What is Wavefunction: Definition and 577 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.

View More On Wikipedia.org
Recent contents Top users
  1. P

    A Graphene wavefunction expressed in tight binding form

    In the framework of tight binding approximation, does the wavefunction for atom A (or B) has two spinorial components(2 components) in "real space"? If so how does this spinorial component propagate in the graphene?
  2. J

    I Correlation between Symmetry number & Total wavefunction

    Some rotational quantum states are not allowed for a rotating particle. At quantum level, these "forbidden" quantum states is based on the requirement of the total wavefunction being either symmetrical or anti-symmetrical, depending on whether the particle is a fermion or boson. The particle's...
  3. 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...
  4. allisrelative

    Doesn't Wigner's Friend Experiment solve the measurement problem?

    If you look at the recent Wigner's Friend experiment, it seems to support Carlo Rovelli's Relational Interpretation which says there's no real measurement. Wiger's Friend carries out a polarization measurement. Before he does, the quantum system is in a superposition of horizontal/vertical...
  5. P

    I Prove that the norm squared of a superposition of two states is +ve

    This is what I have so far: $$ |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + \alpha^*\beta\Psi_1^*\Psi_2 + \alpha\beta^*\Psi_1\Psi_2^* $$ $$=> |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + 2Re(\alpha^*\beta\Psi_1^*\Psi_2) $$ I am...
  6. iymasomhumaan

    B Nuclear wavefunction and Bose-Einstein condensates

    Einstein condensate state or ultra hot fully ionized, removed of all-electron, plasma that is compressed like possibly something like fusion, state (an extraordinarily unattainable state currently; like compressed air into a liquid, but a solid; while even metallic hydrogen is, as far as I know...
  7. Kaguro

    Finding the potential function from the wavefunction

    I would differentiate this twice and plug it into the S.E, but for that I'll need E. Which I don't have. Please provide me some direction.
  8. bob012345

    I Lithium Atom Ground State Radial Wavefunction

    I would like to see what the shape of the ground state radial wavefunction for the Lithium atom is. An approximate function that shows the shape would be fine. Thanks.
  9. K

    I Understanding Nuclear Rotation: Quantum Numbers and Wavefunctions

    Hello! I am a bit confused by the quantum numbers used to describe the rotation of a nucleus. In Wong's book these are J, M and K, which represent the rotational quantum number, its projection along the lab z-axis and its projection along the body intrinsic symmetry axis, respectively. However...
  10. I

    Finding meaning in the Phase of the wavefunction

    Suppose ##\psi_{0}## is a properly normalised wavefunction with ##<x_{\psi_0} >## = ##x_{0}## and ##<p_{\psi_0} >## = ##p_{0}##. Define a new wavefunction ##\psi_{new}##(x) = ##e^{{-iqx}/{\hbar}}## ##\psi_{0}## What is the expectation value ##<\psi_{new}>## in the state given by...
  11. A

    I Master` Equation for the Penrose-Diosi wavefunction collapse

    I'm trying to understand equation 209 on page 81 here: https://arxiv.org/pdf/1204.4325.pdf Here's what I understand so far: - We are to imagine a system in a superposition of |X> and |X'>, where these describe distinct particle configurations. - On the RHS of the eqn, the first term is...
  12. Denselight93

    B The Virtue of Individual Perception RE: Wavefunction Collapse

    Hey guys! New to the forums so unsure if this is exactly the right place to present this idea I have. I hope that you will find it interesting. I am all too familiar with claims of quantum consciousness woo. I am also aware that the role of 'wave function collapse', if such a thing even...
  13. E

    When does the wavefunction wavelength equal the De Broglie wavelength?

    For instance, in the case of the infinite square well, the wavelength of the wavefunction is \frac{2L}{n}. This also turns out to be the De Broglie wavelength, and and we can find the possible energies directly from the Schrodinger equation, or by using the De Broglie relations. However, if the...
  14. olgerm

    I Is the wavefunction always an analytic signal?

    Is wavefunction always analytic signal? Is imagnigray part of wavefunction hilbert transform of real part of wavefunction?
  15. DoobleD

    I How to get the wavefunction of a single particle in QFT?

    Hi folks, I'm trying to get a grasp on some of the basic concepts of QFT. Specifically, I'm trying to picture what are the actual fields of QFT and how they relate to wavefunctions. There are already many helpful posts about those concepts, here and in other places, but some points are fuzzy...
  16. Erik Ayer

    I Does downconversion cause pump photon wavefunction collapse?

    I saw a paper on an experiment where a pump beam first went through a double-slit, then was downconverted with BBO. Recently a friend with a PhD in quantum physics said the downconversion will cause the pump's wavefunction to collapse and the implication for this experiment is that the pump...
  17. B

    Fourier transformation of the Wavefunction in QM

    Hello Physics Forum, I am not sure what to to in this task, because the wavefunction is only given as A_0. Maybe someone can explain it to me. Thanks in Advance, B4ckflip
  18. F

    I How to Determine a Photon's Wavefunction After it Collapses?

    Suppose one measures the position of a photon without destroying it. From my understanding, the wavefunction of the photon should collapse, and will return to a more spread out state over time. How would one calculate this, specifically the rate at which the wavefunction spreads out from the center?
  19. W

    Need help regarding the derivation of a 2-particle wavefunction

    I have an issue trying to understand the derivation of equation 3.40 (screenshot attached) of Blundell's QFT book. Here's my attempt. ##| x,y \rangle = |x\rangle \otimes |y\rangle = \Big( \int dp' \phi_{p'}(x)|p'\rangle \Big) \otimes \Big( \int dq' \phi_{q'}(y)|q'\rangle \Big)## which gives...
  20. dipanshum

    Probability of being in a state is given, Find the normalised wavefunction

    Should I treat ψ1 as ψ and ψ 2 as ψ*?
  21. QuarkDecay

    I How to know where the up and down spin go in the wavefunction?

    We are given the wave function with spin, but it doesn't say in which Ylm each spin X± goes. So how do I know? Examples; (1) Ψ = 1/√3 R21(r) ( Y10 √2Y11 ) Here we have the up Spin X+ to Y10 and the X- to Y11 I notice the X- went to...
  22. A

    A Do QED effects make a huge change to the position of the electrons?

    In https://en.wikipedia.org/wiki/Lamb_shift about the lamb shift, it's mentioned that the change in the electron's frequency due to QED effects (vacuum polarization and self-energy correction) is about 1 GHz, which would translate to an energy change of hf = 6.63E-25 J. This is 3E-7 times of the...
  23. A

    A Do we need stochasticity in a discrete spacetime?

    Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a...
  24. A

    I How does the collision between two atoms work?

    Considering the quantum mechanical model for an atom, what exactly happens when two atoms (say, two Ca2+ ions in a Brownian motion) collide with each other? As I know, this collision is not like a regular elastic or inelastic collision between two macroscopic objects. Is it mainly due to the...
  25. A

    A Does Antony Valentini's "sub-quantum measurement" really work?

    In https://arxiv.org/pdf/quant-ph/0203049.pdf, which is in the realm of Bohmian mechanics, Antony Valentini claims that by having a "non-equilibrium" particle with arbitrarily accurate "known" position, we can measure another particle's position with arbitrary precision, violating Heisenberg's...
  26. fluidistic

    A Is the wavefunction subjective? How?

    I have read Lubos Motl blogposts (https://motls.blogspot.com/2012/11/why-subjective-quantum-mechanics-allows.html and https://motls.blogspot.com/2019/03/occams-razor-and-unreality-of-wave.html) stating that the wavefunction is subjective. This means that it is perfectly valid that two different...
  27. S

    Traveling Wavefunction: Differentiating & Relating $\Psi$

    I am not sure whether I have differentiated correctly $$\frac{\partial\Psi(x-ut,t)}{\partial t}=-u\frac{\partial\Psi(x-ut,t)}{\partial(x-ut)}+\frac{\partial\Psi(x-ut,t)}{\partial t}$$ and $$\frac{\partial^2\Psi(x-ut,t)}{\partial x^2}=\frac{\partial^2\Psi(x-ut,t)}{\partial(x-ut)^2}$$ So, we...
  28. A

    A Why is this Pilot-wave model on a discrete spacetime stochastic?

    Look at the paper in the link below: https://link.springer.com/content/pdf/10.1007%2Fs10701-016-0026-7.pdf It introduces a pilot-wave model on a discrete spacetime lattice. However, the pilot-wave model is not deterministic; the motion of quantum particles is described by a |Ψ|^2-distributed...
  29. A

    A What are Bohmian trajectories for a free electron?

    A free electron, or any other quantum particle, has an uncertain position/momentum, according to Heisenberg uncertainty principle. The squared amplitude of the wavefunction determines the probability of finding the electron at any point of the space. Accordingly, atomic orbitals are attributed...
  30. S

    Solving Particle Motion in a Rigid Box: t=0

    I have question, how can I solve problem of particle in rigid box when one of the wall gets completely destroyed? At time t = 0 the right wall of box gets completely destroyed, left wall is still here( ψ(0) = 0 ), also at t = 0 we know that particle is in ground state. How can I search for...
  31. A

    I What is “quantized momentum transfer” and can it account for interference patterns?

    In https://www.sciencedirect.com/science/article/pii/S0378437109010401, the author claims that the interference pattern obtained in the double-slit experiment does not need a wave description of matter, and can be accounted for by the "quantized momentum transfer" from the slits to the electron...
  32. Arman777

    I Understanding the Role of k in the Wavefunction of Quantum Systems

    I am reading a textbook quantum physics by stephen gasiorowicz. And he defines a wavefunction in this form, $$Ψ(x,t)=\int_{-∞}^{∞}A(k)e^{i:(kx-ωt)}dk$$ I did not understand why its a function of ##k## or why even we are taking integral with respect to ##k## ? Is ##k## actually means momentum...
  33. Arman777

    Finding the Wavefunction for Tunneling,with tunnel lenght L

    Homework Statement Let us suppose we have a particle with energy ##E## and ##E<U## and the potential defined as ##U(x)=0## for ##x<0## (I) ##U(x)=U## for ##0<x<L## (II) ##U(x)=U_0## for ##x>L## (III) In this case ##E>U_0## and ##U>U_0## Homework Equations $$HΨ=EΨ$$ The Attempt at a...
  34. D

    I Derivatives of a normalizable wavefunction

    Hi. In infinite volume a normalizable wavefunction → 0 as r or x,y,z→ 0 but do all the derivatives and higher derivatives → 0 as well ? Thanks
  35. D

    Number operator on wavefunction

    Homework Statement Consider the state $$\psi_\alpha = Ne^{\alpha \hat a^\dagger}\phi_0, $$ where ##\alpha## can be complex, and ##N = e^{-\frac{1}{2}|\alpha|^2}## normalizes ##\psi_\alpha##. Find ##\hat N \psi_\alpha##. Homework Equations $$\hat N = \hat a^\dagger \hat a$$ $$\hat a\phi_n =...
  36. D

    I Evolution of the wavefunction

    Hi . For a system such as an infinite well or a harmonic oscillator if the energy is measured and it returns a value , say E1 corresponding to the ket | 1 > then this evolves according to exp( -iE1t/ħ) | 1 >. So this means that for any time >0 a measurement of the system will always give the...
  37. Mathfan7

    B How to determine the wavefunction of photons?

    Hi, I'm sorry if this question has already been answered somewhere and I'm just too incompetent to find it, buuut: As the title already says, I really do not get that part of quantum physics (if you can even say I'm getting ANY part at all...). As I searched all Google for an answer I just...
  38. D

    Finding Stationary Wavefunction with a Line Potential

    Homework Statement A particle of mass m in one dimension has a potential: $$V(x) = \begin{cases} V_0 & x > 0 \\ 0 & x \leq 0 \end{cases} $$ Find ##\psi(x)## for energies ##0 < E < V_0##, with parameters $$k^2 = \frac{2mE}{\hbar^2}$$ and $$\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$$...
  39. Warda Anis

    Expectation value <p> of the ground state of hydrogen

    Homework Statement How should I calculate the expectation value of momentum of an electron in the ground state in hydrogen atom. Homework Equations The Attempt at a Solution I am trying to apply the p operator i.e. ##-ihd/dx## over ##\psi##. and integrating it from 0 to infinity. The answer I...
  40. D

    I The total derivative of a wavefunction

    I have read that the integral of d3x ∇(ψ*ψ) is zero because the total derivative vanishes if ψ is normalizable. Does this mean that the integral of d3x ∇(ψ*ψ) is ψ*ψ evaluated at the limits where ψ is zero ? Thanks
  41. F

    I Radiofrequncy Photon Wavefunction

    Radio waves are usually not viewed as streams of photons but according to quantum mechanics that is exactly what they are. But what does the wavefunction of an RF photon look like? If we consider a dipole radiator, say of 10 Mhz, that emits a single photon, my guess is that the wavefunction of...
  42. P

    I Why do you need infinite size matrix which commute....

    ...to give a number? https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes5.pdf On page 6, it says, " Matrix mechanics, was worked out in 1925 by Werner Heisenberg and clarified by Max Born and Pascual Jordan. Note that, if we were to write xˆ...
  43. S

    A Size of nuclei wave function in a crystal

    I need to know what is the typical extention of the (spatial) wavefunction of an atomic nucleus in a crystal, in particular I am interested to the case of a Germanium cristal. Please together with the actual number of the size of the nuclei wavefunctions, let me know the references (articles or...
  44. MichPod

    I Is information lost in wavefunction collapse?

    Having in mind the idea that "information is not lost" (referring to the black hole information paradox), is not the same rule violated in the wave function collapse? I.e. during the decoherence process information is not lost as this process of entanglement of some object with its environment...
  45. J

    Measuring momentum using position wavefunction

    I was solving an exercise from Cohen's textbook, but then I got stuck in this question. "Let ψ(x,y,z) = ψ(r) the normalized wave function of a particle. Express in terms of ψ(r) the probability for: b) a measurement of the component Px of the momentum, to yield a result included between p1...
  46. M

    Quantum mechanics: potential steps

    Homework Statement The Attempt at a Solution [/B] Hi All, I'm having trouble answering part (f) of the above question. I have managed parts (d) and (e) fine but am not sure how to proceed with part (f). I am pretty sure that the amplitude of the reflected wave in region 1 will be zero...
  47. P

    I What's the significance of HUP if Ψ collapses?

    If I understand it correctly, a particle doesn't have a definite momentum and a definite position, but is in a superposition of multiple positions and momenta. And when we measure either of the two quantities, say, position, the wavefunction collapses to tell us where the particle is. Now when...
  48. fluidistic

    I How to pick an operator onto which the wavefunction collapses

    Imagine a system of 1 particle in a superposition of eigenstates of some operator(s). If one were to make a measurement of a property of that particle, how is the operator (or observable) "picked" so that the wavefunction collapses into an eigenstate of said operator? In other words, how do one...
  49. E

    I Qualitative plots of harmonic oscillator wave function

    For the harmonic oscillator, I'm trying to study qualitative plots of the wave function from the one-dimensional time independent schrodinger equation: \frac{d^2 \psi(x)}{dx^2} = [V(x) - E] \psi(x) If you look at the attached image, you'll find a plot of the first energy eigenfunction for...
  50. F

    I Does measuring an atom collapse the wavefunction of its parts?

    Suppose you have an experiment that measures the property of an atom as a whole, maybe you can put it through a double-slit or measure its spin, whatever. Presumably that will collapse the wavefunction that you used to describe the atom in that experiment. Would this entail that in the process...
Back
Top