First Quantization

1. From Classical to Quantum State Space

First quantization is the procedure of promoting a classical system to a quantum one while keeping its basic structure intact. Phase space (q,p)(q, p) is replaced by a Hilbert space of state vectors, dynamical quantities become linear operators acting on that space, and the Poisson bracket is replaced by the commutator via {⋅,⋅}→1iℏ[⋅,⋅]\{\cdot,\cdot\} \to \frac{1}{i\hbar}[\cdot,\cdot]. The immediate consequence is that a system's state is no longer a single point specifying definite values for every quantity at once — it is a vector that, in general, does not have a definite value for every observable simultaneously. Everything that follows is really an unpacking of what that shift entails.

2. The Schrödinger Equation

Time evolution of a quantum state is governed by the Schrödinger equation, iℏ ∂Ψ/∂t=H^Ψi\hbar\, \partial \Psi/\partial t = \hat H \Psi, the direct quantum analogue of Hamilton's equations: given the state now and the Hamiltonian operator, the equation determines the state at every later time, deterministically and unitarily. For a Hamiltonian with no explicit time dependence this separates into a time-independent form, H^Ψ=EΨ\hat H \Psi = E\Psi, an eigenvalue equation whose solutions are the stationary states and allowed energies of the system.

iℏ∂Ψ(x,t)∂t=[−ℏ22m∂2∂x2+V(x)]Ψ(x,t)i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)\right]\Psi(x,t)

3. Wave Amplitude

The solution Ψ(x,t)\Psi(x,t) to the Schrödinger equation is a complex-valued function called the wave amplitude, or wavefunction. On its own it is not directly observable — what is physical is ∣Ψ(x,t)∣2|\Psi(x,t)|^2, the probability density for finding the particle at position xx at time tt. Because it is a probability density, Ψ\Psi must be normalized so that ∫∣Ψ∣2 dx=1\int |\Psi|^2\, dx = 1 over all space, and its phase, though unobservable in isolation, is exactly what produces interference when amplitudes are added.

4. Bohr Interpretation

The interpretation most often taught alongside this formalism — usually credited to Bohr and collaborators and known as the Copenhagen interpretation — treats ∣Ψ∣2|\Psi|^2 as a genuine probability rather than a description of hidden, already-determined values. Before a measurement, a system does not possess a definite value for an observable unless it happens to be in an eigenstate of that observable; measurement is taken to force the state to "collapse" onto one eigenstate, with outcome probabilities set by the Born rule. Bohr paired this with the principle of complementarity: quantities such as position and momentum, or wave and particle behavior, are both valid descriptions but can never be jointly and precisely observed in a single experiment.

5. Operators

Every classical dynamical variable is promoted to a linear operator acting on the Hilbert space of states. Position becomes the operator x^\hat x (multiplication by xx in the position representation[1]), and momentum becomes p^=−iℏ ∂/∂x\hat p = -i\hbar\, \partial/\partial x. Operators act on state vectors to produce new state vectors, and the order in which two operators are applied can matter — a departure from classical variables, which are just numbers and always commute.

6. Observables

Not every operator corresponds to something measurable. Physical observables — energy, position, momentum, spin — are represented by Hermitian operators[2] specifically, because Hermiticity guarantees real eigenvalues, and a measurement can only ever return a real number. The Hamiltonian H^\hat H is itself an observable: it represents total energy, and its eigenvalues are the energies the system can actually be measured to have.

7. Eigenvectors and Eigenvalues

For an observable A^\hat A, a state satisfying A^ψ=aψ\hat A \psi = a\psi is an eigenstate, and aa is its eigenvalue. Physically, eigenstates are exactly the states with a definite value for that observable: measuring A^\hat A on the state ψ\psi is guaranteed to return aa. Because A^\hat A is Hermitian, its eigenvectors form a complete orthonormal basis for the Hilbert space — every possible state can be written as a combination of them, which is what makes superposition meaningful.

8. Commutators

The commutator of two operators, [A^,B^]=A^B^−B^A^[\hat A, \hat B] = \hat A \hat B - \hat B \hat A, measures the extent to which order of operation matters, and is the quantum inheritor of the classical Poisson bracket. Position and momentum satisfy the canonical commutation relation [x^,p^]=iℏ[\hat x, \hat p] = i\hbar — never zero — and this single nonzero result is the algebraic root of most distinctly quantum behavior, including the uncertainty principle. Two observables that do commute share a common set of eigenstates and can, in principle, be known simultaneously with arbitrary precision.

9. Superposition

Because the Schrödinger equation is linear, any combination ψ=c1ψ1+c2ψ2+…\psi = c_1\psi_1 + c_2\psi_2 + \dots of valid states is itself a valid state. When the ψi\psi_i are eigenstates of some observable with distinct eigenvalues aia_i, a system in the superposition ψ\psi does not have a definite value of that observable at all — measurement returns aia_i with probability ∣ci∣2|c_i|^2, and only after the measurement is the state left in the corresponding eigenstate. Superposition is the formal statement of what it means for a quantum state to be genuinely indefinite, rather than merely unknown to the observer.

10. Uncertainty Principle

Because position and momentum operators do not commute, no state can be a simultaneous eigenstate of both — there is no state with an exactly definite position and an exactly definite momentum at once. This is formalized by the Heisenberg uncertainty relation,

Δx Δp≥ℏ2\Delta x \, \Delta p \geq \frac{\hbar}{2}

where Δx\Delta x and Δp\Delta p are the standard deviations of position and momentum in a given state. The result generalizes to any pair of non-commuting observables, ΔA ΔB≥12∣⟨[A^,B^]⟩∣\Delta A\, \Delta B \geq \tfrac{1}{2}\left|\langle[\hat A, \hat B]\rangle\right|, making the uncertainty principle a direct algebraic consequence of the commutator structure introduced above, not an added postulate.

Previous: Lesson 01a — Classical Mechanics

  1. A state vector ψ\psi is an abstract object in Hilbert space — it does not inherently "live" anywhere. A representation is a choice of basis that turns that abstract vector into a concrete function. In the position representation you project onto eigenstates of x^\hat x, giving the wavefunction ψ(x)\psi(x): a complex amplitude at each point in space. In that basis x^\hat x acts by multiplying by xx and p^\hat p acts by −iℏ ∂/∂x-i\hbar\,\partial/\partial x. In the momentum representation the roles swap — p^\hat p multiplies by pp and x^\hat x acts by iℏ d/dpi\hbar\,d/dp — and the two pictures are related by a Fourier transform. The form p^=−iℏ ∂/∂x\hat p = -i\hbar\,\partial/\partial x is therefore not a fundamental definition; it is what the abstract momentum operator looks like once you have committed to describing states as functions of position. ↩︎

  2. The linear algebra underlying quantum mechanics has two workhorse classes of operator, each chosen because it preserves something physically essential.

    A Hermitian (or self-adjoint) operator A^\hat A satisfies A^=A^†\hat A = \hat A^\dagger, where the adjoint A^†\hat A^\dagger is defined by demanding ⟨ϕ∣A^†ψ⟩=⟨A^ϕ∣ψ⟩\langle \phi \vert \hat A^\dagger \psi \rangle = \langle \hat A \phi \vert \psi \rangle for all states. In matrix language this means A^†=(A^∗)T\hat A^\dagger = (\hat A^*)^T — conjugate every entry, then transpose. Two consequences follow immediately from this definition. First, all eigenvalues are real: if A^ψ=λψ\hat A \psi = \lambda \psi then λ=⟨ψ∣A^ψ⟩/⟨ψ∣ψ⟩\lambda = \langle \psi \vert \hat A \psi \rangle / \langle \psi \vert \psi \rangle, and Hermiticity forces that ratio to equal its own complex conjugate, so λ∈R\lambda \in \mathbb{R}. Since every measurement outcome must be a real number, observables must be Hermitian — there is no other choice consistent with the formalism. Second, eigenvectors belonging to distinct eigenvalues are orthogonal: if A^ψ1=λ1ψ1\hat A \psi_1 = \lambda_1 \psi_1 and A^ψ2=λ2ψ2\hat A \psi_2 = \lambda_2 \psi_2 with λ1≠λ2\lambda_1 \neq \lambda_2, then (λ1−λ2)⟨ψ2∣ψ1⟩=0(\lambda_1 - \lambda_2)\langle \psi_2 \vert \psi_1 \rangle = 0, forcing ⟨ψ2∣ψ1⟩=0\langle \psi_2 \vert \psi_1 \rangle = 0. Taken together these two facts mean a Hermitian operator always supplies a complete orthonormal basis of eigenstates for the Hilbert space — the spectral theorem — which is exactly what is needed to expand an arbitrary state as a superposition and read off measurement probabilities as squared coefficients.

    A unitary operator U^\hat U satisfies U^†U^=U^U^†=I^\hat U^\dagger \hat U = \hat U \hat U^\dagger = \hat I, the identity. Equivalently, U^†=U^−1\hat U^\dagger = \hat U^{-1}. In matrix language every column (and every row) of a unitary matrix forms an orthonormal set. Unitarity is the condition that preserves the inner product: ⟨U^ϕ∣U^ψ⟩=⟨ϕ∣U^†U^ψ⟩=⟨ϕ∣ψ⟩\langle \hat U \phi \vert \hat U \psi \rangle = \langle \phi \vert \hat U^\dagger \hat U \psi \rangle = \langle \phi \vert \psi \rangle. Because the norm ∥ψ∥2=⟨ψ∣ψ⟩\lVert \psi \rVert^2 = \langle \psi \vert \psi \rangle is the total probability, a norm-preserving map is one that keeps total probability equal to one. Time evolution must therefore be unitary: if H^\hat H is Hermitian, the time-evolution operator U^(t)=e−iH^t/ℏ\hat U(t) = e^{-i\hat H t/\hbar} is unitary, and the Schrödinger equation is precisely the statement that states evolve by unitary maps. Measurement, by contrast, is not unitary — it collapses the state onto an eigenspace, which does not preserve the inner product with the pre-measurement state, and this non-unitarity is the formal expression of the irreversibility of measurement.

    The two classes are related by a simple correspondence: if A^\hat A is Hermitian then eiA^e^{i\hat A} is unitary, and conversely every unitary operator near the identity can be written in this exponential form with a Hermitian generator. In physics this connection runs everywhere — the Hamiltonian generates time translations, the momentum operator generates spatial translations, angular momentum generates rotations — and in each case the generator is Hermitian (an observable) while the finite transformation it produces is unitary (a symmetry operation that preserves probability). ↩︎

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Contributors: Hanh Huynh Huu