Probability amplitude

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 ∈ X, and the probability that the system is "in the state x" will always be zero. An archetypical example of this is the L²(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⋅⟨x | Ψ⟩, x ∈ 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 ∈ L²(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 L²(X); see Normalization, below.

When the set X is discrete (see above), vectors |Ψ⟩ represented with the Hilbert space L²(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 ∈ 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 or 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.

|ψ(x)| = 1 if and only if |x⟩ is the same quantum state as |Ψ⟩. ψ(x) = 0 if and only if |x⟩ and |Ψ⟩ are orthogonal (see inner product space). Otherwise the modulus of ψ(x) is between 0 and 1.

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.

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 ${\displaystyle |H\rangle }$ or the vertical state ${\displaystyle |V\rangle }$. Until its polarization is measured the photon can be in a superposition of both these states, so its state ${\displaystyle |\psi \rangle }$ could be written as:

${\displaystyle |\psi \rangle =\alpha |H\rangle +\beta |V\rangle }$

The probability amplitudes of ${\displaystyle |\psi \rangle }$ for the states ${\displaystyle |H\rangle }$ and ${\displaystyle |V\rangle }$ are ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ 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 ${\displaystyle |\alpha |^{2}}$, and the probability of being vertically polarized is ${\displaystyle |\beta |^{2}}$.

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

Another meaningful example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (${\textstyle \sigma _{z}}$), the following must be true for the measurement of spin "up" and "down":

${\displaystyle \sigma _{z}|u\rangle =(+1)|u\rangle }$

${\displaystyle \sigma _{z}|d\rangle =(-1)|d\rangle }$

If one assumes that system is prepared, so that +1 is registered in ${\textstyle \sigma _{x}}$ and then the apparatus is rotated to measure ${\textstyle \sigma _{z}}$, the following holds:

${\displaystyle {\begin{aligned}\langle r|u\rangle &=\left({\frac {1}{\sqrt {2}}}\langle u|+{\frac {1}{\sqrt {2}}}\langle d|\right)\cdot |u\rangle \\&=\left({\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\end{pmatrix}}+{\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\1\end{pmatrix}}\right)\cdot {\begin{pmatrix}1\\0\end{pmatrix}}\\&={\frac {1}{\sqrt {2}}}\end{aligned}}}$

The probability amplitude of measuring spin up is given by ${\textstyle \langle r|u\rangle }$, since the system had the initial state ${\textstyle |r\rangle }$. The probability of measuring ${\textstyle |u\rangle }$ is given by

${\displaystyle P(|u\rangle )=\langle r|u\rangle \langle u|r\rangle =\left({\frac {1}{\sqrt {2}}}\right)^{2}={\frac {1}{2}}}$

Which agrees with experiment.

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 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.

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

for some t₀, then ψ = ψ₀/a is called the normalized wavefunction. 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(x, t₀)|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

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

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

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

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.

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 (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:

$${\displaystyle P=\left|\psi _{\text{first}}+\psi _{\text{second}}\right|^{2}=\left|\psi _{\text{first}}\right|^{2}+\left|\psi _{\text{second}}\right|^{2}+2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2}).}$$

Here, ${\displaystyle \varphi _{1}}$ and ${\displaystyle \varphi _{2}}$ 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 ${\textstyle 2\left|\psi _{\text{first}}\right|\left|\psi _{\text{second}}\right|\cos(\varphi _{1}-\varphi _{2})}$ 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, due to wavefunction collapse, 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]

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

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

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

Then the current satisfies the equation

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

The probability density is ${\displaystyle \rho =|\psi |^{2}}$, 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.

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 .

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 L²s on non-discrete spaces as well.

• Free particle
• Finite potential barrier
• Matter wave
• Uncertainty principle
• Ward's probability amplitude
• Wave packet
• Phase space formulation

• Feynman, R. P.; Leighton, R. B.; Sands, M. (1989). "Probability Amplitudes". The Feynman Lectures on Physics. Vol. 3. Redwood City: Addison-Wesley. ISBN 0-201-51005-7.
• Gudder, Stanley P. (1988). Quantum Probability. San Diego: Academic Press. ISBN 0-12-305340-4.