Wave Function and Hilbert Space

Summary

In quantum mechanics, the state of a system is fully described by a wave function — a normalized vector in a complex Hilbert space. Physical quantities (observables) correspond to Hermitian operators, and measurement outcomes are eigenvalues drawn from a probability distribution given by the Born rule.

Overview

Classical physics describes a particle by its exact position and momentum. Quantum mechanics replaces this with a state vector $\psi$ that encodes probability amplitudes for all possible measurement outcomes. The mathematical arena is a separable complex Hilbert space $\mathcal{H}$, which may be infinite-dimensional. This framework successfully explains atomic spectra, the photoelectric effect, spin, and the discrete energy levels of bound systems.

Mathematical Formulation

The State Postulate

Quantum State

The state of a quantum mechanical system is a vector $\psi$ belonging to a separable complex Hilbert space $\mathcal{H}$. The state is normalized:

$$⟨\psi ,\psi ⟩=1$$

and is defined up to a global phase: $\psi$ and $e^{i\alpha }\psi$ represent the same physical system.

The possible states form the projective space of $\mathcal{H}$.

Examples of Hilbert spaces:

• Position/momentum of a particle: $\mathcal{H}=L^2(\mathbb{C})$, the space of square-integrable functions
• Spin-$\frac{1}{2}$ particle: $\mathcal{H}={\mathbb{C}}^2$ with the standard inner product

Observables

Observable

A physical quantity (position, momentum, energy, spin) is represented by a Hermitian (self-adjoint) linear operator $\hat{A}$ acting on $\mathcal{H}$.

A quantum state $\psi$ is an eigenstate of $\hat{A}$ with eigenvalue $\lambda$ if $\hat{A}\psi =\lambda \psi$. In general $\psi$ is a superposition of eigenstates.

Born Rule

Born Rule

When observable $\hat{A}$ is measured on state $\psi$:

• If eigenvalue $\lambda$ is non-degenerate, the probability of obtaining $\lambda$ is $|⟨\vec{\lambda },\psi ⟩{|}^2$, where $\vec{\lambda }$ is the unit eigenvector.
• If eigenvalue $\lambda$ is degenerate, the probability is $⟨\psi ,P_{\lambda }\psi ⟩$, where $P_{\lambda }$ is the projector onto the eigenspace.
• For continuous spectra, these formulas give a probability density.

Wavefunction Collapse

After a measurement yields result $\lambda$:

• Non-degenerate case: state collapses to $\vec{\lambda }$
• General case: state collapses to $\frac{P_{\lambda }\psi }{\sqrt{⟨\psi ,P_{\lambda }\psi ⟩}}$

This collapse is the source of the measurement problem in quantum foundations.

Superposition

A quantum state may be a linear combination (superposition) of eigenstates:

$$\psi =\sum _n{c}_n{\varphi }_n,\sum _n|c_n{|}^2=1$$

where $|c_n{|}^2$ is the probability of measuring eigenvalue ${\lambda }_n$.

Connections

• Schrödinger Equation and Time Evolution: $\psi$ evolves deterministically between measurements according to $i\hslash {\partial }_t\psi =H\psi$.
• Uncertainty Principle: Non-commuting observables cannot both be precisely defined simultaneously.
• Quantum Entanglement: Composite systems live in tensor-product Hilbert spaces; entanglement arises when the state cannot be factored.
• Canonical Quantization of Fields: QFT promotes classical fields to operator-valued distributions, generalizing this single-particle formalism.

See Also

• Schrödinger Equation and Time Evolution — time dynamics of the state vector
• Uncertainty Principle — limits on simultaneous eigenvalues of non-commuting observables
• Quantum Entanglement — non-separable states of composite systems