What is Wave function: Definition and 873 Discussions

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively).
The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.
For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).
According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves.In Born's statistical interpretation in non-relativistic quantum mechanics,
the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured—its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.

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  1. Tony Hau

    I The derivative of the complex conjugate of the wave function

    It is a rather simple question: In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$ $$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...
  2. JD_PM

    Computing a wave function through a (non-relativistic) propagator

    We know that the non-relativistic propagator describes the probability for a particle to go from one spatial point at certain time to a different one at a later time. I came across an expression (lecture notes) relating ##\Psi(x,t)##, an initial wave function and the propagator. Applying the...
  3. E

    A Feynman solution for the radial wave function of the hydrogen atom

    Reading the classical Feynman lectures, I encounter the formula(19.53) that gives the radial component of the wave function: $$ F_{n,l}(\rho)=\frac{e^{-\alpha\rho}}{\rho}\sum_{k=l+1}^n a_k \rho^k $$ that, for ##n=l+1## becomes $$ F_{n,l}=\frac{e^{-\rho/n}}{\rho}a_n\rho^n $$ To find ##a_n## I...
  4. Martian2020

    B Does the wave function change from the viewpoint of the photon?

    The question arose when watching Sean Carroll video: The Biggest Ideas in the Universe _ Q&A 6 - Spacetime 3:50 - 13:30 Because photons follow null geodesic in spacetime the question arose from viewers: "photons do they really experience no time this is a question" And in the answer: "but if...
  5. H

    I Are electro-magnetic waves the same as the wave function?

    When studying classical mechanics we are told that light is the propogation of electromagnetic waves. This makes perfect sense, as I can imagine these fields behaving this way, and in turn have an associated wave length. When learning about QM, I have heard that the wavelength of a (any)...
  6. I

    I Quantum Mechanics Wave Function in 3D

    I was wondering if it's possible to plot a wave function that is a function of 3 coordinates, such as (x, y, z). The text my class uses calls this Quantum Mechanics in 3 dimensions, but wouldn't this technically by four dimensions?
  7. danielhaish

    I Why quantum wave function collapse is not consider a signal?

    I read about the non-communication theorem and I understand why when changing one practical will not change the other . But suppose that there is two observers that doing the double slit experiment, but using it with two entanglement practicals. observer one should send signal of yes or...
  8. T

    Finding the wave function when given the momentum eigenstate

    For part a, I have the following $$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$ but I am totally lost on how to proceed.
  9. I

    I Complex Conjugate of Wave Function

    I've been studying quantum mechanics this semester in school and have ran into an issue I can't find an answer for. I understand why we take the complex conjugate of the wave function, such as when calculating expectation values. I'm a little confused though as to why we take the complex...
  10. omegax241

    A strange wave function of the Hydrogen atom

    I am trying to solve the following exercise. In a H atom the electron is in the state described by the wave function in spherical coordinates: \psi (r, \theta, \phi) = e^{i \phi}e^{-(r/a)^2(1- \mu\ cos^2\ \theta)} With a and \mu positive real parameters. Tell what are the possible values...
  11. F

    I Why QFT still goes well while it lacks the notion of wave function?

    In QM by virtue of wave function we calculate any things. But in QFT it seems that there is a lacking of notion of wave function.I do not understand why QFT still goes well(it is a good theory to calculate any things)?
  12. jjson775

    Normalization of the wave function for the electron in a hydrogen atom

  13. StevieTNZ

    B "Test of wave function collapse suggests gravity is not the answer"

    https://phys.org/news/2020-09-function-collapse-gravity.html An interesting article I saw yesterday. However, both Nature articles (the summary [https://www.nature.com/articles/s41567-020-1026-2] and the actual technical paper [https://www.nature.com/articles/s41567-020-1008-4]) are behind a...
  14. J

    The collapse of the wave function?

    I never took any physics courses nor don't have a background in mathematics never the less I became very interested in quantum physics after reading Sean Carroll's book Something deeply hidden. One of the difficult things for me to wrap my head around was the concept of superposition and...
  15. R

    I How do you normalize this wave function?

    I have a basic question in elementary quantum mechanics: Consider the Hamiltonian $$H = -\frac{\hbar^2}{2m}\partial^2_x - V_0 \delta(x),$$ where ##\delta(x)## is the Dirac function. The eigen wave functions can have an odd or even parity under inversion. Amongst the even-parity wave functions...
  16. J

    Question about an Eqn. in Shankar - wave function probability

    I don't see why it is not ##P(\omega)\propto |\langle \psi | \mathbb{P}_{\omega}|\psi\rangle |^2.## After all, the wavefunction ends up collapsing from ##|\psi\rangle## to ##\mathbb{P}_{\omega}|\psi\rangle.##
  17. wolfy

    B When the wave function collapses, how long is it collasped?

    When wave function collapses how long is it collasped... Shooting electrons at a double slit and observing the electrons before they reach the 2 slits collasped the wave function...so is its behavior particle like forever? Quantum mechanics is simple however wrapping ones head around it is...
  18. Terrycho

    I The wave function in the finite square well

    Hello! I have been recently studying Quantum mechanics alone and I've just got this question. If the potential function V(x) is an even function, then the time-independent wave function can always be taken to be either even or odd. However, I found one case that this theorem is not applied...
  19. G

    A Quantum Mechanic/wave function/junction condition

    May i know how do i eliminate C and D and how do i obtain the last two equations? Are there skipping of steps in between 4th to 5th equation? What are the intermediate steps that i should take to transit from 4th equation to the 5th equation?
  20. Zouatine

    What units for the wave function of a string?

    hello , hope all of you are doing well , i have question about the unit of the function of waves of string fixed in both boundary , the function of waves is function of two variables x and t , so it's function describe the displacement in function of place and time , Ψ(x,t)=φ(x)*sin(ωt+α)...
  21. S

    I Splitting of a one-particle wave function

    Hello all, I am a newcomer here. Not a physicist, just an enthusiast. ;) I was thinking whether it is possible to separate a one-particle wave function into two, "completely disjoint" parts. The following thought experiment explains better what I am thinking about. Let us suppose, that there...
  22. CrosisBH

    Computing the wave function of a square potential

    The book's procedure for the "shooting method" The point of this program is to compute a wave function and to try and home in on the ground eigenvalue energy, which i should expect pi^2 / 8 = 1.2337... This is my program (written in python) import matplotlib.pyplot as plt import numpy as...
  23. Zack K

    Infinite Square Well with polynomial wave function

    Some questions: Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well. Since we have no complex components. I am guessing that the ##\psi *=\psi##. If...
  24. hilbert2

    A Can Bloch Waves Reveal Periodic Potentials in Quantum Mechanics?

    I was thinking about a problem I had considered a long time ago in some thread, finding an example of a wave function ##\displaystyle \psi (x) =e^{iax}\phi (x)## with ##\displaystyle\phi (x)## being periodic with period ##\displaystyle L## and the corresponding Schrödinger equation...
  25. TopologyisGeometry

    Plotting the radial wave function of Deuteron in a finite well

    To plot ##u(r)## we need to find the solutions for each region. Which is in the relevant equations part. Now, I have to do this numerically. Using python 3.7 I made an ##u## which is filled with zeros and a for loop with if/elseif statement, basically telling it to plot values for whenever...
  26. Z

    Spatial Wave function of two indistinguishable particles

    Hi, I just need someone to check over my work. I am having trouble with the next part of this question and I just wanted to check that this part was correct first. I have two particles in an infinite square well (walls at x=0 and x=L). I need write an expression for the spatial wave...
  27. G

    Find the time-dependent wave function Ψ(x, t).

    I thought I could start somewhere along the lines of ##\psi(x,t)= \psi(x,0)e^{-iE_nt/\hbar}##, but I'm not sure what ##E_n## would be. I also thought about doing the steps listed below in the picture, but I'm not sure how to decompose ##\psi(x,0)## like it says to in the first step. Any help...
  28. G

    What is the complex conjugate of this wave function?

    I was planning to find the value of N by taking the integral of φ*(x)φ(x)dx from -∞ to ∞ = 1. However, this wave function doesn't have a complex number so I'm not sure what φ*(x) is. I was thinking φ*(x) is exactly the same φ(x), but with x+x0 instead of x-x0. Thank you
  29. F

    I Macroscopic object wave function

    A composite object made of many atoms has a large mass hence a small de Broglie wavethength...and we know that recent experiments succeeded to obtain interference patterns even for such objects (for instance the C60 molecule). Did theoretician understood how a wavefunction with such a small...
  30. P

    I Collapse of the wave function under simultaneous measurement

    Considering Bell’s theorem and the expected correlations between entangled particles or photons. In a measurement setup e.g. Like Alain Aspect‘s with 2 entangled photons. If we could make a setup that guarantees that the measurement on both photons is done at exactly the same moment, what...
  31. Quantum Alchemy

    I Questions about QFT and the reality of subatomic particles

    I've been reading about Quantum Field Theory and what it says about subatomic particles. I've read that QFT regards particles as excited states of underlying quantum fields. If this is the case, how can particles be regarded as objective? It seems to me that this also removes some of the...
  32. DEvens

    I Wave function of a laser beam?

    Summary:: Wave function of a laser beam before it hits the diffraction grating So I'm reading "Foundations of Quantum Mechanics" by Travis Norsen. And I've just read Section 2.4 on diffraction and interference. And he derives a lovely formula for the wave function of a particle after it leaves...
  33. T

    I Equal or larger/smaller versus larger/smaller in boundary conditions

    Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...
  34. Haynes Kwon

    I Fourier Transform of the Wave function

    Given that the wave function represented in momentum space is a Fourier transform of the wave function in configuration space, is the conjugate of the wave function in p-space is the conjugate of the whole transformation integral?
  35. olgerm

    I Wave function for a helium atom

    Can you say whether I understood these things correctly? to get condition on wavefunction ##\Psi## for a system that consists of 2 electrons(without taking spin into account) and helium nuclei I can solve schrödinger equation: ##i*\frac{\partial \Psi}{\partial...
  36. Haynes Kwon

    I Why does the square of the amplitude of a wave function represent P?

    Born's postulate suggests if a particle is described a wave function ψ(r,t) the probability of finding the particle at a certain point is ψ*ψ. How does this work and why?
  37. dRic2

    I Wave function of a scattered particle and cross section

    In Landau-Lifsits's book about non relativistic QM it is said that if I have a particle described by a plane wave ##\phi = e^{ikz}## (I think he choses the ##z## direction for simplicity) the wave function after the scattering event is (far from the scattering event) $$\psi \approx e^{ikz} +...
  38. Haorong Wu

    How to calculate a wave function in time t

    I use the equation ##\psi \left ( x, t \right ) = e^{-iEt/\hbar} \psi \left ( x,0 \right )## to calculate ##\psi \left ( x , t \right)##, and the result is ##\psi \left ( x , t \right) = \frac 1 {\sqrt {2 \pi \hbar}} exp \left [ \frac {ip_0 x} {\hbar} - \frac {i p^2 t} {2m \hbar} \right...
  39. nomadreid

    I Quantum states: only vectors?

    Elementary question: Is there ever a case where the solutions for a wave equation turn out not to be a vector (in Hilbert space of infinite complex-valued dimensions, or a restriction to a subspace thereof) , but something else -- say, (higher-order) tensors or bivectors, or some such? My...
  40. H

    I Wave function when there is coupling between spin and position

    Why can't the general state, in the presence of coupling, take the form $$\psi_-(r)\chi_++\psi_+(r)\chi_-$$ where ##\psi_+(r)## and ##\psi_-(r)## are respectively the symmetric and anti-symmetric part of the wave function, and ##\chi_+## and ##\chi_-## are respectively the spinors representing...
  41. Lynch101

    B Entanglement & Wave Function Collapse

    This thread is to look at the notion of wave function collapse and relativity of simultaneity. The other thread I started on QFT has helped to clarify a lot, so hopefully this one can do the same. I may have this all wrong, but I will outline my question and hopefully someone can point out...
  42. Jamister

    I How can the photon wave function be described?

    Fermions such as the electron and proton can be described by wave function in momentum and in position, and it is possible to get the momentum wavefunction from space wave function and vice versa by Fourier Transform. what about photons? can photons be described by position wave function? If...
  43. L

    I Can we truly ignore the existence of quantum objects between measurements?

    We always think in terms of isolated particles. It's better to analyze it with solids. If wave functions were just calculational tools. Molecules like the following still interact by wave functions, right? So how can it be calculational tool? And if it is, then what model do you use to...
  44. L

    I Objective Wave Function and Non-locality

    In interpretations where the wave function represents something real, like Many worlds, Copenhagen with objective wave function and spontaneous objective collapses. I'd like to understand which of them has true non-locality. First. Is Many Worlds not having true non-locality due to the...
  45. E

    Compute the energy for 2D wave function with discontinuous derivatives

    I have calculated the normalization constant, but I'm struggling with the discontinuities in the derivatives of the wave function. Due to the symmetry, it should suffice to consider the first two cases. The results should be (according to WolframAlpha): \left( \frac{\partial^{2}}{\partial...
  46. QuasarBoy543298

    I Particle in free space - what happens to the wave function after measurement?

    If I'm trying to solve the problem of a particle in free space (H = P2/2m ). the eigenfunctions of the Hamiltonian cannot be normalized. now assume I have a legitimate wave function expressed in terms of the eigenfunction of H and I want to measure its energy. what will happen to the...
  47. J

    I Electron wave funtion harmonic oscillator

    As we see in this Phet simulator, this is only the real part of the wave function, the frequency decreases with the potential, so lose energy as moves away the center. we se this real-imaginary animation in Wikipedia, wave C,D,E,F. Because with less energy, the frequency of quantum wave...
  48. S

    The Eigenfunction of a 2-electron system

    Hello! I am stuck at the following question: Show that the wave function is an eigenfunction of the Hamiltonian if the two electrons do not interact, where the Hamiltonian is given as; the wave function and given as; and the energy and Born radius are given as: and I used this for ∇...
  49. Silvio Macedo

    I Wave function of particle / quantum field in space, also in time?

    Quantum fields have wave functions that determine a particle position in space. It solves non-locality, double-slit paradox, tunnel effect, etc. What if the wave function is also in time? Won't it solve the breaking of causality at quantum level? (Delayed Choice/Quantum Eraser/Time) Not much...
  50. T

    I Locality and Wave Function Collapse Implications

    OK, so I'm trying to work out a few ideas regarding locality. I've studied at the undergrad level in the past (including quantum), but with professors that slaved away at proving math constructs and never bothered to indulge in clarifying the context of any concepts, so I'm pretty weak here...
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