Quantum Physics Part-1

Explore the double slit experiment, showing how amplitude superposition creates interference patterns and links quantum states to probability and intensity, with electrons and light. Examine how observation alters outcomes.

During the continued double-slit experiment, electrons in a translation superposition state show interference when unobserved, but measuring which slit they pass through collapses the state and removes the interference.

Explore how measurement disturbs quantum systems and drives the uncertainty between position and momentum, illustrated through single slit diffraction concepts and the Herzenberg gamma ray microscope.

Explore how superposition yields indeterminacy, how measurement disturbs quantum systems, and how the wavefunction encodes probabilistic outcomes via the linear Schrödinger equation with a Hamiltonian operator.

Define probability for simple events, explain normalization, and connect discrete and continuous distributions with standard deviation and probability density.

Explore the inverse relation between differentiation and integration, and practice key techniques such as integration by parts, substitution, and using trigonometric and hyperbolic forms with common integrals.

Define and apply the gamma function through zero-to-infinity integrals, derive properties like gamma(n+1)=n!, and use substitutions to evaluate infinite integrals in quantum mechanics.

Explore how the wave function provides probabilistic information, with |ψ|^2 as the position probability density, normalization to one, and measurement leading to collapse in identically prepared systems.

Explore how normalization fixes the wavefunction constant so that the probability density |ψ(x)|^2 integrates to one in one- and two-dimensional Schrödinger problems, distinguishing normalizable from non-normalizable solutions.

Explore how quantum probability densities require normalization, define expectation values, and how position and momentum operators yield observable measurements for identically prepared systems.

Explore the time-independent Schrödinger equation, derive separable solutions, and identify stationary states as energy eigenstates of the Hamiltonian, with normalization and a discussion of superposition.

Explore linear combinations of energy states and the resulting superposition, and how measurements yield definite eigenvalues with probabilities. Discuss normalization, probability amplitudes, and the energy expectation value.

Explore ordinary and partial differential equations, distinguishing linear from nonlinear forms. Learn about order, degree, and superposition, with applications to physics like a mass spring system.

Explore linear differential equations with constant coefficients, including homogeneous and nonhomogeneous forms, solving via characteristic roots, and constructing complete solutions using exponentials, sines, and cosines.

Explore how any function within a region of interest can be expanded into sine and cosine harmonics via Fourier series, with coefficients found by inner products and orthogonality.

Explore Fourier series representations of a piecewise function using sine, cosine, and the complex form; compute coefficients with integrals, examine convergence and discontinuities.

Explore the 1D box with properties of energy eigenstates, including normalization, orthogonality, and completeness, and learn how any state expands in this basis and how measurements yield eigenvalues.

Examine 1D box quantum states in symmetric wells, explore origin shifts, even/odd wavefunctions, and stationary-state energies, with emphasis on probability densities, expectation values, and independence of coordinate choices.

Explore how a normalized wave function for a particle in a 1d box encodes probability amplitudes and how measurement collapses superpositions to energy eigenstates with probabilities given by squared amplitudes.

Explore the sudden-wall expansion in a one-dimensional box, express the initial state in the final Hamiltonian basis, and compute energy probabilities for the ground and excited states.

Explore how a particle in a one-dimensional box forms a linear combination of eigenstates. Apply normalization and probabilities to predict energy measurements and expected values.

Explore how stable equilibrium yields small-amplitude simple harmonic motion via Taylor expansion of a general potential, and compare classical and quantum oscillator energy and origin shifts.

Explore the harmonic oscillator using ladder operators to derive energy eigenvalues and stationary states, compute expectation values and uncertainties, and analyze measurement outcomes and wave-function symmetry.

Explore late-stage problem hints for the linear harmonic oscillator, including frequency changes, energy eigenvalues, and measurement probabilities, with emphasis on wavefunctions, normalization, and postulates.

Explore how to compute Fourier transforms through worked examples, exploring real integrals, even/odd function properties, and the relationship between time and frequency domains.

Explore a free particle with zero potential, deriving plane-wave traveling solutions. Discuss energy and momentum, the non-normalizability of definite momentum, and wave packets via momentum-space Fourier transforms.

Explore how the free particle challenges definite position and momentum, derive the momentum-space wave function via a Fourier transform, and interpret the uncertainty principle with normalization and probability.

Explore how superposing plane waves form a moving wave packet for a free particle, revealing group velocity vs phase velocity and the envelope.

Explore the Dirac delta as a distribution, its defining properties, normalization, spikes, and relation to the unit step, including delta prime and scaling concepts.

Define the probability current density and derive its continuity with probability density from the Schrödinger equation, highlighting the flow interpretation and real versus complex wavefunctions.

Explore bound and scattering states by contrasting classical turning points and classically allowed regions with quantum tunneling, highlighting discrete versus continuous spectra across various potentials.

Analyze a one-dimensional step potential in quantum mechanics. Match wavefunctions at x=0, derive reflection and transmission coefficients, and discuss current densities for E>V0 vs E<V0.

Analyze the finite potential well, with bound and scattering states, using symmetry to form even and odd solutions. Match at boundaries and count bound states graphically.