Probability amplitude

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

Probability amplitudes provide a relationship between the

—topics that continue to be debated even today.

Physical overview

Neglecting some technical complexities, the problem of

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

.

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 ${\displaystyle t}$, by a state vector |Ψ⟩ belonging to a separable complex Hilbert space. Using bra–ket notation the relation between state vector and "position basis" ${\displaystyle \{|x\rangle \}}$ of the Hilbert space can be written as^[1]

${\displaystyle \psi (x)=\langle x|\Psi \rangle }$.

Its relation with an observable can be elucidated by generalizing the quantum state ${\displaystyle \psi }$ to a measurable function and its domain of definition to a given σ-finite measure space ${\displaystyle (X,{\mathcal {A}},\mu )}$. This allows for a refinement of Lebesgue's decomposition theorem, decomposing μ into three mutually singular parts

${\displaystyle \mu =\mu _{\mathrm {ac} }+\mu _{\mathrm {sc} }+\mu _{\mathrm {pp} }}$

where μ_ac is absolutely continuous with respect to the Lebesgue measure, μ_sc is singular with respect to the Lebesgue measure and atomless, and μ_pp is a pure point measure.^[2][3]

Continuous amplitudes

A usual presentation of the probability amplitude is that of a wave function ${\displaystyle \psi }$ belonging to the L² space of (equivalence classes of) square integrable functions, i.e., ${\displaystyle \psi }$ belongs to L²(X) if and only if

${\displaystyle \|\psi \|^{2}=\int _{X}|\psi (x)|^{2}\,dx<\infty }$.

If the norm is equal to 1 and ${\displaystyle |\psi (x)|^{2}\in \mathbb {R} _{\geq 0}}$ such that

${\displaystyle \int _{X}|\psi (x)|^{2}\,dx\equiv \int _{X}\,d\mu _{ac}(x)=1}$,

then ${\displaystyle |\psi (x)|^{2}}$ is the

.

Whereas a Hilbert space is separable if and only if it admits a

${\displaystyle x}$ is an uncountable set (i.e. the probability that the system is "at position ${\displaystyle x}$" will always ${\displaystyle {\hat {\mathrm {x} }}}$ defined as

${\displaystyle \langle x|{\hat {\mathrm {x} }}|\Psi \rangle ={\hat {\mathrm {x} }}\langle x|\Psi \rangle =x_{0}\psi (x),\quad x\in \mathbb {R} ,}$

whose eigenfunctions are Dirac delta functions

${\displaystyle \psi (x)=\delta (x-x_{0})}$

which clearly do not belong to L²(X). By replacing the state space by a suitable rigged Hilbert space, however, the rigorous notion of eigenstates from spectral theorem as well as spectral decomposition is preserved.^[4]

Discrete amplitudes

Let ${\displaystyle \mu _{pp}}$ be atomic (i.e. the set ${\displaystyle A\subset X}$ in ${\displaystyle {\mathcal {A}}}$ 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

of Q. Then ${\displaystyle \psi (x)}$ is the probability amplitude for the eigenstate |x⟩. If it corresponds to a non-degenerate eigenvalue of Q, then ${\displaystyle |\psi (x)|^{2}}$ gives the probability of the corresponding value of Q for the initial state |Ψ⟩.

|ψ(x)| = 1 if and only if |x⟩ is

. 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.^{[citation needed]} Discrete dynamical variables are used in such problems as a particle in an idealized reflective box and quantum harmonic oscillator.^{[clarification needed]}

Examples

An example of the discrete case is a quantum system that can be in

. 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

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

with ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ the probability amplitudes for the states ${\displaystyle |H\rangle }$ and ${\displaystyle |V\rangle }$ 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 ${\displaystyle |\beta |^{2}}$.

Hence, 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

of measurements are made. The order of such results, is, however, completely random.

Another 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.

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

.

The

if

${\displaystyle \int |\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =a^{2}<\infty .}$

After normalization the wave function still represents the same state and is therefore equal by definition to^[5][6]

${\displaystyle \psi (\mathbf {x} ,t):={\frac {\psi (\mathbf {x} ,t)}{a}}.}$

Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Hence, ρ(x) = |ψ(x, t)|² is a probability density function and the probability that the particle is in the volume V at fixed time t is given by

${\displaystyle P_{\mathbf {x} \in V}(t)=\int _{V}|\psi (\mathbf {x} ,t)|^{2}\,\mathrm {d\mathbf {x} } =\int _{V}\rho (\mathbf {x} )\,\mathrm {d\mathbf {x} } .}$

The probability density function does not vary with time as the evolution of the wave function is dictated by the Schrödinger equation and is therefore entirely deterministic.^[7] This is key to understanding the importance of this interpretation: for a given particle constant mass, initial ψ(x, t₀) and potential, the Schrödinger equation fully determines subsequent wavefunctions. The above then gives probabilities of locations of the particle at all subsequent times.

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

of the probability amplitude, then, follows the interference pattern under the requirement that amplitudes are complex: 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

, 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.^[8]

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

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

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.

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

.

Amplitudes in operators

The concept of amplitudes is also used in the context of

specifies transition probabilities between a finite number of states.

The "transitional" interpretation may be applied to L²s on non-discrete spaces as well.^[clarification needed]

See also

• Expectation value (quantum mechanics)
• Free particle
• Finite potential barrier
• Matter wave
• Phase space formulation
• Uncertainty principle
• Ward's probability amplitude
• Wave packet

Notes

References

• Bäuerle, Gerard G. A.; de Kerf, Eddy A. (1990). Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics. Studies in Mathematical Physics. Amsterdam: North Holland.
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• de la Madrid Modino, R. (2001). Quantum mechanics in rigged Hilbert space language (PhD thesis). Universidad de Valladolid.
• Feynman, R. P.; Leighton, R. B.; Sands, M. (1989). "Probability Amplitudes".
  ISBN 0-201-51005-7
  .
• Gudder, Stanley P. (1988). Quantum Probability. San Diego: Academic Press.
  ISBN 0-12-305340-4
  .
• MR 2105088
  .
• Teschl, G. (2014). Mathematical Methods in Quantum Mechanics. Providence (R.I): American Mathematical Soc.
  ISBN 978-1-4704-1704-8
  .
• Zwiebach, Barton (2022). Mastering Quantum Mechanics. Cambridge, Mass: MIT Press.
  ISBN 978-0-262-04613-8
  .