turning points in hydrogen
- Derivation of turning-point formula for hydrogen orbits
- Degeneracy structure of hydrogen energy levels
- Motivation for the Simplest Quantum System
- Requirements on the Hamiltonian and Inner Product
- Two-Component Wave Function and State Space
- Inner Product and Hermitian Condition for Two-State Systems
- Most General 2×2 Hermitian Matrix
- Pauli Matrices as Basis for Hamiltonians
- Spin Angular Momentum from Pauli Matrices
- Spin-1/2 System and Its Physical Hamiltonian
- Spin-½ eigenstates of S_z
- Two-dimensional spin state space and superposition
- Eigenstates of S_x: spin along the X direction
- Eigenstates of S_y and the role of complex numbers
Syllabus
Lectures
- Quantum mechanics as a framework. Defining linearity
- Linearity and nonlinear theories. Schrödinger's equation
- Necessity of complex numbers
- Photons and the loss of determinism
- The nature of superposition. Mach-Zehnder interferometer
- More on superposition. General state of a photon and spin states
- Entanglement
- Mach-Zehnder interferometers and beam splitters
- Interferometer and interference
- Elitzur-Vaidman bombs
- The photoelectric effect
- Units of h and Compton wavelength of particles
- Compton Scattering
- de Broglie’s proposal
- de Broglie wavelength in different frames
- Galilean transformation of ordinary waves
- The frequency of a matter wave
- Group velocity and stationary phase approximation
- Motion of a wave-packet
- The wave for a free particle
- Momentum operator, energy operator, and a differential equation
- Free Schrödinger equation
- The general Schrödinger equation. x, p commutator
- Commutators, matrices, and 3-dimensional Schrödinger equation
- Interpretation of the wavefunction
- Normalizable wavefunctions and the question of time evolution
- Is probability conserved? Hermiticity of the Hamiltonian
- Probability current and current conservation
- Three dimensional current and conservation
- Wavepackets and Fourier representation
- Reality condition in Fourier transforms
- Widths and uncertainties
- Shape changes in a wave
- Time evolution of a free particle wavepacket
- Fourier transforms and delta functions
- Parseval identity
- Three-dimensional Fourier transforms
- Expectation values of operators
- Time dependence of expectation values
- Expectation value of Hermitian operators
- Eigenfunctions of a Hermitian operator
- Completeness of eigenvectors and measurement postulate
- Consistency condition. Particle on a circle
- Defining uncertainty
- Uncertainty and eigenstates
- Stationary states: key equations
- Expectation values on stationary states
- Comments on the spectrum and continuity conditions
- Solving particle on a circle
- Energy eigenstates for particle on a circle
- Infinite square well energy eigenstates
- Nodes and symmetries of the infinite square well eigenstates
- Finite square well. Setting up the problem
- Finite square well energy eigenstates
- Nondegeneracy of bound states in 1D. Real solutions
- Potentials that satisfy V(-x) = V(x)
- Qualitative insights: Local de Broglie wavelength
- Correspondence principle: amplitude as a function of position
- Local picture of the wavefunction
- Energy eigenstates on a generic symmetric potential. Shooting method
- Delta function potential I: Preliminaries
- Delta function potential I: Solving for the bound state
- Node Theorem
- Harmonic oscillator: Differential equation
- Behavior of the differential equation
- Recursion relation for the solution
- Quantization of the energy
- Algebraic solution of the harmonic oscillator
- Ground state wavefunction
- Number operator and commutators
- Excited states of the harmonic oscillator
- Creation and annihilation operators acting on energy eigenstates
- Scattering states and the step potential
- Step potential probability current
- Reflection and transmission coefficients
- Energy below the barrier and phase shift
- Wavepackets
- Wavepackets with energy below the barrier
- Particle on the forbidden region
- Waves on the finite square well
- Resonant transmission
- Ramsauer-Townsend phenomenology
- Scattering in 1D. Incoming and outgoing waves
- Scattered wave and phase shift
- Incident packet and delay for reflection
- Phase shift for a potential well
- Excursion of the phase shift
- Levinson's theorem, part 1
- Levinson's theorem, part 2
- Time delay and resonances
- Effects of resonance on phase shifts, wave amplitude and time delay
- Modelling a resonance
- Half-width and time delay
- Resonances in the complex k plane
- Translation operator. Central potentials
- Angular momentum operators and their algebra
- Commuting observables for angular momentum
- Simultaneous eigenstates and quantization of angular momentum
- Associated Legendre functions and spherical harmonics
- Orthonormality of spherical harmonics
- Effective potential and boundary conditions at r=0
- Hydrogen atom two-body problem
- Center of mass and relative motion wavefunctions
- Scales of the hydrogen atom
- Schrödinger equation for hydrogen
- Series solution and quantization of the energy
- Energy eigenstates of hydrogen
- Energy levels and diagram for hydrogen
- Degeneracy in the spectrum and features of the solution
- Rydberg atoms
- Orbits in the hydrogen atom
- More on the hydrogen atom degeneracies and orbits
- The simplest quantum system
- Hamiltonian and emerging spin angular momentum
- Eigenstates of the Hamiltonian