Talk:Probability amplitude

One-dimensional quantum tunnelling

It is certainly a classical problem in quantum mechanics with far-reaching conclusions such as an S-matrix. Though, these A/B/C_r/l have a trouble: they do not match the currently formulated definition of a probability amplitude. They are not values of the wave function, not even such values up to rescale, but only coefficients in a linear combination. They would be something like probability amplitudes (although non-normalized) were the particle free, in a momentum space, but it isn’t free and the Fourier transform on the entire line, certainly, will conflate A with C. Which decomposition of states can define these 6 parameters as probability amplitudes? Incnis Mrsi (talk) 11:53, 11 January 2014 (UTC)

P.S. After an intensive meditation I concluded that interpretation of B_r/l as probability amplitudes is misleading (unrelated to the actual problem), and interpretation of A/C_r/l as probability amplitudes cannot be explained in the way the section currently presents. So I propose to move the section to wave function and replace it here with a stub extracted from S-matrix. Opinions? Incnis Mrsi (talk) 12:11, 11 January 2014 (UTC)

Support. YohanN7 (talk) 15:53, 11 January 2014 (UTC)

Some basics probability theory rules that i think should be taking account

I thought about some anomaly stuff that exists in the basics of probability theory , that could have some implication in QM: Max Born probabilistic interpretation and Uncertainty principle. And it will insert some set theory based math to physics( measure theory which is basis of modern probability is based on set theory) .

QM physics and any statistical theory presume strong law of large number holds all the time ( when n->inf average=mean ) But The strong law of large number holds only when the expected value of probability density function converges surely ( by the mean of Lebesgue integration), there are many probability density function that either the expected value or second moment does not hold this condition( they can converge by other types if integral such as improper reiman or gauge integral ( see here the status on integration definition in math http://www.math.vanderbilt.edu/~schectex/ccc/gauge/ gauge integral also has some connection with QM path integration)

so if some wave function has no first or second moment according to Lebesgue integration then what is the SD and expected value of position for example?

And if there is no such wave functions ( I don’t think it’s true because Cauchy/Lorentzian distribution is in use now) , then such limiting conditions should be taken into account . ( adding to the demand that integral |Psy|^2 dx =1, integral |x|*|Psy|^2 dx < inf and integral x^2*|Psy|^2 dx < inf ) — Preceding unsigned comment added by Itaijj (talk • contribs) 12:53, 19 April 2014 (UTC)

"mysterious consequences"

"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."

What is this sentence trying to say? Is "mysterious" an adequate word? —Giratina000 ^¡Hola! 19:19, 20 May 2016 (UTC)

Values of coefficients α and β in "A basic example"

Given the equation

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

and the example

$|\psi \rangle ={\sqrt {1 \over 3}}|H\rangle -i{\sqrt {2 \over 3}}|V\rangle ,$

I read it to say $\alpha ={\sqrt {1 \over 3}}$ and $\beta =-i{\sqrt {2 \over 3}}$. So

$(\beta )^{2}=(-i{\sqrt {2 \over 3}})^{2}=-{2 \over 3}$

But the text implies a non-negative value of $\beta ^{2}={2 \over 3}$.

Is the text correct? If it is, I think it would be worth some additional explanatory text. BC Graham (talk) 17:11, 30 July 2018 (UTC)

An example with ψ_first and ψ_second

Please, could you give an example of ψ_first and ψ_second in the section: In the context of the double-slit experiment.

Paragraph moved to disc page because of foundational issues

I removed the following paragraph from the article because it has foundational issues, as described in more detail below.

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:

1. 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: $P=|\phi |^{2}$.
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:
   $$\phi =\sum _{i}\phi _{i};P=|\phi |^{2}=\left|\sum _{i}\phi _{i}\right|^{2}.$$
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. Non-entangled states of a composite quantum system have amplitudes equal to the product of the amplitudes of the states of constituent systems:
   $$\phi _{\text{system}}(\alpha ,\beta ,\gamma ,\delta ,\ldots )=\phi _{1}(\alpha )\phi _{2}(\beta )\phi _{3}(\gamma )\phi _{4}(\delta )\cdots .$$
   See § Composite systems 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.

Problems with the above paragraph

General

• Citations are missing.
• Notation is unclear: It is not explained what $P,\phi ,\phi _{i},\phi _{\text{system}},\phi _{APB}$ and so on are.

Content-wise

• The paragraph describes "the probability of an event to occur", without defining what an "event" is, or what "occur" means. In the Copenhagen interpretation, we only predict measurement outcomes by definition, so that an "event to occur" actually means the observation of a measurement outcome that lies in the spectrum of some self-adjoint operator that models the measurement made. However, the paragraph talks about events ocurring "when the system is not subjected to measurement". By definition this makes no sense in the Copenhagen interpretation, and can only be meaningful in non-standard interpretations of quantum mechanics such as stochastic collapse. (For more information on the Copenhagen interpretation, see Counterfactual_definiteness#Copenhagen_interpretation. According to the Copenhagen interpretation, unperformed experiments have no results. In particular, it makes no sense to speak of events occurring when no measurement is made if you adhere to the Copenhagen interpreation.)
• The connection to Feynman's path integral formulation is unclear: The latter is used as a calculational tool to compute inner products of different elements in the Hilbert space of the quantum theory. This information is not retrievable from the paragraph.

Maximilian Janisch (talk) 22:05, 23 April 2023 (UTC)