Probability amplitude

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

The principal use of probability amplitudes is the 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,^[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.

Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws. For example, in the classic double-slit experiment where electrons are fired randomly at two slits, an intuitive interpretation is that $P (hit either slit) = P (hit first slit) + P (hit second slit)$ , 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"), one runs into trouble in explaining the probability-distribution on the screen if he assumes the above to be true. The particle cannot be said to go through either slit and this simplistic explanation does not work. The probability distribution on the screen, on the other hand, follows a simple law: that of interference. There exist complex numbers corresponding to each event, which are our probability amplitudes, $ψ (event)$ . 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, as a purely real formulation has too few dimensions to describe the system's state when superposition is taken into account.^[2]

Formalism and 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 $|\psi \rangle$ in 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 $V$ , we need to do the following: we need to write an arbitrary state as a linear superposition of the eigen-states of that quantity, $\{|v\rangle \}$ :

|\psi \rangle =\int _{v}C_{v}|v\rangle \,\mathrm {d} v,

where $C_{v}$ is the weight of the state $|v\rangle$ belonging to the eigenvalue $v$ . These coefficients $\{C_{v}\}$ are the probability amplitudes of the states $\{|v\rangle \}$ 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 corresponding to that state. 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 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 $\psi (x)$ .

A basic example

Discrete components A_k of a complex vector |A⟩ = ∑_k A_k|e_k⟩.

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 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 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. 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.

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 $ψ 0 (x, t)$ 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} } =:k^{2}<\infty

for some $t 0$ , then $ψ = ψ 0 / k$ 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 $t 0$ , $ρ (x) = | ψ (x, t 0)| 2$ 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)}{k}}\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 relationship between probability amplitudes and probabilities

For more details on this topic and the proof, see probability current.

Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relation 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}\cdot {1 \over {2i}}\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 quantum continuity equation

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

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

For more details on this example, see finite potential barrier.

In the one-dimensional case of particles with energy less than $U > 0$ in the square potential

V(x)={\begin{cases}U&|x|<a\\0&{\text{otherwise,}}\end{cases}}

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

\chi (x)={\begin{cases}A\exp(ikx)+B\exp(-ikx)&x<-a,\\C\exp(\kappa x)+D\exp(-\kappa x)&|x|\leq 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 $A$ , $C$ , and $E$ signify motion to the right, while $B$ , $D$ , and $F$ to the left. Under this beam interpretation, put $F = 0$ since no particles are coming from the right. By applying 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.

References

^ http://nobelprize.org/nobel_prizes/physics/laureates/1954/#
^ Example taken from Raymond, David (2006-04-07). "Sense and Nonsense in Quantum Mechanics" (PDF). Retrieved 2011-02-23.

[1] ttp://nobelprize.org/nobel_prizes/physics/laureates/1954/#

[2] Example taken from Raymond, David (2006-04-07). "Sense and Nonsense in Quantum Mechanics" (PDF). Retrieved 2011-02-23.

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