Quantum computing and physics

Quantum mechanics combines linear algebra, calculus, complex numbers and probability: wave functions, the Schrödinger equation, Fourier duality and unitary evolution.

4 topics

Linear algebra  +  Calculus  +  Complex numbers  +  Probability  =  Quantum mechanics

In wave mechanics the state is a complex function $\psi(x, t)$ , its evolution a partial differential equation, and probabilities are integrals of $|\psi|^2$ . Quantum computing works in finite dimensions, where calculus recedes and linear algebra takes over — see Math of Quantum for that side.

Topics

Wave function

A complex-valued function $\psi(x)$ whose squared modulus is a probability density (Born rule): the probability of finding the particle in $[a,b]$ is $\int_a^b|\psi|^2\,\dd x$ , and normalization requires $\int|\psi|^2 = 1$ .

AdvancedApplication

Schrödinger equation

The PDE for the time evolution of a wave function: $i\hbar\,\partial_t\psi = \hat H\psi$ , with the Hamiltonian built from the Laplacian (kinetic energy) and a potential. Its operators — derivatives acting on functions — are where calculus and linear algebra meet; in quantum computing the same equation becomes $|\psi(t)\rangle = e^{-iHt/\hbar}|\psi(0)\rangle$ .

SpecializationApplication

Uncertainty principle

Position and momentum wave functions are Fourier transforms of each other, and a function and its transform cannot both be narrow: $\sigma_x\sigma_p \ge \hbar/2$ . The same inequality limits time–frequency resolution in signal processing (spectrograms, radar).

AdvancedApplication