A probability amplitude is a [complex number](Quantum%20Mechanics%20(index).md#Complex%20numbers%20in%20quantum%20mechanics) composed of the [inner product](inner%20products%20between%20state%20vectors.md) of a [state vector](State%20vector.md) $|\psi\rangle,$ and another state vector $\langle a_i|,$ $\langle a_i|\psi\rangle,$ where here the subscript just denotes the fact $a_i$ belongs to [a set of other vectors like it.](State%20vector.md#Eigenstate%20decomposition%20of%20state%20vectors) When squared this quantity gives us the probability that $|\psi\rangle$ will be measured as being in state $|a_i\rangle.$ We may thus think of this as a complex number that relates an initial and final condition for a state vector where the bra is a final condition and the ket is an initial condition. The fact that squaring this number gives rise to a proability is [one aspect](Born%20rule#^628928) of the [Born rule.](Born%20rule.md) ^80d783 %%This way of defining probability amplitudes is inspired by Feynman's lectures in quantum mchanics where he explains this notion in terms of the double slit experiment, see here https://www.feynmanlectures.caltech.edu/III_03.html%% # Wavefunctions If instead of a single ket $|a_i\rangle,$ we have a [continuous variable ket](State%20vector.md#State%20vectors%20with%20continuous%20eigenstates) $|a\rangle,$ we have a [wavefunction,](Wavefunction.md) [$\Psi(a)=\langle a|\psi\rangle,$](wavefunction#^0a0844) which is a function of $a$ that gives [probability amplitudes](probability%20amplitude.md). #QuantumMechanics/QuantumMeasurement