nLab probability amplitude

Definition

Where a probability density function (on a measure space) is a real-valued function $\rho$ (satisfying certain conditions), a probability amplitude is a complex-valued function $\psi$ such that its pointwise absolute value squared

$$\rho(x) \coloneqq \psi^\ast(x) \psi(x)$$

is a probability density.

In quantum physics

General

Probability amplitudes appear as pure states of quantum mechanical systems in the form of wave functions $\psi$ on the phase space of a corresponding classical system. The Born rule of quantum physics says that the probability density $\rho = \psi^\ast \psi$ describes the probability to find the physical system in a given classical state (in a given region of its phase space).

The fact that probability amplitudes are complex-valued means that under addition (“superposition”) they exhibit quantum interference (the addition of their complex phases) in stark contrast to the addition of probability densities, for which this cannot happen.

For instance on some probability space $(X,\mu)$ there are the probability amplitudes $\exp(i \pi/2) \mu$ and $\exp(-i \pi/2)\mu$ whose associated probability densities are both just $\mu$ itself again. But the sum of these two probability amplitudes vanishes, in contrast to the sum of their associated probability densities. This is known as “destructive” quantum interference.

Scattering amplitudes

In perturbative quantum field theory the key probability amplitudes considered are scattering amplitudes, which encode the probability for a given configuration of field quanta to come in from the far past, interact with each other and hence “scatter off” each other, and then re-emerge as some other set of field quanta in the far future.

These amplitudes are collected in the scattering matrix of the perturbative quantum field theory.

Related concepts

• quantum logic

• interpretation of quantum mechanics

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem