Probability amplitude

5d1 atomic orbital of an electron in a hydrogen atom. The rigid body shows the places where the electron's probability density is above a certain value (here 0.02 nm−3): this is calculated from the probability amplitude. The color shows the complex phase of the wavefunction.

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 or probability density.

Probability amplitudes provide a physical meaning of the wave function, a link first proposed by Max Born, and this is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the wave function were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation was offered. Born was awarded half of the 1954 Nobel Prize in physics for this understanding (see Reference 1), though it was vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. Therefore, the probability thus calculated is sometimes called the "Born probability", and the relationship used to calculate probability from the wave-function is sometimes called the Born rule.

Formalism: relationship between wave-function and probability amplitude

In a formal setup, any system in quantum mechanics is described by a state, which is a vector $| \psi \rangle$, residing in an abstract vector space, called a Hilbert space. The dimension of the space may be infinite, i.e., in general, an infinite number of linearly independent basis vectors are required to describe the state $| \psi \rangle$. To describe the effect of measurement of a physical quantity, say $V$, which in quantum mechanics is represented by an operator acting on vectors in the Hilbert space, one needs to do the following: write an arbitrary state $| \psi \rangle$ as a linear superposition of the eigen-states of that quantity, $\{ | v \rangle \}$:

$| \psi \rangle =\int_v C_v | v \rangle\, \mathrm dv ,$

where $C_v$ is the weight of the state $| v \rangle$ belonging to the eigenvalue $v$, $C_v=\langle v|\psi\rangle$. These coefficients $\{ C_v \}$ are the probability amplitudes of the state $| \psi \rangle$ in the $v$-basis. When a measurement of $V$ is made, the system jumps to one of the above eigen-states, returning the eigen-value to which the state belongs. Which of the above eigen-states the system jumps to is given by a probabilistic law: the probability of the system jumping to the state $| v \rangle$ is proportional to $|C_v|^2$, explaining the name. In the above description, one may also have a discrete summation instead of an integral (when the eigen-values of the dynamical variable considered are finite discrete: e.g. a particle in an idealized infinite box) . When the state of a system is described in the position basis, the eigen-vectors being $| x \rangle$, the corresponding probability amplitude is called the wave function $\psi (x) = \langle x|\Psi \rangle$. It is to be noted that the position basis is a continuous basis, since a particle may be found anywhere within a given range. Moreover, in three dimensions, the wave-function depends on three space-variables.

A basic example

Discrete components Ak of a complex vector |A = ∑k Ak|ek.
Continuous components ψ(x) of a complex vector |ψ = ∫dx ψ(x)|x.
Components of complex vectors plotted against index number; discrete k and continuous x. Two particular probability amplitudes out of infinitely many are highlighted.

Take a quantum system that can be in two possible states: for example, the polarisation of a photon. When the polarisation is measured, it could be horizontal, labelled as state $|H \rangle$, or vertical, state $|V\rangle$. Until its polarisation 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 α and β respectively. When the photon's polarisation is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarised is $| \alpha |^2$, and the probability of being vertically polarised is $| \beta |^2$.

Therefore, a photon in a state $| \psi \rangle = \sqrt{1\over 3} |H\rangle - i \sqrt{2\over 3}|V \rangle,\,$ whose polarisation was measured. It would have a probability of 1/3 to come out horizontally polarised, and a probability of 2/3 to come out vertically polarised, on measurement, when an ensemble of measurements are made. The order of such results, is, however, completely random.

Normalisation

The measurement must give either $|H \rangle$ or $|V \rangle$, so the total probability of measuring $|H \rangle$ or $|V \rangle$ must be 1. This leads to a constraint that $| \alpha |^2 + | \beta |^2 = 1 \,$; more generally the sum of the squared moduli of the probability amplitudes of all the possible states is equal to one. Wavefunctions that fulfill this constraint are called normalised wavefunctions.

The laws of calculating probabilities of events

A. Provided a system is not subjected to measurement, the following laws apply:

1. The probability of an event to occur is the absolute squared of the probability amplitude for the event: $P=|\phi|^2$
2. If there are several mutually exclusive, indistinguishable alternatives in which an event might occur, the probability amplitude of all these possibilities add to give the probability amplitude for that event: $\phi=\sum_i\phi_i; P=|\phi|^2=|\sum_i\phi_i|^2$ .
3. 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:$\phi_{APB}=\phi_{AP}\phi_{PB}$.
4. Consider a system of non-interacting multiple particles. Say, an event for the system corresponds to a set of independent events for the constituent particles. The product of the probability amplitudes of the independent events corresponding to the constituent particles gives the above probability amplitude for the above event:$\phi_{system} (\alpha,\beta,\gamma,\delta,...)=\phi_1(\alpha)\phi_2(\beta)\phi_3(\gamma)\phi_4(\delta)...$.

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.

B. When an experiment is performed to decide between the several alternatives, the same laws hold true for the corresponding probabilities and not the probability amplitudes: $P=\sum_i|\phi_i|^2$.

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

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

Main article: 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 though (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 ($\psi_{first}$ and $\psi_{second}$) follow the law of precisely the form expected: $\psi_{total}=\psi_{first} + \psi_{second}$. 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: $P = |\psi_{first} + \psi_{second}|^2 = |\psi_{first}|^2 + |\psi_{second}|^2 + 2 |\psi_{first}| |\psi_{second}| \cos (\alpha_1-\alpha_2)$. Here, $\alpha_1$ and $\alpha_2$ are the arguments of $\psi_{first}$ and $\psi_{second}$ 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 $2 |\psi_{first}| |\psi_{second}| \cos (\alpha_1-\alpha_2)$ is called the "interference term", and this would be missing if we had added the probabilities. However, one may choose to device an experiment in which he 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 he gets rid of this "which-path information" by a "quantum eraser." Then the case A applies again and the interference pattern is restored.

Normalizable states

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 ψ0(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 t0, then ψ = ψ0/a is called the normalized wave function. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, at a given time t0, ρ(x) = |ψ(xt0)|2 is the probability density function of the particle's position. Thus the probability that the particle is in the volume V at t0 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 ψ0 to the wave equation is normalisable at some time t0, 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 initial ψ(x, 0), the Schrödinger equation fully determines subsequent wavefunction, and the above then gives the probable location of the particle at all subsequent times.

Conservation of probabilities and the Continuity equation

Main article: Probability current.

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

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

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.

Wave function of a free particle

The wave function of a free particle is given by $\psi(\mathbf{r})=\frac{1}{\sqrt{4\pi}}e^\frac{i\mathbf{p}.\mathbf{r}}{\hbar}$, which describes a plane wave. This implies that the probability of finding a particle which is not interacting with anything else in the universe (idealized situation) is constant, since $|\psi(\mathbf{r})|^2 =\frac{1}{{4\pi}}$ is a constant. It has a definite momentum, given by $\mathbf p = \hbar \mathbf k$, where $\hbar$ (pronounced "h-bar") is the reduced Planck's constant. This is the de Broglie hypothesis. However, its position is completely uncertain, as dictated by Heisenberg's Uncertainty Principle.

Discrete amplitudes

While the wave function describes the state of a system for the continuous variable position, there are also many discrete variables to which probabilities may also be attached, which in quantum mechanics are found from complex amplitudes.

Example: One-dimensional quantum tunnelling

Main article: Finite Potential Barrier.
Potential barrier in one dimension

In the one-dimensional case of particles with energy less than $V_0$ in the square potential

$V(x)=\begin{cases}V_0 & |x|

the steady-state solutions to the wave equation have the form (for some constants $k, \kappa$)

$\psi (x) = \begin{cases} A\exp(ikx)+B\exp(-ikx) & x<-a, \\ C\exp(\kappa x)+D\exp(-\kappa x) & |x|\le a, \\ E\exp(ikx)+F\exp(-ikx) & x>a. \end{cases}$

The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting A =1 corresponds to firing particles singly; the terms containing A and E signify motion to the right, while B and F to the left. Under this beam interpretation, put F = 0 since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.

The conclusion is that the complex value B is a probability amplitude, with a real interpretation in the problem. The corresponding probability |B|2 describes the probability of a particle fired from the left being reflected by the potential barrier. Note that, very neatly, |B|2 +|E|2 =1 just as expected.

Probability frequency

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