Probability amplitude

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability density.

Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born. 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.

Overview

Physical

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 (see below) 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

In a formal setup, any system in quantum mechanics is described by a state, which is a vector $|Ψ⟩$ , residing in an abstract complex vector space, called a Hilbert space. It may be either infinite- or finite-dimensional. A usual presentation of that Hilbert space is a special function space, called $L 2 (X)$ , on certain set $X$ , that is either some configuration space or a discrete set.

For a measurable function $\psi$ , the condition $\psi \in L^{2}(X)$ specifies that a finitely bounded integral must apply:

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

this integral defines the square of the norm of $ψ$ . If that norm is equal to 1, then

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

It actually means that any element of $L 2 (X)$ of the norm 1 defines a probability measure on $X$ and a non-negative real expression $| ψ (x) | 2$ defines its Radon–Nikodym derivative with respect to the standard measure $μ$ .

If the standard measure $μ$ on $X$ is non-atomic, such as the Lebesgue measure on the real line, or on three-dimensional space, or similar measures on manifolds, then a real-valued function $| ψ (x) | 2$ is called a probability density; see details below. If the standard measure on $X$ consists of atoms only (we shall call such sets $X$ discrete), and specifies the measure of any $x \in X$ equal to 1,^[1] then an integral over $X$ is simply a sum^[2] and $| ψ (x) | 2$ defines the value of the probability measure on the set ${x$ }, in other words, the probability that the quantum system is in the state $x$ . How amplitudes and the vector are related can be understood with the standard basis of $L 2 (X)$ , elements of which will be denoted by $| x ⟩$ or $⟨ x |$ (see bra–ket notation for the angle bracket notation). In this basis

\psi (x)=\langle x|\Psi \rangle

specifies the coordinate presentation of an abstract vector $|Ψ⟩$ .

Mathematically, many $L 2$ presentations of the system's Hilbert space can exist. We shall consider not an arbitrary one, but a convenient one for the observable $Q$ in question. A convenient configuration space $X$ is such that each point $x$ produces some unique value of $Q$ . For discrete $X$ it means that all elements of the standard basis are eigenvectors of $Q$ . In other words, $Q$ shall be diagonal in that basis. Then $\psi (x)$ is the "probability amplitude" for the eigenstate $⟨ x |$ . If it corresponds to a non-degenerate eigenvalue of $Q$ , then $|\psi (x)|^{2}$ gives the probability of the corresponding value of $Q$ for the initial state $|Ψ⟩$ .

For non-discrete $X$ there may not be such states as $⟨ x |$ in $L 2 (X)$ , but the decomposition is in some sense possible; see spectral theory and Spectral theorem for accurate explanation.

Wave functions and probabilities

If the configuration space $X$ is continuous (something like the real line or Euclidean space, see above), then there are no valid quantum states corresponding to particular $x \in X$ , and the probability that the system is "in the state $x$ " will always be zero. An archetypical example of this is the $L 2 (R)$ space constructed with 1-dimensional Lebesgue measure; it is used to study a motion in one dimension. This presentation of the infinite-dimensional Hilbert space corresponds to the spectral decomposition of the coordinate operator: $⟨ x | Q | Ψ⟩ = x \cdot ⟨ x | Ψ⟩, x \in R$ in this example. Although there are no such vectors as $⟨ x |$ , strictly speaking, the expression $⟨ x | Ψ⟩$ can be made meaningful, for instance, with spectral theory.

Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function $ψ$ is the wave function.

If the function $ψ \in L 2 (X), ‖ ψ ‖ = 1$ represents the quantum state vector $|Ψ⟩$ , then the real expression $| ψ (x) | 2$ , that depends on $x$ , forms a probability density function of the given state. The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in $X$ to obtain probability values – as was stated above, the system can't be in some state $x$ with a positive probability. It gives to both amplitude and density function a physical dimension, unlike a dimensionless probability. For example, for a 3-dimensional wave function, the amplitude has the dimension [L^−3/2], where L is length.

Note that for both continuous and infinite discrete cases not every measurable, or even smooth function (i.e. a possible wave function) defines an element of $L 2 (X)$ ; see Normalization, below.

Discrete amplitudes

When the set $X$ is discrete (see above), vectors $|Ψ⟩$ represented with the Hilbert space $L 2 (X)$ are just column vectors composed of "amplitudes" and indexed by $X$ . These are sometimes referred to as wave functions of a discrete variable $x \in X$ . Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator. Components of the vector will be denoted by $ψ (x)$ for uniformity with the previous case; there may be either finite of infinite number of components depending on the Hilbert space. In this case, if the vector $|Ψ⟩$ has the norm 1, then $| ψ (x) | 2$ is just the probability that the quantum system resides in the state $x$ . It defines a discrete probability distribution on $X$ .

A discrete probability amplitude may be considered as a fundamental frequency^{[citation needed]} in the Probability Frequency domain (spherical harmonics) for the purposes of simplifying M-theory transformation calculations.

Examples

Take the simplest meaningful example of the discrete case: a quantum system that can be in two possible states: for example, the polarization of a photon. When the polarization is measured, it could be the horizontal state $|H\rangle$ or the vertical state $|V\rangle$ . Until its polarization is measured the photon can be in a superposition of both these states, so its state $|\psi \rangle$ could be written as:

|\psi \rangle =\alpha |H\rangle +\beta |V\rangle

The probability amplitudes of $|\psi \rangle$ for the states $|H\rangle$ and $|V\rangle$ are $\alpha$ and $\beta$ respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is $\alpha ^{2}$ , and the probability of being vertically polarized is $\beta ^{2}$ .

Therefore, for example, a photon in a state $|\psi \rangle ={\sqrt {\frac {1}{3}}}|H\rangle -i{\sqrt {\frac {2}{3}}}|V\rangle$ would have a probability of ${\frac {1}{3}}$ to come out horizontally polarized, and a probability of ${\frac {2}{3}}$ to come out vertically polarized when an ensemble of measurements are made. The order of such results, is, however, completely random.

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 $α 2 + β 2 = 1$ ; 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 $L 2 (X)$ , though. Wave functions that fulfill this constraint are called normalizable.

The Schrödinger wave equation, describing states of quantum particles, has solutions that describe a system and determine precisely how the state changes with time. Suppose a wavefunction $ψ 0 (x, t)$ is a solution of the wave equation, giving a description of the particle (position $x$ , for time $t$ ). If the wavefunction is square integrable, i.e.

\int _{\mathbf {R} ^{n}}|\psi _{0}(\mathbf {x} ,t_{0})|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty

for some $t 0$ , then $ψ = ψ 0 / a$ is called the normalized wavefunction. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, at a given time $t 0$ , $ρ (x) = | ψ (x, t 0) | 2$ is the probability density function of the particle's position. Thus the probability that the particle is in the volume $V$ at $t 0$ is

\mathbf {P} (V)=\int _{V}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } =\int _{V}|\psi (\mathbf {x} ,t_{0})|^{2}\,\mathrm {d\mathbf {x} } .

Note that if any solution $ψ 0$ to the wave equation is normalisable at some time $t 0$ , then the $ψ$ defined above is always normalised, so that

\rho _{t}(\mathbf {x} )=\left|\psi (\mathbf {x} ,t)\right|^{2}=\left|{\frac {\psi _{0}(\mathbf {x} ,t)}{a}}\right|^{2}

is always a probability density function for all $t$ . This is key to understanding the importance of this interpretation, because for a given the particle's constant mass, initial $ψ (x, 0)$ and the potential, the Schrödinger equation fully determines subsequent wavefunction, and the above then gives probabilities of locations of the particle at all subsequent times.

The laws of calculating probabilities of events

A. Provided a system evolves naturally (which under the Copenhagen interpretation means that the system is not subjected to measurement), the following laws apply:

The probability (or the density of probability in position/momentum space) of an event to occur is the square of the absolute value of the probability amplitude for the event: $P=|\phi |^{2}$ .
If there are several mutually exclusive, indistinguishable alternatives in which an event might occur (or, in realistic interpretations of wavefunction, several wavefunctions exist for a space-time event), the probability amplitudes of all these possibilities add to give the probability amplitude for that event: $\phi =\sum _{i}\phi _{i};P=|\phi |^{2}=\left|\sum _{i}\phi _{i}\right|^{2}$ .
If, for any alternative, there is a succession of sub-events, then the probability amplitude for that alternative is the product of the probability amplitude for each sub-event: $\phi _{APB}=\phi _{AP}\phi _{PB}$ .
Non-entangled states of a composite quantum system have amplitudes equal to the product of the amplitudes of the states of constituent systems: $\phi _{\rm {system}}(\alpha ,\beta ,\gamma ,\delta ,\ldots )=\phi _{1}(\alpha )\phi _{2}(\beta )\phi _{3}(\gamma )\phi _{4}(\delta )\ldots$ . See the #Composite systems section for more information.

Law 2 is analogous to the addition law of probability, only the probability being substituted by the probability amplitude. Similarly, Law 4 is analogous to the multiplication law of probability for independent events; note that it fails for entangled states.

B. When an experiment is performed to decide between the several alternatives, the same laws hold true for the corresponding probabilities: $P=\sum _{i}|\phi _{i}|^{2}$ .

Provided one knows the probability amplitudes for events associated with an experiment, the above laws provide a complete description of quantum systems in terms of probabilities.

The above laws give way to the path integral formulation of quantum mechanics, in the formalism developed by the celebrated theoretical physicist Richard Feynman. This approach to quantum mechanics forms the stepping-stone to the path integral approach to quantum field theory.

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 nature does not have a way to distinguish which slit the electron has gone through (a much more stringent condition than simply "it is not observed"), 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. This is an example of the case A as described in the previous article. The complex amplitudes which represent the electron passing each slit ( $ψ first$ and $ψ second$ ) follow the law of precisely the form expected: $ψ total = ψ first + ψ second$ . 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=|\psi _{\rm {first}}+\psi _{\rm {second}}|^{2}=|\psi _{\rm {first}}|^{2}+|\psi _{\rm {second}}|^{2}+2|\psi _{\rm {first}}||\psi _{\rm {second}}|\cos(\varphi _{1}-\varphi _{2}).

Here, $\varphi _{1}$ and $\varphi _{2}$ are the arguments of $ψ first$ and $ψ second$ respectively. A purely real formulation has too few dimensions to describe the system's state when superposition is taken into account. That is, without the arguments of the amplitudes, we cannot describe the phase-dependent interference. The crucial term $2|\psi _{\rm {first}}||\psi _{\rm {second}}|\cos(\varphi _{1}-\varphi _{2})$ is called the "interference term", and this would be missing if we had added the probabilities.

However, one may choose to devise an experiment in which the experimenter observes which slit each electron goes through. Then case B of the above article applies, and the interference pattern is not observed on the screen.

One may go further in devising an experiment in which the experimenter gets rid of this "which-path information" by a "quantum eraser". Then, according to the Copenhagen interpretation, the case A applies again and the interference pattern is restored.^[3]

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

\mathbf {j} ={\hbar  \over m}{1 \over {2i}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)={\hbar  \over m}\operatorname {Im} \left(\psi ^{*}\nabla \psi \right),

measured in units of (probability)/(area × time).

Then the current satisfies the equation

\nabla \cdot \mathbf {j} +{\partial  \over \partial t}|\psi |^{2}=0.

The probability density is $\rho =|\psi |^{2}$ , this equation is exactly the continuity equation, appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where $j$ corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local conservation of charges.^{[clarification needed]}

Composite systems

For two quantum systems with spaces $L 2 (X 1)$ and $L 2 (X 2)$ and given states $|Ψ 1 ⟩$ and $|Ψ 2 ⟩$ respectively, their combined state $|Ψ 1 ⟩ \otimes |Ψ 2 ⟩$ can be expressed as $ψ 1 (x 1) ψ 2 (x 2)$ a function on $X 1 \times X 2$ , 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 described above is relevant to quantum state vectors. It is also used in the context of unitary operators that are important in the 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. Note that columns of a unitary matrix, as vectors, have the norm 1.

The "transitional" interpretation may be applied to $L 2$ s on non-discrete spaces as well.

Footnotes

^ The case of an atomic measure on $X$ with $μ ({x}) \neq 1$ is not interesting, because such $x$ that $μ ({x}) = 0$ are unused by $L 2 (X)$ and can be dropped, whereas for $x$ of positive measures the value of $μ ({x})$ is virtually the question of rescaling of $ψ (x)$ . Due to this trivial fix this case was hardly ever considered by physicists.
^ If $X$ is countable, then an integral is the sum of an infinite series.
^ A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain ensemble-alike realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. Schmidt, L. Ph. H.; et al. (2013). "Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates" (PDF). Physical Review Letters. 111 (10): 103201. Bibcode:2013PhRvL.111j3201S. doi:10.1103/PhysRevLett.111.103201. PMID 25166663. S2CID 2725093.

References

Feynman, R. P.; Leighton, R. B.; Sands, M. (1989). "Probability Amplitudes". The Feynman Lectures on Physics. Vol. Volume 3. Redwood City: Addison-Wesley. ISBN 0-201-51005-7. {{cite book}}: |volume= has extra text (help)
Gudder, Stanley P. (1988). Quantum Probability. San Diego: Academic Press. ISBN 0-12-305340-4.

[1] The case of an atomic measure on $X$ with $μ ({x}) \neq 1$ is not interesting, because such $x$ that $μ ({x}) = 0$ are unused by $L 2 (X)$ and can be dropped, whereas for $x$ of positive measures the value of $μ ({x})$ is virtually the question of rescaling of $ψ (x)$ . Due to this trivial fix this case was hardly ever considered by physicists.

[2] If $X$ is countable, then an integral is the sum of an infinite series.

[3] A recent 2013 experiment gives insight regarding the correct physical interpretation of such phenomena. The information can actually be obtained, but then the electron seemingly went through all the possible paths simultaneously. (Certain ensemble-alike realistic interpretations of the wavefunction may presume such coexistence in all the points of an orbital.) Cf. Schmidt, L. Ph. H.; et al. (2013). "Momentum Transfer to a Free Floating Double Slit: Realization of a Thought Experiment from the Einstein-Bohr Debates" (PDF). Physical Review Letters. 111 (10): 103201. Bibcode:2013PhRvL.111j3201S. doi:10.1103/PhysRevLett.111.103201. PMID 25166663. S2CID 2725093.

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