In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square modulus of this quantity at a point in space represents a probability density at that point.
Probability amplitudes provide a relationship between the quantum state of a system and the results of observations of that system, a link that was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability".
These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
Contents
Physical overview
Neglecting some technical complexities, the problem of quantum measurement is the behaviour of a quantum state, for which the value of the observable Q to be measured is uncertain. Such a state is thought to be a coherent superposition of the observable's eigenstates, states on which the value of the observable is uniquely defined, for different possible values of the observable.
When a measurement of Q is made, the system (under the Copenhagen interpretation) jumps to one of the eigenstates, returning the eigenvalue belonging to that eigenstate. The system may always be described by a linear combination or superposition of these eigenstates with unequal "weights". Intuitively it is clear that eigenstates with heavier "weights" are more "likely" to be produced. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. These numerical weights are called probability amplitudes, and this relationship used to calculate probabilities from given pure quantum states (such as wave functions) is called the Born rule.
Clearly, the sum of the probabilities, which equals the sum of the absolute squares of the probability amplitudes, must equal 1. This is the normalization requirement.
If the system is known to be in some eigenstate of Q (e.g. after an observation of the corresponding eigenvalue of Q) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of Q (so long as no other important forces act between the measurements). In other words, the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. If the set of eigenstates to which the system can jump upon measurement of Q is the same as the set of eigenstates for measurement of R, then subsequent measurements of either Q or R always produce the same values with probability of 1, no matter the order in which they are applied. The probability amplitudes are unaffected by either measurement, and the observables are said to commute.
By contrast, if the eigenstates of Q and R are different, then measurement of R produces a jump to a state that is not an eigenstate of Q. Therefore, if the system is known to be in some eigenstate of Q (all probability amplitudes zero except for one eigenstate), then when R is observed the probability amplitudes are changed. A second, subsequent observation of Q no longer certainly produces the eigenvalue corresponding to the starting state. In other words, the probability amplitudes for the second measurement of Q depend on whether it comes before or after a measurement of R, and the two observables do not commute.
Mathematical formulation
In a formal setup, the state of an isolated physical system in quantum mechanics is represented, at a fixed time
t
{\displaystyle t}
, by a state vector |Ψ⟩ belonging to a separable complex Hilbert space. Using bra–ket notation the relation between state vector and "position basis"
{
|
x
⟩
}
{\displaystyle \{|x\rangle \}}
of the Hilbert space can be written as
ψ
(
x
)
=
⟨
x
|
Ψ
⟩
{\displaystyle \psi (x)=\langle x|\Psi \rangle }
.
Its relation with an observable can be elucidated by generalizing the quantum state
ψ
{\displaystyle \psi }
to a measurable function and its domain of definition to a given σ-finite measure space
Continuous amplitudes
A usual presentation of the probability amplitude is that of a wave function
ψ
{\displaystyle \psi }
belonging to the L2 space of (equivalence classes of) square integrable functions, i.e.,
ψ
{\displaystyle \psi }
belongs to L2(X) if and only if
‖
ψ
‖
2
=
∫
X
|
ψ
(
x
)
|
2
d
x
<
∞
{\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }
.
If the norm is equal to 1 and
|
ψ
(
Discrete amplitudes
Let
μ
p
p
{\displaystyle \mu _{pp}}
be atomic (i.e. the set
A
⊂
X
{\displaystyle A\subset X}
in
A
{\displaystyle {\mathcal {A}}}
is an atom); specifying the measure of any discrete variable x ∈ A equal to 1. The amplitudes are composed of state vector |Ψ⟩ indexed by A; its components are denoted by ψ(x) for uniformity with the previous case. If the ℓ2-norm of |Ψ⟩ is equal to 1, then |ψ(x)|2 is a probability mass function.
A convenient configuration space X is such that each point x produces some unique value of the observable Q. For discrete X it means that all elements of the standard basis are eigenvectors of Q. Then
ψ
(
x
)
{\displaystyle \psi (x)}
is the probability amplitude for the eigenstate |x⟩. If it corresponds to a non-degenerate eigenvalue of Q, then
|
ψ
(
x
Examples
An example of the discrete case is a quantum system that can be in two possible states, e.g. the polarization of a photon. When the polarization is measured, it could be the horizontal state
|
H
⟩
{\displaystyle |H\rangle }
or the vertical state
|
V
⟩
{\displaystyle |V\rangle }
. Until its polarization is measured the photon can be in a superposition of both these states, so its state
|
ψ
⟩
{\displaystyle |\psi \rangle }
could be written as
|
ψ
⟩
=
α
|
H
⟩
+
β
|
V
⟩
{\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }
Normalization
In the example above, the measurement must give either | H ⟩ or | V ⟩, so the total probability of measuring | H ⟩ or | V ⟩ must be 1. This leads to a constraint that α2 + β2 = 1; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. If to understand "all the possible states" as an orthonormal basis, that makes sense in the discrete case, then this condition is the same as the norm-1 condition explained above.
One can always divide any non-zero element of a Hilbert space by its norm and obtain a normalized state vector. Not every wave function belongs to the Hilbert space L2(X), though. Wave functions that fulfill this constraint are called normalizable.
The Schrödinger equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time. Suppose a wave function ψ(x, t) gives a description of the particle (position x at a given time t). A wave function is square integrable if
∫
|
ψ
(
x
,
t
)
|
2
d
x
=
a
2
<
∞
.
{\displaystyle \int |\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty .}
In the context of the double-slit experiment
Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic double-slit experiment, electrons are fired randomly at two slits, and the probability distribution of detecting electrons at all parts on a large screen placed behind the slits, is questioned. An intuitive answer is that P(through either slit) = P(through first slit) + P(through second slit), where P(event) is the probability of that event. This is obvious if one assumes that an electron passes through either slit. When no measurement apparatus that determines through which slit the electrons travel is installed, the observed probability distribution on the screen reflects the interference pattern that is common with light waves. If one assumes the above law to be true, then this pattern cannot be explained. The particles cannot be said to go through either slit and the simple explanation does not work. The correct explanation is, however, by the association of probability amplitudes to each event. The complex amplitudes which represent the electron passing each slit (ψfirst and ψsecond) follow the law of precisely the form expected: ψtotal = ψfirst + ψsecond. This is the principle of quantum superposition. The probability, which is the modulus squared of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex:
P
=
|
ψ
first
+
ψ
second
|
2
=
|
ψ
first
|
2
Conservation of probabilities and the continuity equation
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.
Define the probability current (or flux) j as
j
=
ℏ
m
1
2
i
(
ψ
∗
∇
ψ
−
ψ
∇
ψ
∗
)
=
ℏ
m
Im
(
ψ
∗
∇
ψ
)
,
Composite systems
For two quantum systems with spaces L2(X1) and L2(X2) and given states |Ψ1⟩ and |Ψ2⟩ respectively, their combined state |Ψ1⟩ ⊗ |Ψ2⟩ can be expressed as ψ1(x1) ψ2(x2) a function on X1 × X2, that gives the
product of respective probability measures. In other words, amplitudes of a non-entangled composite state are products of original amplitudes, and respective observables on the systems 1 and 2 behave on these states as independent random variables. This strengthens the probabilistic interpretation explicated above.
Amplitudes in operators
The concept of amplitudes is also used in the context of scattering theory, notably in the form of S-matrices. Whereas moduli of vector components squared, for a given vector, give a fixed probability distribution, moduli of matrix elements squared are interpreted as transition probabilities just as in a random process. Like a finite-dimensional unit vector specifies a finite probability distribution, a finite-dimensional unitary matrix specifies transition probabilities between a finite number of states.
The "transitional" interpretation may be applied to L2s on non-discrete spaces as well.