Quantum Mechanics · Chapter 01

Quantum ideas and the state space

Why the classical picture fails, and what replaces it: amplitudes that add before they are squared, states that superpose, and the complex vector space every quantum state lives in. The chapter ends with Dirac’s notation and the expansion of a state in a basis, which every later chapter speaks.

E = h f

z = x + i y

P = ∣ ψ ∣^{2}

∣ ψ ⟩ = n \sum c_{n} ∣ n ⟩

Lessons

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05Hilbert space intuitionStates can be added and multiplied by complex numbers, so they are vectors. The space they live in is called a Hilbert space: a vector space over the complex numbers with a way to measure how much two vectors overlap. For a qubit it is just pairs of complex numbers. For a particle on a line it is a space of functions, one complex number for every point: a vector with infinitely many components.Read lesson →

06Bras and ketsDirac’s notation gives every state two faces. The ket is the state as a column of amplitudes. The bra is its partner, the row of conjugated amplitudes, and a bra meeting a ket makes a single number. The notation is built so that the brackets close exactly when a number comes out, and it makes most of the algebra of quantum mechanics write itself.Read lesson →

07Inner product and normThe bracket of two states measures how much they overlap. A state has overlap one with itself when it is normalised, and overlap zero with a state it can be told apart from with certainty. In between, the overlap is an amplitude: its modulus squared is the probability that a system prepared in one state passes a test for the other. The Cauchy–Schwarz inequality is the guarantee that this never exceeds one.Read lesson →

08Orthonormal bases and completenessA basis is a complete set of mutually exclusive answers to one question: horizontal or vertical, diagonal or antidiagonal. Its states are orthogonal to each other, normalised, and together they can build any state. The fact that they can is written as one line — the sum of all the projections onto the basis states is the identity — and that line, inserted into any expression, is the most used trick in quantum mechanics.Read lesson →

09Expanding a state in a basisOnce a basis is chosen, a state is nothing but its list of coefficients, one amplitude per basis state. The coefficient of a basis state is the bracket of that state with the one being described, the modulus squared of each coefficient is a probability, and the probabilities add to one because the state is normalised. Inner products and norms can then be computed from the lists alone.Read lesson →

10Chapter practiceBring the chapter together: photon energies and matter waves, complex arithmetic, amplitudes that interfere, superpositions and their phases, and the vectors, brackets and bases that turn all of it into linear algebra. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Quantum ideas and the state space

Build the foundation