Probability Amplitudes and the History of Science

[DaTario]

Hi All,

I would like to know who was the first scientist to use probability amplitudes in solving either math or physics problems.

Best wishes,
DaTario

 

[fresh_42]

What is a "probability amplitude"?

 

[martinbn]

What is a "probability amplitude"?

It is a misnomer. It is the "square root" of the probability. It probably comes from the other misnomer the wave function.

https://en.wikipedia.org/wiki/Probability_amplitude

 

[fresh_42]

It is a misnomer. It is the "square root" of the probability. It probably comes from the other misnomer the wave function.

So the question is basically: Who wrote a $\sqrt{}$ sign above a probability for the first time? Yes, that makes a lot of more sense now!

 

[PeroK]

So the question is basically: Who wrote a $\sqrt{}$ sign above a probability for the first time? Yes, that makes a lot of more sense now!

"Probability amplitude", like any terminology, is not perfect, but it's not simply "the square root of a probability".

You can find an insightful discussion of probability amplitudes here:

https://www.scottaaronson.com/democritus/lec9.html

 

[stevendaryl]

I think "probability amplitude" makes perfect sense in talking about quantum mechanics.

In a stochastic theory, the basic question is to compute: $P(A, t_1, B, t_2)$. That's the probability of the system getting to state $B$ at time $t_2$ given that it was in state $A$ at time $t_1$.

In quantum mechanics, the basic question is to compute: $\Psi(A, t_1, B, t_2)$. That's the probability amplitude of the system getting from $A$ at time $t_1$ to $B$ at time $t_2$. If $A$ and $B$ are points in configuration space, then probability amplitude is the Green function.

A probability amplitude is not simply the square root of the probability, because it also has a phase. You can compute probabilities from amplitudes, but going in the reverse direction is not one-to-one.

 

[fresh_42]

I only wanted to point at the insufficiency of the question. Sorry, for using sarcasm - should have used tags ...

• Does the OP mean the terminology or the term?
• If they meant the term, in which context?
• As they mentioned mathematics as well as physics, any of these isn't clear at all.
• @stevendaryl is right, it makes sense in QM, so what does the reference to math mean?

I bet all of these questions will be silently answered within every response and thus provide a whole set of hidden assumptions. This post is substandard as it stands.

 

[Nugatory]

his post is substandard as it stands.

"Needs to be phrased more precisely" would be less pejorative, but... yes.

 

[DaTario]

Hi All,
I will try to better the question. The probability amplitude $\psi$ is usually, in the Schroedinger picture, a complex function of, say, position and time. So we usually have $\psi(x,t)$ and frequently we call it the wave function. By taking the modulus squared of $\psi$ ( i.e. $\psi(x,t) \psi^*(x,t)$) one gets a probability density, so that $\psi(x_0,t) \psi^*(x_0,t) dx$ is the probability of finding the particle between positions $x_0$ and $x_0 + dx$ and at a given time $t$.
Having introduced the context in which probablity amplitudes commonly appears, I would like to ask who was the scientist that introduced this concept. I have joined mathematics in this OP for $\psi$ is part of a development in probability theory, so may be it was a mathematician who first introduced this tool.

Best Regards,
DaTario

 

[fresh_42]

What you call a "tool" is an interpretation. The "tool" is a Hilbert space. As you do not want to talk about inner products and orthonormal basis, I suggest to restrict your question to physics, where it makes sense. From a mathematical point of view, it is still unclear what you want to know. There is no mathematics involved so far.

 

[PeterDonis]

Having introduced the context in which probablity amplitudes commonly appears, I would like to ask who was the scientist that introduced this concept.

As far as I know, it was Max Born, who introduced the probabilistic interpretation of the Schrodinger Equation. Schrodinger originally viewed the wave function $\psi (x, t)$ that appeared in his equation as describing a real wave, perhaps of charge density or something like it (since he was originally trying to solve problems like the bound states of electrons in the hydrogen atom). Born was the one who realized that $| \psi (x, t) |^2$ (the squared modulus of the wave function) represented the probability of finding a particle at a particular position $x$ at a particular time $t$. I don't know exactly when the term "probability amplitude" was invented to describe $\psi$ itself, but the concept was there as soon as Born introduced his interpretation.

This Wikipedia article gives a brief summary (and references the original paper by Born that introduced this interpretation):

https://en.wikipedia.org/wiki/Born_rule

 

[Buzz Bloom]

I would like to ask who was the scientist that introduced this concept.

Hi DaTario:
I recommend the book "What is Real" by Adam Becker. It is an excellent history of the development of QM, going back to the 1920s As I remember it, (perhaps incorrectly) it was either Schrodinger or Heisenberg.

Regards,
Buzz

 

[DaTario]

What you call a "tool" is an interpretation. The "tool" is a Hilbert space. As you do not want to talk about inner products and orthonormal basis, I suggest to restrict your question to physics, where it makes sense. From a mathematical point of view, it is still unclear what you want to know. There is no mathematics involved so far.

But don´t you agree that when we use $\psi$ the probability formalism is conducted differently? Classical probability sums probabilities when facing alternatives like "from this window OR from that window" (evaluating the probability of entering a ball in my room and brake a statue). With amplitudes we are told to sum the amplitudes of probabilities of the ball entering the room from both windows and then square the sum (which, BTW, produces the interference terms). I am not saying that mathematics uses amplitudes, but as it is probability theory in some sense, I decided to leave it open in the OP.

 

[DaTario]

Hi DaTario:
I recommend the book "What is Real" by Adam Becker. It is an excellent history of the development of QM, going back to the 1920s As I remember it, (perhaps incorrectly) it was either Schrodinger or Heisenberg.

Regards,
Buzz

Thank you, Buzz Bloom!

 

[DaTario]

As far as I know, it was Max Born, who introduced the probabilistic interpretation of the Schrodinger Equation. Schrodinger originally viewed the wave function $\psi (x, t)$ that appeared in his equation as describing a real wave, perhaps of charge density or something like it (since he was originally trying to solve problems like the bound states of electrons in the hydrogen atom). Born was the one who realized that $| \psi (x, t) |^2$ (the squared modulus of the wave function) represented the probability of finding a particle at a particular position $x$ at a particular time $t$. I don't know exactly when the term "probability amplitude" was invented to describe $\psi$ itself, but the concept was there as soon as Born introduced his interpretation.

This Wikipedia article gives a brief summary (and references the original paper by Born that introduced this interpretation):

https://en.wikipedia.org/wiki/Born_rule

Thank you, Peter.

 

[DaTario]

What you call a "tool" is an interpretation. The "tool" is a Hilbert space. As you do not want to talk about inner products and orthonormal basis, I suggest to restrict your question to physics, where it makes sense. From a mathematical point of view, it is still unclear what you want to know. There is no mathematics involved so far.

I would say that the Hilbert space is the tool box.

 

[fresh_42]

... but as it is probability theory in some sense, I decided to leave it open in the OP.

That's why I asked in the first place: we have probability theory, measure theory, functional analysis all being mathematically involved. However, it is not a mathematical term or even of mathematical interest. The concept behind is a certain Hilbert space (late 19th, early 20th century). If you want to tie it to mathematics, you will need to specify a mathematical relevance. The origins of probability theory date back to the 16th and 17th century - and only if we disregard ancient treatments about gambling, which are far older.

 

[DaTario]

That's why I asked in the first place: we have probability theory, measure theory, functional analysis all being mathematically involved. However, it is not a mathematical term or even of mathematical interest. The concept behind is a certain Hilbert space (late 19th, early 20th century). If you want to tie it to mathematics, you will need to specify a mathematical relevance. The origins of probability theory date back to the 16th and 17th century - and only if we disregard ancient treatments about gambling, which are far older.

I would like to emphazise that I have not said the math makes use of amplitudes. I have just left room for the answer be a mathematician.

 

[PeterDonis]

don´t you agree that when we use $\psi$ the probability formalism is conducted differently?

I think a better way of putting it is that you use a different formalism from "the probability formalism". Probabilities themselves in QM behave the same as probabilities in any other branch of science. The difference is in how probabilities are computed in QM, since those computations involve probability amplitudes and vectors in Hilbert spaces, which are mathematical tools that are not really used outside QM.

I am not saying that mathematics uses amplitudes

The math of QM most certainly does make use of amplitudes; it has to, since it's using $\psi$ and the Schrodinger Equation.

 

[fresh_42]

I would like to emphazise that I have not said the math makes use of amplitudes. I have just left room for the answer be a mathematician.

In this case, the spark could have been Fredholm's paper (1900) about integral equations $\varphi(s)=f(s)+\int K(s,t)f(t)\,dt\;$ followed by Hilbert, who investigated symmetric kernels, which provided easier solutions to Fredholm's equations.

Sorry, my book only tells the story and doesn't name a specific source in this case.

 

[PeterDonis]

@DaTario you might also want to read Scott Aaronson's lecture about QM and probability theory:

https://www.scottaaronson.com/democritus/lec9.html

 

[DaTario]

@DaTario you might also want to read Scott Aaronson's lecture about QM and probability theory:

https://www.scottaaronson.com/democritus/lec9.html

Thank you, once more, Peter.

 

[PeterDonis]

Thank you, once more, Peter.

You're welcome!

 

[fresh_42]

This is again an example how different developments led to specific results. It started with linear algebra and differential equations. Some geniuses saw parallels and patterns and developed functional analysis, and Hilbert spaces, and now we have QED and QCD. The years between say 1890 and 1940 or so have been in my opinion the most productive ones in recent history of math and physics.

 

[DaTario]

Thank you all. Now just a silly comment: In trying to understand the need for probability amplitudes I often try to appeal to some examples like this one:

Suppose you have a snack room in your workplace and it has one of those machines that delivers chocolate bar, coca cola, peanuts, candy bar and bottles of water. These five products are displayed side by side and all one has to do is to insert a coin and choose the product by a selecting its number. Supose this room has two doors (see figure). During some time only door 1 was opened. And a mean pattern of selection of those snacks was reported (fig. a). Then someone closed door 1 and opened door 2. A similar patern was reported, with some differences (the employees took more coke than before, but every product was taken at least one time. Now one opens the two doors and we observe that some products were supressed from the list of wanted snacks (fig c). By offering an extra way to find the machine, the only reasonable effect would be an increase in the frequency of choice of some of the snacks. The decrease of the frequency is an effect not explicable by classical probability theory as long as we keep believing in the randomicity of these events. The use of amplitudes allows for either the decrease or the supression of the frequency of events when more ways for them to happen are provided.