Overview Probability amplitude

physical

neglecting technical complexities, problem of quantum measurement behaviour of quantum state, value of observable q measured uncertain. such state thought coherent superposition of observable s eigenstates, states on value of observable uniquely defined, different possible values of observable.

when measurement of q made, system (under copenhagen interpretation) jumps 1 of eigenstates, returning eigenvalue state belongs. superposition of states can give them unequal weights . intuitively clear eigenstates heavier weights more produced. indeed, of above eigenstates system jumps given probabilistic law: probability of system jumping state proportional absolute value of corresponding numerical factor squared. these numerical factors called probability amplitudes, , relationship used calculate probabilities given pure quantum states (such wave functions) called born rule.

different observables may define incompatible decompositions of states. observables not commute define probability amplitudes on different sets.

mathematical

in formal setup, system in quantum mechanics described state, vector |Ψ⟩, residing in abstract complex vector space, called hilbert space. may either infinite- or finite-dimensional. usual presentation of hilbert space special function space, called l(x), on set x, either configuration space or discrete set.

for measurable function



ψ


{\displaystyle \psi }

, condition



ψ
∈

l

2


(
x
)


{\displaystyle \psi \in l^{2}(x)}

reads:

∫

x



|

ψ
(
x
)


|


2




d

μ
(
x
)
<
∞
;


{\displaystyle \int \limits _{x}|\psi (x)|^{2}\,\mathrm {d} \mu (x)<\infty ;}

this integral defines square of norm of ψ. if norm equal 1, then

∫

x



|

ψ
(
x
)


|


2




d

μ
(
x
)
=
1.


{\displaystyle \int \limits _{x}|\psi (x)|^{2}\,\mathrm {d} \mu (x)=1.}

it means element of l(x) of norm 1 defines probability measure on x , non-negative real expression |ψ(x)| defines radon–nikodym derivative respect standard measure μ.

if standard measure μ on x non-atomic, such lebesgue measure on real line, or on three-dimensional space, or similar measures on manifolds, real-valued function |ψ(x)| called probability density; see details below. if standard measure on x consists of atoms (we shall call such sets x discrete), , specifies measure of x ∈ x equal 1, integral on x sum , |ψ(x)| defines value of probability measure on set {x}, in other words, probability quantum system in state x. how amplitudes , vector related can understood standard basis of l(x), elements of denoted |x⟩ or ⟨x| (see bra–ket notation angle bracket notation). in basis

ψ
(
x
)
=
⟨
x

|

Ψ
⟩


{\displaystyle \psi (x)=\langle x|\psi \rangle }

specifies coordinate presentation of abstract vector |Ψ⟩.

mathematically, many l presentations of system s hilbert space can exist. shall consider not arbitrary one, convenient 1 observable q in question. convenient configuration space x such each point x produces unique value of q. discrete x means elements of standard basis eigenvectors of q. in other words, q shall diagonal in basis.



ψ
(
x
)


{\displaystyle \psi (x)}

probability amplitude eigenstate ⟨x|. if corresponds non-degenerate eigenvalue of q,




|

ψ
(
x
)


|


2




{\displaystyle |\psi (x)|^{2}}

gives probability of corresponding value of q initial state |Ψ⟩.

for non-discrete x there may not such states ⟨x| in l(x), decomposition in sense possible; see spectral theory , spectral theorem accurate explanation.