Contents

General principles

• No-cloning theorem

• Lower bound on survival probability

Uncertainty relations

• Cauchy-Schwarz inequality

• Robertson-Schrödinger uncertainty relation

• Mandelstam-Tamm uncertainty relation

Mixed states

• Density operator of a subsystem

• Density operator at thermal equilibrium

One-dimensional systems

• Free-particle Gaussian wave packets

• Motion under a spatially uniform time-dependent force

• Harmonic oscillator with a time-dependent frequency

Two-dimensional systems

• Two-dimensional hydrogen atom

• Two-dimensional hydrogen atom as a harmonic oscillator

Momentum representation

• Schrödinger equation in momentum space

Hydrodynamic representation

• Madelung equations

Phase-space representation

• Wigner function: States

• Wigner function: Observables

Approximations

• Free particle: Long time evolution

Numerical methods

• Chebyshev polynomial expansion of the time-evolution operator