What is Probability amplitudes: Definition and 21 Discussions

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding, and the probability thus calculated is sometimes called the "Born probability". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. 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.

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  1. AshIsH_0001

    Probabilities out of non-normalizable functions?

    a and b were fairly easy to solve; but the c part which actually demands the probability! How are we suppose to fetch the value if the function can't even be normalized; I tried to make some assumptions like making the system bounded; but I don't think that it's the right way to do so... What...
  2. Ahmed1029

    I Can I find a particle in two states simultaneously?

    If I want to get the spin angular momentum of a particle using the Stem-Gerlach experiment, I think I will find the spin 1/2 particle either spin up or spin down, but not both. I however want to ask this : Is there a non-zero probability that a particle which is spin-up in the z direction to be...
  3. entropy1

    I Why are there no probability amplitudes in MWI?

    If I consider the MWI, one of the notions for what happens during measurement is that the initial wavefunction, if I use Dirac notation and two dimensions, ##|A\rangle+|B\rangle## undergoes the transformation ##(|A\rangle+|B\rangle)|E_{before}\rangle \rightarrow...
  4. I

    What is the Relationship Between Probability Amplitudes and the Sum of Terms?

    I am not sure what I can do with the equation. I realize that ## \vert c_1 \vert ^2 = \vert c_2 \vert ^2 = \frac{1}{2} ## does not mean that ## c_1 ^2 = c_2 ^2 = \frac{1}{2} ## or that ## c_1 = c_2 ##, so I don't know how to use it. I think ideally I might have something like ##P = \vert c_1...
  5. DaTario

    I Probability Amplitudes and the History of Science

    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
  6. W

    I Time-dependence in probability amplitudes

    Hi all, I am rather confused about the following concept. Assistance is greatly appreciated! A time-dependent probability amplitude can be written as $$\langle a_k| e^{-\frac{i}{\hbar}\hat{H}t} |\psi\rangle$$ where ##a_k## is an eigenvalue. Suppose I want the x-representation of the ket, I can...
  7. redtree

    I Normalization and the probability amplitude

    Given two probability amplitude wavefunctions, one in position space ##\psi(r,k)## and one in wavenumber space ##\phi(r,k)##, where ##r## and ##k## are Fourier conjugates, how is it possible for the modulus squared, i.e., probability density, of BOTH wavefunctions to be normalized? It seems...
  8. redtree

    A Deriving Probability Amplitude from Markov Density Function

    1. Given a Markov state density function: ## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ## ##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
  9. Z

    Why wave function is probability of particle existence

    Homework Statement I post here to check if I am in the right way to understand this point in the book. The wave function of free particle is ##Ae^{\frac{i}{\hbar}(px-Et)}##.This could be regarded as ##{\phi}(x,t)=Ae^{\frac{i}{\hbar}S(x,t)}##. ##S(x,t)## is the free particle's least action...
  10. I

    B How does a Bell State lead to a probability in the CHSH game

    Hi, I am a student in the Netherlands, and I'll be attending university next year. However, I am doing some form of research on Quantum Computing with another student for our so-called "profielwerkstuk" but my understanding of Quantum Physics and math is sometimes not at the level that is...
  11. DaTario

    A Amplitudes of probability in Mathematics before QM

    Hi All, Was there any use of the concept of amplitudes of probability before their use in quantum mechanics? In connection to this question, who invented or was the first to use this resource? Best wishes, DaTario
  12. N

    Probability amplitudes & light / particle wavelengths

    So this is basic question but the more I read the more I am confusing myself! I was assuming that the wavelength of a photon was the same wavelength as the associated probability amplitude (although a complex number). So to make constructive interference it means one path takes say ten...
  13. B

    Elektromagnetic field ~ probability amplitudes

    Given an electromagnetic field by its components E and B. How is this related to probability amplitudes of a Schrödinger wave function for the same field. Trying the same or at least a similar question from a different angle: given the E and B field, can we derive from it, in principle, not...
  14. C

    Why Does Quantum Mechanics Use Complex Probability Amplitudes?

    Im new to quantum mechanics and prof. James Binney writes the probability amplitude of going through two paths s and t, is the mod square of A(s)+A(t). Then he writes this as The mod square of A(s)+ mod square of A(t) + A(s)A(t)* + A(t)A(s). Why is it expanded like this? Could someone please...
  15. nomadreid

    Short question about probability amplitudes

    When one says the probability amplitude for a quantity when it has probability P, which one does one take, the a+bi or the a-bi?
  16. P

    Probability amplitudes and photons

    Suppose a Fock state contains 2 photons, both in the same spacetime mode and having the save (vertical) polarization. So we can write this state as |2>, or, if we want to emphacize its vertical polarization, we may write |2v> or |2v,0h>. Suppose now we want to measure polarization in the...
  17. S

    Time independence of probability amplitudes

    What's wrong with the following: \frac {d \left< E_{k} | \Psi \right>}{dt} = \frac{\partial \left< E_{k} \right|}{\partial t} \left| \Psi \right> + \left< E_{k} \right| \frac{\partial \left| \Psi \right>}{\partial t} = \frac{i}{\hbar} \left< E_{k} | H | \Psi \right> - \frac{i}{\hbar} \left<...
  18. B

    Unraveling the Mystery of Probability Amplitudes

    Homework Statement What physical phenomenon requires us to work with probability amplitudes rather than just with probabilities, as in other fields of endeavour? Homework Equations The Attempt at a Solution Not sure. The wording of the q throws me... phenomenon i.e...
  19. redtree

    Probability amplitudes, de Broglie and Schrödinger

    What is the relationship between the "matter waves" described by de Broglie, the probability amplitude function and Schrödinger's wave equation? I've read the following: "The wavelengths postulated by de Broglie to be associated with the motions of particles are in reality the wavelengths...
  20. Y

    How Does the Uncertainty Principle Relate to the Number of Waves in a Box?

    I have this problem: The total number of waves Nw in a box is somewhat uncertain beause of the way the amplitude falls off. For a region of size DeltaX, call the uncertainty in the number of waves DeltaNw. a.) Relate DeltaNw to the uncertainty principle in the wavelength Delta(lambda)...