Physics / advanced
Quantum states, observables, and representations
Deepen quantum kinematics through operator domains, basis changes, density matrices, tensor products, and operationally defined measurement probabilities.
Your route through physics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 advanced
Quantum states, observables, and representations
Deepen quantum kinematics through operator domains, basis changes, density matrices, tensor products, and operationally defined measurement probabilities.
- 01
Wavefunctions and position representation
Treat a wavefunction as coordinates of a pure state, with normalization, phase, and continuous probabilities interpreted precisely.
29 min - 02
Momentum representation and Fourier duality
Transform amplitudes between position and momentum with consistent units, normalization, and continuous-spectrum conventions.
30 min - 03
Self-adjointness and physical boundaries
Specify an unbounded observable by its action and domain, and see how boundary conditions determine its spectrum and probability flow.
33 min - 04
Compatible observables and complete bases
Resolve degeneracy using commuting observables and distinguish a shared eigenbasis from an arbitrary set of state coordinates.
31 min - 05
Density matrices and mixed preparations
Combine preparation probabilities at the operator level and distinguish statistical mixtures from coherent superpositions in a chosen basis.
31 min - 06
Partial traces and reduced states
Build a local density operator from a composite state and distinguish discarding a subsystem from conditioning on its measurement result.
32 min - 07
Schmidt decomposition and bipartite purity
Use singular values to expose entanglement in a pure bipartite state and calculate its subsystem purity independently of local bases.
32 min - 08
Generalized measurements and POVMs
Predict outcomes using positive effects that sum to identity, including noisy two-outcome and nonorthogonal multi-outcome measurements.
32 min - 09
Quantum instruments and postmeasurement states
Specify both an outcome probability and its conditional output state, and compare distinct state updates with identical measurement effects.
33 min - 10
Wigner functions and quasiprobability
Represent a continuous quantum state in phase space while keeping true marginals, normalization conventions, and negative quasiprobability distinct.
33 min