Quantum Mechanics I
The Schrödinger equation, wave functions, operators and simple systems, developed as a full first course in quantum mechanics: states, probability amplitudes and measurement in the Stern–Gerlach experiment, superposition, interference and Bell's theorem; the wave function, Born's statistical interpretation, normalization, the probability current and conservation of probability, expectation values, the momentum operator and Ehrenfest's theorem; the time-dependent Schrödinger equation, separation of variables, stationary states and the expansion of any state in energy eigenstates, with direct Crank–Nicolson integration; the infinite square well, orthonormality and completeness, Fourier's trick, sloshing superpositions and quantum revivals; the harmonic oscillator solved both with ladder operators and with the Hermite power series, zero-point energy and molecular vibrations; free-particle wave packets, the Fourier transform and momentum space, group velocity and spreading, and the delta-function potential with its single bound state and its scattering; the finite square well by transcendental equations, reflection and transmission at steps, wells and barriers, transfer matrices, the Ramsauer–Townsend effect and resonant tunnelling; the formalism of Hilbert space, Dirac notation, Hermitian operators with real eigenvalues and orthogonal eigenvectors, continuous spectra and matrix mechanics; the generalized statistical interpretation, commutators and compatible observables, the generalized uncertainty principle, energy–time uncertainty and conservation laws; the Schrödinger equation in three dimensions, degeneracy in boxes and oscillators, separation in spherical coordinates, the radial equation and the infinite spherical well; angular momentum from its commutation relations and ladder operators, spherical harmonics and molecular rotation; and a capstone solving the hydrogen atom exactly, with its energies, degeneracies, radial functions and spectral series, with runnable Python examples throughout.
PHYS-372 Year 3 3 credits 12 lessons · 4 h 25 min ▶ Video lessons coming soon
Offered by HSM University · Physics
What you'll learn
- Describe quantum states as normalized vectors of amplitudes, apply the Born rule to measurement probabilities, and explain interference and the effect of measurement in spin experiments.
- Normalize wave functions, compute expectation values and standard deviations of position and momentum, and prove conservation of probability with the probability current.
- Separate the time-dependent Schrödinger equation, state the properties of stationary states, and build general solutions as superpositions of energy eigenstates.
- Solve the infinite square well, prove orthonormality of its eigenfunctions, expand arbitrary initial states by projection, and follow their time evolution and revivals.
- Solve the harmonic oscillator by ladder operators and by power series, and use the ladder algebra to compute expectation values and uncertainties.
- Construct free-particle wave packets with the Fourier transform, explain group velocity and spreading, and solve the delta-function potential for bound and scattering states.
- Find bound states of the finite well from transcendental equations and compute reflection and transmission at steps, wells and barriers, including resonant tunnelling.
- Work in Hilbert space with Dirac notation, represent operators as matrices, and prove that Hermitian operators have real eigenvalues and orthogonal eigenvectors.
- Apply the generalized statistical interpretation, use commutators to identify compatible observables, and prove and apply the generalized uncertainty principle.
- Separate the three-dimensional Schrödinger equation in Cartesian and spherical coordinates, derive the radial equation with its centrifugal term, and solve the spherical well.
- Derive the angular momentum spectrum from commutation relations and ladder operators, and use spherical harmonics and their properties in calculations.
- Solve the hydrogen atom exactly, derive its energies, degeneracies and radial functions, and predict its spectral lines and selection rules.
Course content
12 lessons · 4 h 25 min total · final exam
- 1 Why quantum mechanics: states, probabilities and measurement 22 min · quiz
- 2 The wave function and the Born rule 22 min · quiz
- 3 The time-dependent Schrödinger equation and stationary states 23 min · quiz
- 4 The infinite square well and the superposition principle 21 min · quiz
- 5 The quantum harmonic oscillator: algebraic and analytic methods 22 min · quiz
- 6 The free particle, wave packets and the delta-function potential 22 min · quiz
- 7 The finite square well and scattering in one dimension 21 min · quiz
- 8 The formalism: Hilbert space, Hermitian operators and eigenvectors 25 min · quiz
- 9 Measurement, commutators and the generalized uncertainty principle 22 min · quiz
- 10 Quantum mechanics in three dimensions: separation in spherical coordinates 20 min · quiz
- 11 Angular momentum: operators, ladder methods and spherical harmonics 23 min · quiz
- 12 Capstone: the hydrogen atom solved exactly 22 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
The Schrödinger equation, wave functions, operators and simple systems, developed as a full first course in quantum mechanics: states, probability amplitudes and measurement in the Stern–Gerlach experiment, superposition, interference and Bell's theorem; the wave function, Born's statistical interpretation, normalization, the probability current and conservation of probability, expectation values, the momentum operator and Ehrenfest's theorem; the time-dependent Schrödinger equation, separation of variables, stationary states and the expansion of any state in energy eigenstates, with direct Crank–Nicolson integration; the infinite square well, orthonormality and completeness, Fourier's trick, sloshing superpositions and quantum revivals; the harmonic oscillator solved both with ladder operators and with the Hermite power series, zero-point energy and molecular vibrations; free-particle wave packets, the Fourier transform and momentum space, group velocity and spreading, and the delta-function potential with its single bound state and its scattering; the finite square well by transcendental equations, reflection and transmission at steps, wells and barriers, transfer matrices, the Ramsauer–Townsend effect and resonant tunnelling; the formalism of Hilbert space, Dirac notation, Hermitian operators with real eigenvalues and orthogonal eigenvectors, continuous spectra and matrix mechanics; the generalized statistical interpretation, commutators and compatible observables, the generalized uncertainty principle, energy–time uncertainty and conservation laws; the Schrödinger equation in three dimensions, degeneracy in boxes and oscillators, separation in spherical coordinates, the radial equation and the infinite spherical well; angular momentum from its commutation relations and ladder operators, spherical harmonics and molecular rotation; and a capstone solving the hydrogen atom exactly, with its energies, degeneracies, radial functions and spectral series, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.