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- Adding a constant to the potential introduces a phase factor
- Adiabatic approximation in quantum mechanics
- Adiabatic approximation: higher order corrections
- Airy functions and the bouncing electron
- Alpha decay using the WKB approximation
- Angular equation – alternative solution
- Angular momentum: adding 2 spins
- Angular momentum: adding 3 spins
- Angular momentum: adding spins in arbitrary directions
- Angular momentum: addition and Clebsch-Gordan coefficients
- Angular momentum of circular motion
- Angular momentum – commutators
- Angular momentum commutators in hydrogen
- Angular momentum – commutators with position and momentum
- Angular momentum: commutators of added spins
- Angular momentum – eigenfunctions
- Angular momentum – eigenvalues
- Angular momentum as an eigenvector problem
- Angular momentum as a generator of rotations
- Angular momentum and parity
- Angular momentum – Poisson bracket to commutator
- Angular momentum: probabilities of eigenvalues in two dimensions
- Angular momentum – raising and lowering operators
- Angular momentum – raising and lowering operators from rectangular coordinates
- Angular momentum: restriction to integer values
- Angular momentum in three dimensions
- Angular momentum in 3-d: expectation values and uncertainty principle
- Angular momentum and torque
- Anti-hermitian operators
- Atomic wavefunctions: symmetrization
- Average and standard deviation
- Band structure of solids: degeneracy of states
- Band structure of solids: numerical solution
- Band structure of solids: negative energies
- Berry’s phase: definition and value for a spin-1 particle in a magnetic field
- Blackbody radiation
- Born approximation in one dimension
- Born approximation of delta function well and finite square well
- Born approximation for a spherical delta function shell
- Bosons in the infinite square well
- Buffon’s needle: estimating pi
- Bulk modulus in the electron gas model
- Canonical transformations in 2-d: rotations and polar coordinates
- Canonical transformations: a few more examples
- Changing the position basis with a unitary transformation
- Classical limit of quantum mechanics; Ehrenfest’s theorem
- Clebsch-Gordan coefficients for higher spin
- Combining translations and rotations
- Commutators: a few theorems
- Complex exponentials and trig functions
- Compound systems of fermions and bosons
- Conditions for a transformation to be canonical
- Continuity of the wave function – Born’s conditions revisited
- Continuous probability distribution: needle on a pivot
- Correspondence between classical and quantum transformations
- Coupled masses on springs – a solution using matrix diagonalization
- Coupled masses on springs – properties of the propagator
- Cyclic coordinates and Poisson brackets
- Decoupling the two-particle Hamiltonian
- Degeneracy pressure in a solid
- Degenerate eigenvalues and Gram-Schmidt orthogonalization
- Degenerate perturbation theory: two states
- Degenerate perturbation in 3 state system
- Degenerate solutions don’t exist in one dimension
- Delta-function well – bound state
- Delta function well: bound state – uncertainty principle
- Delta-function well – scattering
- Delta function well: statistics
- Delta function potential – moving delta function
- Delta function well as limit of finite square well
- Delta function – Fourier transform
- Delta function in time perturbation
- Determinant and trace of normal operators
- Determinate states
- Diagonalization of matrices
- Differential operators – matrix elements and hermiticity
- Differential operator – eigenvalues and eigenstates
- Dirac delta function
- Dirac delta function in three dimensions
- Dirac delta function – simple examples
- Direct product of two vector spaces
- Direct product of vector spaces: 2-dim examples
- Double delta function well
- Double delta function well – scattering states
- Dysprosium electron configuration
- Earth-Sun system as a quantum atom
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors – examples
- Eigenvalues and eigenvectors of the 2-d rotation operator
- Eigenvalues of two-dimensional angular momentum
- Eigenvalues of angular momentum
- Electrodynamics in quantum mechanics: gauge transformations
- Electromagnetic force law in quantum mechanics
- Electromagnetic Lagrangian
- Electromagnetism in quantum mechanics: example
- Electron as a classical spinning sphere
- Electron gas: a crude model of a solid
- Electron gas in a 2-d infinite square well
- Electron in a precessing magnetic field
- Electron pressure in a neutron star
- Electron pressure in a white dwarf star
- Energy levels of hydrogen: Bohr’s semi-classical derivation
- Energy states: bound and scattering states
- Energy & wave functions: a few theorems
- Energy-time uncertainty relation
- Energy-time uncertainty: an alternative definition
- Energy-time uncertainty principle – example
- Energy-time uncertainty principle: Gaussian free particle
- Energy-time uncertainty principle: infinite square well
- Every attractive 1-dimensional potential has a bound state
- Exchange force: harmonic oscillator
- Exchange force: infinite square well
- Exponentials of operators – Baker-Campbell-Hausdorff formula
- Exponentials of operators – Hadamard’s lemma
- Extended uncertainty principle
- Fermions and bosons: counting states
- Fermions and bosons: n-particle systems
- Fermions and bosons in the infinite square well
- A few statistics on the first 25 digits of pi
- Feynman-Hellmann theorem and the harmonic oscillator
- Feynman-Hellmann theorem: hydrogen atom mean values
- Fine structure constant
- Fine structure of hydrogen: Dirac formula
- Fine structure of hydrogen: relativistic correction
- Fine structure of hydrogen: spin-orbit coupling
- Fine structure of hydrogen: spin-orbit eigenstates and final formula
- Fine structure of a spectral line in hydrogen
- Finite drop potential
- Finite rotations about an arbitrary axis in three dimensions
- Finite spherical well
- Finite square barrier – scattering
- Finite square well: bound states & even wave functions
- Finite square well: bound states & odd wave functions
- Finite square well – normalization
- Finite square well – numerical solution
- Finite square well – scattering
- Finite step potential – scattering
- Finite transformations: correspondence between classical and quantum
- First Born approximation: soft-sphere scattering
- First order non-degenerate perturbation theory
- Forbidden transitions in the harmonic oscillator and hydrogen
- Forced harmonic oscillator: exact solution and adiabatic approximation
- The free particle
- Free particle in momentum space
- Free particle in the position basis
- Free particle in spherical coordinates – finding the solutions
- The free particle as a wave packet
- Free particle: Gaussian wave packet
- Free particle – travelling wave packet
- The free particle: probability current
- Free particle moving in the z direction
- Free particle revisited: solution in terms of a propagator
- Free particle propagator from a complete path integral
- Functions of hermitian operators
- Fusion with a muon-deuteron system
- Gaussian distribution
- Geometric phase is always zero for real wave functions
- Gram-Schmidt orthogonalization – a couple of examples
- Green’s function for one dimensional Schrödinger equation
- Half-harmonic oscillator
- Half life of a beer can
- Hamiltonian matrix elements
- Hamiltonian in non-rectangular coordinates
- Hamiltonian in two-level system
- Hamiltonian for three-state system
- Hamiltonian and observables in three-state system
- Hamiltonian formalism and Legendre transformations
- Hamiltonian for the electromagnetic force
- Hamiltonians for harmonic oscillators
- Hamiltonian for the two-body problem
- Hamilton’s equations of motion under a regular canonical transformation
- Harmonic oscillator: algebraic solution; raising and lowering operators
- Harmonic oscillator: algebraic normalization of raising and lowering operators
- Harmonic oscillator – asymptotic solution
- Harmonic oscillator – change in spring constant
- Harmonic oscillator: coherent states
- Harmonic oscillator in an electric field
- Harmonic oscillator – probability of being outside classical region
- Harmonic oscillator ground state – numerical solution
- Harmonic oscillator – eigenfunctions in momentum space
- Harmonic oscillator energies and eigenfunctions derived from the propagator
- Harmonic oscillator excited states – numerical solution
- Harmonic oscillator: first order perturbation
- Harmonic oscillator – Hermite polynomials
- Harmonic oscillator: Hermite polynomials and orthogonality of eigenfunctions
- Harmonic oscillator in a magnetic field
- Harmonic oscillator: matrix elements
- Harmonic oscillator: matrix elements using Hermite polynomials
- Harmonic oscillator – mixed initial state
- Harmonic oscillator – mixed initial state and Ehrenfest’s theorem
- Harmonic oscillator: mixture of two lowest states
- Harmonic oscillator – mean position and momentum
- Harmonic oscillator: momentum space functions and Hermite polynomial recursion relations from raising and lowering operators
- Harmonic oscillator – position, momentum and energy
- Harmonic oscillator – raising and lowering operator calculations
- Harmonic oscillator – raising and lowering operators as functions of time
- Harmonic oscillator: relativistic correction
- Harmonic oscillator: Schrödinger’s exact solution
- Harmonic oscillator – example starting state
- Harmonic oscillator – series solution
- Harmonic oscillator – series solution revisited
- Harmonic oscillator: statistics
- Harmonic oscillator – summary
- Harmonic oscillator – three lowest stationary states
- Harmonic oscillator – zero-point energy from uncertainty principle
- Harmonic oscillator in 2-d and 3-d, and in polar and spherical coordinates
- Harmonic oscillator in 2 dimensions: comparison with rectangular coordinates
- Harmonic oscillator in 3-d – rectangular coordinates
- Harmonic oscillator in 3-d: spherical coordinates
- Helium atom
- Helium atom: electron-electron interaction
- Helium atom: parahelium and orthohelium
- Helium atom using the variational principle
- Hermite polynomials – recursion relations
- Hermite polynomials – generation
- Hermite polynomials – the Rodrigues formula
- Hermitian conjugate of an operator
- Hermitian operators
- Hermitian operators: common eigenfunctions implies they commute
- Hermitian operators – equivalence of conditions
- Hermitian matrices – example with 4 matrices
- Hermitian operators – a few examples
- Hermitian operators – a few theorems
- Hermitian operators – a few more theorems
- Hermitian operators: periodic function
- Hilbert space – power functions
- Hund’s rules
- Hybrid infinite-finite square well
- Hydrogen atom: coincident spectral lines
- Hydrogen atom: combined position and spin state
- Hydrogen atom in a crystal lattice
- Hydrogen atom – Laguerre polynomials example
- Hydrogen atom – mean radius of electron position
- Hydrogen atom – mixed initial state and mean potential energy
- Hydrogen atom – most probable distance of electron
- Hydrogen atom: powers of the momentum operator
- Hydrogen atom: probability of finding electron inside the nucleus
- Hydrogen atom -radial equation
- Hydrogen atom – radial function examples
- Hydrogen atom: radial functions for large l
- Hydrogen atom – radial function at large r
- Hydrogen atom – a sample wave function
- Hydrogen atom – series solution and Bohr energy levels
- Hydrogen atom – spectrum
- Hydrogen atom – wave function example 1
- Hydrogen atom – complete wave function & example 2
- Hydrogen atom: wave function example 3
- Hydrogen-like atoms
- Hydrogen molecule ion
- Hydrogen molecule ion – different trial function
- Hydrogen molecule ion – oscillation of the protons
- Hyperfine splitting and the 21 cm line of hydrogen
- Hyperfine splitting in deuterium
- Identical particles: fermions and bosons
- Identical particles – bosons and fermions revisited
- Impulse approximation in scattering theory
- Infinite spherical well – numerical solutions
- Infinite spherical well – spherical Bessel functions
- The infinite square well (particle in a box)
- Infinite square well – average energy
- Infinite square well – centered coordinates
- Infinite square well – change in well size
- Infinite square well – combination of two lowest states
- Infinite square well – cubic sine initial state
- Infinite square well – expanding well
- Infinite square well – force to decrease well width
- Infinite square well – minimum energy
- Infinite square well: momentum
- Infinite square well: momentum space wave functions
- Infinite square well – numerical solution
- Infinite square well – particle in left half
- Infinite square well – phase difference
- Infinite square well in three dimensions
- Infinite square well – triangular initial state
- Infinite square well with triangular initial state using delta function
- Infinite square well: 2 particle systems
- Infinite square well – uncertainty principle
- Infinite square well with delta function barrier
- Infinite square well with variable delta function barrier: ground state energy
- Infinite square well with variable delta function barrier: location of the particle
- Infinitesimal rotations in canonical and noncanonical transformations
- Inner products and Hilbert spaces
- Inner product of two wave functions is constant in time
- Integral form of the Schrödinger equation
- Integral form of the Schrödinger equation: ground state of hydrogen
- Invariance of Euler-Lagrange and Hamilton’s equations under canonical transformations
- Invariance of symmetric and antisymmetric states; exchange operators
- Inverses of linear operators
- Isotropic harmonic oscillator in 3-d – use of spherical harmonics
- Kramers’s relation for averages of radial powers in hydrogen
- Kramers’s relation: application to hydrogen mean values
- Lagrangians for harmonic oscillators
- Lagrangian for a spherically symmetric potential energy function
- Lagrangian for the two-body problem
- Laguerre polynomials – normalization
- Legendre polynomials: generation by Gram-Schmidt process
- Levi-Civita antisymmetric tensor, vector products and systems of 3 fermions
- Linear combinations of spherical harmonics; probabilities
- Linear functionals and adjoint operators
- Linear operators & commutators
- Linear operators: null space, range, injectivity and surjectivity
- Magnetic resonance
- Matrix elements: example
- Matrix representation of linear operators; matrix multiplication
- Matrix representation of linear operators: change of basis
- Momentum: eigenvalues and normalization
- Momentum space representation of finite wave function
- Momentum space: harmonic oscillator
- Momentum space: mean position
- Momentum space: another example
- Momentum space in 3-d
- The need for quantum theory
- Nondegenerate states in 3-d spherically symmetric systems
- Non-denumerable basis: position and momentum states
- Normal operators
- Optical theorem
- Orthonormal basis and orthogonal complement
- Parity transformations
- Partial waves in three dimensions: hard sphere scattering
- Particle on a circular wire
- Passive, regular and active transformations. Invariance of the Hamiltonian and generators of transformations
- Path integral formulation of quantum mechanics: free particle propagator
- The path integral is equivalent to the Schrödinger equation
- Path integral to Schrödinger equation for a vector potential
- Path integrals for special potentials; use of classical action
- Periodic potentials: Bloch’s theorem and the band structure of solids
- Periodic table
- Perturbation due to finite size of the proton in hydrogen
- Perturbation theory for higher-level degenerate systems
- Perturbation theory and the variational principle
- Perturbing the 3-d harmonic oscillator
- Perturbation of 3-d square well
- Perturbing a particle on a circular wire
- Perturbing the wave function (Stark effect and proton electric dipole moment)
- Phase shift in one-dimensional scattering
- Phase shift in the spherical delta function shell
- Phases in the adiabatic approximation
- Phases in the adiabatic theorem: delta function well
- Plancherel’s theorem
- Poisson brackets are invariant under a canonical transformation
- Poisson brackets to commutators: classical to quantum (X and P in momentum space)
- Position and momentum
- Position and momentum unit operators
- Position operator: eigenfunctions
- Position operator in momentum space
- Postulates of quantum mechanics: momentum
- Postulates of quantum mechanics: states and measurements
- Postulates of quantum mechanics: Schrödinger equation and propagators
- Probability current
- Probability current in 3-d
- Probability current with complex potential
- Probability current: a few examples
- Projection operators
- Projection operators (linear algebra)
- Propagator for a Gaussian wave packet for the free particle
- Quantum versus classical mechanics in solids and gases
- Quantum dots
- Quantum revival time
- Quantum scattering: partial wave analysis
- Quantum scattering: scattering amplitude and differential cross section
- Radial function for large r
- Radial function for small r
- Radially symmetric potentials, angular momentum and centrifugal force
- Reflectionless potential
- Relation between action and energy
- Rigid rotor in quantum mechanics
- Rotation matrices – matrix elements
- Rotation of a vector wave function
- Rotational invariance in two dimensions
- Rotational transformations using passive transformations
- Rotations through a finite angle; use of polar coordinates
- Rotations in 3-d: classical and quantum rotations compared
- Rotations in 3-d: Euler angles
- Rubber band helium
- Runge-Lenz vector and closed orbits
- Rutherford scattering
- Scattering matrix
- Scattering from the Yukawa potential
- The Schrödinger equation
- The Schrödinger equation: the motivation
- Schrödinger equation – a few theorems
- Schrödinger equation – minimum energy
- Schrödinger equation in three dimensions – spherical harmonics
- Schrödinger equation in three dimensions – radial equation
- Schrödinger equation for 2 particles – separation of variables
- Second order Born approximation in scattering theory
- Second order non-degenerate perturbation theory
- Selection rules for spontaneous emission of radiation
- Selection rules in spontaneous emission: transition between spherically symmetric states not allowed
- Self-adjoint differential equations
- Sequential measurements
- Simultaneous diagonalization of hermitian matrices
- Sinusoidal perturbations in time
- Spectral decomposition of operators
- Spectral theorem for normal operators
- Spherical Bessel functions – behaviour for small arguments
- Spherical harmonics – examples
- Spherical harmonics – more examples
- Spherical harmonics: normalization
- Spherical harmonic at the top of the ladder
- Spherical harmonic using the raising operator
- Spherical harmonics using the lowering operator
- Spherical harmonics from power series – examples for m=0
- Spherical harmonics: rotation about the x axis
- Spherically symmetric potentials – a simple example
- Spherically symmetric potentials: hermiticity of the radial function
- Spin – expectation values of components
- Spin – introduction
- Spin ½
- Spin 1/2 along an arbitrary direction
- Spin 1/2: minimum uncertainty
- Spin 1/2 particle in a magnetic field
- Spin 1/2 particle in time-varying magnetic field
- Spin 1/2: spin components
- Spin 1
- Spin 3/2
- Spin matrices: general case
- Spin and quarks
- Spin – statistical calculations
- Spin: the y component
- Spontaneous emission: Einstein’s argument
- Spontaneous emission rates for the hydrogen atom
- Spontaneous emission rates for hydrogen: general solution
- Spontaneous emission from n=3 to n=1 in hydrogen
- Spontaneous emission from the zero point field
- Stark effect in hydrogen for n = 1 and n = 2
- Stark effect in hydrogen for n = 3; hydrogen wave functions in Maple
- Stark effect: tunnelling probability
- Statistical mechanics in quantum theory: 3-d harmonic oscillator
- Statistical mechanics in quantum theory: Bose condensation
- Statistical mechanics in quantum theory: counting states
- Statistical mechanics in quantum theory: counting boson states
- Statistical mechanics in quantum theory: counting states, general case
- Statistical mechanics in quantum theory: energy probabilities
- Statistical mechanics in quantum theory: most probable state
- Statistical mechanics in quantum theory: most probable state for fermions
- Stefan-Boltzmann law
- Stimulated emission of radiation: lasers
- Stimulated emission of radiation at high frequencies
- Subspaces and direct sums
- Thermodynamics of harmonic oscillators – classical and quantum
- Time-dependent perturbation theory: iterative solution
- Time-dependent perturbation theory: general two-state solution
- Time-dependent perturbation theory: switching a perturbation on and off
- Time-dependent perturbation theory for a multi-level system
- Time-dependent perturbation theory: transition probabilities in a multi-state system
- Time-dependent perturbation uniform in space
- Time-dependent perturbation of the infinite square well
- Time-dependent propagators
- Time dependent Schrödinger equation: two-state systems
- Time-dependent Schrödinger equation: switching a perturbation on and off
- The time-independent Schrödinger equation
- The time-independent Schrödinger equation – general solutions
- Time reversal, antiunitary operators and Wigner’s theorem
- Time translation and conservation of energy
- Total angular momentum – matrix elements and commutation relations
- Total angular momentum – finite rotations
- Total angular momentum is Hermitian
- Transfer matrix
- Translational invariance in quantum mechanics
- Translational invariance and conservation of momentum
- Translation invariance in two dimensions
- Translation operator from passive transformations
- Translations in space and time
- Triangle inequality as an equality
- Triangular wave function: probabilities
- Two-dimensional harmonic oscillator – Part 1
- Two-dimensional harmonic oscillator – Part 2: Series solution
- Uncertainty principle; position-momentum commutator
- Uncertainty principle – Shankar’s more general treatment
- Uncertainty principle: an example
- Uncertainty principle – examples
- Uncertainty principle: condition for minimum uncertainty
- Uncertainty principle: rates of change of operators
- Uncertainty principle in three dimensions
- Uncertainty principle and an estimate of the ground state energy of hydrogen
- Uncertainties in the harmonic oscillator and hydrogen atom
- Unitary matrices – some examples
- Unitary operators
- Unitary operators: active and passive transformations of an operator
- Unstable particles: a crude model
- Van der Waals interaction
- The variational principle in quantum mechanics
- Variational principle and the delta function well
- Variational principle and the electron in a magnetic field
- Variational principle and the first excited state
- Variational principle and the harmonic oscillator
- Variational principle and the harmonic oscillator – 2
- Variational principle and harmonic oscillator: a more general trial function
- Variational principle and the hydrogen atom
- Variational principle and the hydrogen ion: two parameters
- Variational principle with a two-state hamiltonian
- Variational principle and the Yukawa potential
- Vector operators; transformation under rotation
- Vector spaces: definitions and examples
- Vector spaces – number of dimensions
- Vector spaces: span, linear independence and basis
- Vector spaces & linear independence – some examples
- Vector spaces and Hilbert space
- Vibrating string – normal mode analysis
- Vibrational states in a diatomic molecule (HCl)
- Virial theorem
- Virial theorem in 3-d
- Virial theorem in classical mechanics; application to harmonic oscillator
- WKB approximation
- WKB approximation – alternative derivation
- WKB approximation – analysis of the overlap region near a turning point
- WKB approximation for a barrier with sloping sides
- WKB approximation of a double potential well: turning points
- WKB approximation of double-well potential: wave functions
- WKB approximation of the harmonic oscillator
- WKB approximation and the hydrogen atom
- WKB approximation and the power law potential
- WKB approximation and the radial equation
- WKB approximation and the reflectionless potential
- WKB approximation: tunneling
- WKB approximation: turning points
- WKB approximation at a turning point with decreasing potential
- The wave function as a probability
- The wave function: Born’s conditions
- The wave function: making energy measurements
- Wave-particle duality
- Zeeman effect: degenerate perturbation theory for n = 2
- Zeeman effect for n = 3: general case
- Zeeman effect for n = 3: strong field
- Zeeman effect for n = 3; weak field
- Zeeman effect for l=0
- Zeeman effect: the n=2 line in hydrogen
- Zeeman effect: strength of magnetic field
- Zeeman effect: strong field
- Zeeman effect: weak field

AnonymousI am going through a QM course and not having an easy time of it. Thanks for putting so much effort into this blog, Its much more comprehensive that many.

Cheers from BC 🙂

AnonymousI’m new here and I like your stuff. However, some of your blogs are sequential, but the index is alphabetic. It would be nice if under a topic, say harmonic oscillator, the sub-topics were listed sequentially. Or maybe related blogs grouped into chapters.

growescienceThere are already various ways you can get posts related to a given topic. Use the ‘Search’ box in the left panel, or on the index page use your browser’s search option (Control+F on Windows). The problem indexes for the various textbooks I’m working through provide a grouping of posts into chapters. I’ve also tried to tag each post with relevant keywords, so you can do a tag search.

Links to articles referenced in a post are to be found within that post so it shouldn’t be too hard to find a sequence.

Kazi Parvez IslamSir can you please suggest few best reference books on undergraduate quantum physics and atomic and molecular physics . I know there are so many links available in the internet suggesting a lot( it’s preety confusing because it’s really a lot!!) of books. But if you give few best amongt them…it will be very much helpful.

Thanks.

gwrowePost authorIt depends on your background, but if you’re familiar with basic calculus and Newtonian physics, Griffiths’s book (Introduction to Quantum Mechanics – the one I’ve worked through) is excellent. At a slightly more advanced level, the book by R. Shankar (Principles of Quantum Mechanics), which I’m currently working on, appears to be good as well.

I can’t really offer any suggestions on atomic or molecular physics as I haven’t studied those.