What is Wavefunctions: Definition and 145 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. J

    Quantum Mechanics homework question on wavefunctions

    Absolutely no clue on how to even begin this question due to the exceptionally poor quality of our lectures, who has also flatly refused to give out any solutions, which I could have used to understand what is going on. I assume the energy has to be obtained by using the eigenfunction equation...
  2. M

    A Normalization of Morse potential wavefunctions

    Hello! I am trying to use the wavefunctions of a Morse potential as defined in the link provided. They define a parameter ##z## and the wavefunctions are in terms of z. In my particular case, given their definitions, I have ##\lambda = 132.19377##, ##a=1.318 A^{-1}## and ##R_e = 2.235 A##. I am...
  3. D

    I Inner products and wavefunctions

    Hi As far as i am aware taking the inner product of a bra and a ket results in a (complex) number or scalar. So why does < x | ψ > give ψ ( x ) , the wavefunction in position space ? Surely < x | ψ > should give a number not a function ? Thanks
  4. CuriousLearner8

    A Many-Particle Wavefunction Question

    Hello, I hope you are well. I have been doing a lot of readings on the wavefunction and have a question I did not see asked anywhere else in these forums. I was wondering if someone could shed some light on this for me? I know the wavefunction is in 3N coordinate space and could be used to...
  5. Dario56

    I Inner and Outer Product of the Wavefunctions

    Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$ where $v_1,w_1...
  6. Q

    A Representation of Z2 acting on wavefunctions

    If I have a wavefunction ##\Psi(X)## that is invariant under the group ##Z_2##, what specifically does that mean? There can be several operators that are representations of the group ##Z_2##, for example the operators $$\mathbb{Z}_2=\{ \mathbb{I}, -\mathbb{I} \},$$ or $$\mathbb{Z}_2=\{...
  7. P

    A Phasing 2 Electrons' Wavefunctions

    Can anyone tell me how to mathematically represent 2 phased electrons mathematically? Thank you in advance.
  8. I

    I Fourier's Trick and calculation of Cn

    I understand that the solutions to the time-independent Schrodinger equation are complete, so a linear combination of the wavefunctions can describe any function (i.e. ##f(x) = \sum_{n = 1}^{\infty}c_n\psi_n(x) = \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} c_n\sin\left(\frac{n\pi}{a}x\right)## for...
  9. Jd_duarte

    I Orthonormal Basis of Wavefunctions in Hilbert Space

    Hello, I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...
  10. WeiShan Ng

    [QM] Two-Particle Systems: overlapping/non-overlapping wavefunctions

    Homework Statement Hi, I was reading Griffiths and stumble upon some questions. This is from 5.1.2 Exchange Forces. The section is trying to work out the square of the separation distance between two particles, $$\langle (x_1 - x_2)^2 \rangle = \langle x_1^2 \rangle + \langle x_2^2 \rangle -...
  11. O

    Wavefunctions and the de Broglie wavelength

    The position wavefunction makes a spatial probability amplitude wave right? And it is the combination of different frequencies My question is that if these frequencies are the spatial frequencies in the debrogile relations
  12. dumpling

    A On tempered distributions and wavefunctions

    Very often in standard QM books, certain states, like exponentially growing ones are rejected on the basis that they are not in L^2 space. On the other hand, scattered states are also not in L^2 spaces. This dichotomy can be repelled by using Rigged Hilbert spaces, and allowing tempered...
  13. rocdoc

    I Quantum Field Configurations and Wavefunctions

    Could anyone explain what a quantum field configuration is, and any relation this concept may have to the idea of a wavefunction? Perhaps for a scalar, quantum field?
  14. A

    I State Vectors vs. Wavefunctions

    Hi physicsforums, I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc. I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
  15. SemM

    I Non-Hermitian wavefunctions and their solutions

    I was wondering if anyone has worked with non-Hermitian wavefunctions, and know of an approach to derive real and trivial values for their observables using numerical calculations? Cheers
  16. K

    I Wavefunctions in first order degenerate pertubation theory

    Suppose we want to solve the Hamiltonian ##H=H_0+\lambda V## pertubatively. Let ##E_1,...,E_n## be the eigenvalues of ##H_0## and ##S_1,...,S_n## the eigenspaces that belong to them. In order to do that, one usually choses an orthonormal Basis ##|\psi_{i,j}>## of each ##S_i## with the property...
  17. B

    I The Relation Between Wavefunctions in Dirac Equation

    Can the wave function in four dimensions be expressed as e^i(kx+ky+kz-wt)?
  18. M

    I Measurement Values for z-component of Angular Momentum

    Given a wave function $$\Psi(r,\theta,\phi)=f(r)\sin^2(\theta)(2\cos^2(\phi)-1-2i*\sin(\phi)\cos(\phi))$$ we are trying to find what a measurement of angular momentum of a particle in such wave function would yield. Attempts were made using the integral formula for the Expectation Value over a...
  19. T

    Expectation of energy for a wave function

    Homework Statement At ##t = 0##, a particle of mass m in the harmonic oscillator potential, ##V(x) = \frac1 2 mw^2x^2## has the wave function:$$\psi(x,0)=A(1-2\sqrt\frac{mw} {\hbar} x)^2e^{\frac{-mw}{2\hbar}x^2}$$ where A is a constant If we make a measurement of the energy, what possible...
  20. W

    QM: Difference between these Initial Wavefunctions

    Homework Statement I've been asked as a part of some school project to find the Fourier transform, and time evolution of the following initial wavefunctions: 1. ##\Psi(x,0) = Ae^{\frac{-x^2}{2\sigma ^2}}## 2. ##\Psi(x,0) = Be^{\frac{-x^2}{2\sigma ^2}}e^{\frac{ipx}{\hbar}}## What physical...
  21. durant35

    I Expanding universe and quantum wavefunctions

    In a dark energy dominated universe, it seems that all the particles get away from each other and that the final state will be one with one or zero particles per horizon. This sounds very intuitive, but it is based on classical physics and GR. Particles have wavefunctions and this is whar...
  22. DeathbyGreen

    A Zero Energy Wavefunctions in 1D superconductor

    \bf{Setup} Hi! I am trying to derive the wavefunctions of the zero energy solutions of the Schrodinger equation in a 1D p-wave superconductor (Kitaev model). I am starting with the Hamiltonian $$ \begin{equation} H = \left[\begin{array}{cc} \epsilon_k & \Delta^{\ast}_k\\ \Delta_k & -\epsilon_k...
  23. Cocoleia

    Confusion with formulas for wavefunctions

    Homework Statement I am working on this problem: The formula I am given in my notes is: I found a website which explains this problem, and they give this formula: But then, their answer is: And I don't see how they went from sqrt(8) to 2 on the numerator. I am confused which formula to use...
  24. V

    How to solve for D and sketch the wavefunctions

    Homework Statement See attached image. The potential in question is, ##-V_0## for ##0<r<a,## and ##0## for ##r\geq a.## Homework Equations $$\sinh(x)=\frac{e^x+e^{-x}}{2}$$ $$\cosh(x)=\frac{e^x-e^{-x}}{2}$$ The Attempt at a Solution I know that the wavefunction for ##r<a## is given by...
  25. Jilang

    I The nature of wavefunctions collapse

    This post is a result of reading RKaster's links and I am wondering if there is some evidence that supports measurement being a two stage process. In the Stern Gerlach experiment the particles are in superposition until measured. But is the superposition ended once they enter the magnetic field...
  26. D

    I Linear equations and superposition of wavefunctions

    Hi. I have read many times that the Schrodinger equation is a linear equation and so if Ψ1 and Ψ2 are both solutions to the equation then so is Ψ1 + Ψ2. Is this use of the word linear the same as generally used for differential equations ? As the Schrodinger equation is also an eigenvalue...
  27. H

    I Still confused about (anti)symmetrization of wavefunctions

    I am currently confused about anti-symmetrization of wavefunctions. In a thread "Still confused about super position and mixed state", kith responded that anti-symmetrization was not done thus the none of the bras and kets shows any properties of a possible state of a molecule I have mentioned...
  28. Isaac0427

    I Form of Wavefunctions: Exploring Outside of e^ikx

    Hi all! I'm currently watching MIT 8.04 (Quantum Physics I) on MIT open courseware. I have just finished lecture 5. In the past 2 lectures, they introduced opperators, specifically momentum, energy and position. To prove/derive (I'm not sure what the correct term is) the momentum and energy...
  29. sciencejournalist00

    I Is there a difference between EM waves and photon wavefunctions?

    Matter has a wavefunction associated to it. But what about light? Does it have both a electromagnetic wave described by Maxwell's equation and a wavefunction described by Schroedinger's equation? Or is the electromagnetic wave considered to be the wavefunction of the photon? I read somewhere...
  30. J

    How to Normalize a Wavefunction with Modulus x?

    Homework Statement Normalise the wavefunction in the diagram which is given by: psi(x) = A {x} < a {x} is supposed to be mod x.Homework Equations None specifically The Attempt at a Solution I know that the square integral of the wavefunction needs to be set equal to 1. I am unsure exactly...
  31. W

    Rotations of spins and of wavefunctions

    This is a question regarding the intrinsic angular momentum S of a particle of spin 1. Assuming S = s(s+1)I = 2I and I is the identity operator. In our case s = 1. Let |z> be a ket of norm 1 such that Sz |z> = 0, and let |x> and |y> be the ket vectors obtained from it by rotations of + 1/2 Pi...
  32. M

    Inner Product/Wavefunction Proof

    Homework Statement Show that ∫ ψ1(x)*ψ2(x) dx = ∫φ1(k)*φ2(k) dk (Where the integrations are going from -∞ to ∞) Homework Equations 1. Plancherel Theorem: ψ(x) = 1/√2π∫φ(k)eikx dk ⇔ φ(k) = 1/√2π∫ψ(x)e-ikx dx The Attempt at a Solution It is clear that Plancherel's theorem must be used to...
  33. Molar

    Why do we need a normalisation constant for wavefunctions?

    ψ = Ae-kx2 ; A = normalisation constant For normalising, -inf∫inf A2ψ°ψ dx = A2M (say) = 1 so we put A = 1/√M My question is why we need 'A' ?? The thing is either we find a particle or we do not and if we think of a simple waveform...'A' gives the amplitude part...so can we put it in this...
  34. J

    What is the significance of normalising wavefunctions in quantum mechanics?

    Homework Statement I am struggling to understand all the steps in a derivation involving a normalisation of a particular wavefunction. I get most of the steps. I have attached the derivation and put a star next to the steps I don't fully understand. Homework Equations Listed on attachment...
  35. F

    Wavefunctions of Singlet and Triplet States

    If we have a system of two electrons, addition of angular momentum tells us that the spin states of the composite system can be decomposed into those of the two electrons as follows |1,1>=|+>|+> |1,0>=(|+>|-> + |->|+>)√2 |1,-1>=|->|-> |0,0>=(|+>|-> - |->|+>)√2 where the states are |s,ms> for the...
  36. S

    Photon Wavefunctions: Is There a Schrodinger Equation?

    Do photons have quantum mechanical wave functions like other particles do? If so, would I use some alternate version of Schrodinger's equation when deriving said wave function? I ask this because as we know, the Schrodinger equation is as follows: (-ħ2/2m)∇2 + v(x,y,z)Ψ = EΨ Photons however...
  37. L

    How do I normalize a wavefunction in three dimensions?

    Homework Statement 2. Homework Equations [/B] Uploaded as a picture as it's pretty hard to type out The Attempt at a Solution So to normalise a wavefunction it has to equal 1 when squared. A is the normalisation factor so we have: A.x2e-x/2a0.x2e-x/2a0 = 1 ∫ψ*ψdx = A2∫x4e-axdx = 1 Then I've...
  38. P

    Wavefunctions that don't satisfy Schrödinger equation

    If there exists some normalized wavefunction ##\psi## that is not a solution to the Schrödinger equation (1D), what does this mean? You can still presumably use the square of the wave function to ascertain the probability it exists at some interval in space, but does it provide any other useful...
  39. J

    Two particle wavefunctions of helium

    Homework Statement The two-particle wavefunctions are given: Where |V> is the spin down state and |Λ> is the spin down state ½(Ψ3p(r1)Ψ1s(r2) + Ψ1s(r1)Ψ3p(r2))(|V>|Λ> - |V>|Λ>) ½(Ψ3p(r1)Ψ1s(r2) - Ψ1s(r1)Ψ3p(r2))(|V>|Λ> + |V>|Λ>) 1/√2 (Ψ3p(r1)Ψ1s(r2) - Ψ1s(r1)Ψ3p(r2))(|Λ>|Λ>) 1/√2...
  40. ChrisVer

    Proton Mixed A and S wavefunctions charge

    I have the proton wavefunctions of mixed symmetry: |MA> = \frac{1}{\sqrt{2}} ( u_1 d_2 u_3 - d_1 u_2 u_3 ) and |MS> = \frac{1}{\sqrt{6}} (2u_1 u_2 d_3 - u_1 d_2 u_3 - d_1u_2u_3 ) If the charge defined as: qu =\frac{2}{3} u and qd= -\frac{1}{3} d , I need to show what are the...
  41. R

    Question about the best method to use for finding wavefunctions and eigenvalues

    We have been covering the annhilation and creation operators in class. You can use the annihilation operator to find the groundstate wavefunction, and then use the hamiltonian in terms of annhiliation and creation operators to find the energy eigen value of that state. (or you could put the...
  42. M

    What Constitutes a Quantum Measurement in Liquid Helium Experiments?

    I just found out about this via Twitter: http://phys.org/news/2014-10-function-electron.html I'm too tired to have got my head around all the details, but it looks as if there's a fascinating new experimental perspective on what a "measurement" in QM actually is. DOI for the original journal...
  43. T

    Question on angular momentum of triplet and singlet wavefunctions

    Hi, I'm relatively new to QM so just a basic explanation of my problem would be amazing! I'm doing some internet research on superfluidity over my summer holiday, and was looking specifically at 3He, and the way it forms Cooper pairs. Having read a classical analogy to why the relative angular...
  44. D

    Confused about wavefunctions and kets

    Hi. I am confused about the following problems. Any help would be appreciated. Thanks 1. I don't understand what ψ (r)= <r|ψ> means. What is the difference between the wavefunction ψ and the ket |ψ> ? 2. A similar equation is ψ (p) = <p|ψ>. Is this ket |ψ> the same as the one above or is...
  45. D

    Do wavefunctions always need to be normalised?

    I understand the meaning of normalising wavefunctions, but what instance would a wavefunction NOT need to be normalised?
  46. F

    Talmi-Moshinsky coefficients for momentum space wavefunctions

    I've found a number of papers about how to calculate Talmi-Moshinsky coefficients. For example W. Tobocman Nucl. Phys A357 (1981) 293-318 and FORTRAN code base on it Y.-P. Gan et al. Comput. Phys. Commun. 34 (1985) 387. This works well if I want to calculate matrix elements that only depend on...
  47. D

    Normalization of wavefunctions

    Hi I have been working my way through some past papers and then checking the solutions but I am confused about the following. One question asked for the normalisation constant for the following wavefunction ψ( r, θ , ø ) = Aexp(-r/R) where R is a constant. The solution requires a triple volume...
  48. T

    Complex wavefunctions and electromagnetic waves

    Hi Everyone I am wondering about something. As everyone here knows, electromagnetic waves obviously possesses an electrical component and a magnetic component. Firstly, can electromagnetic waves be considered to be a sort of complex wavefunction? If yes then do the two components of...
  49. T

    Complex wavefunctions and probability

    Hi Everyone I am a bit confused about something. I have been taught that wavefunctions are basicaly the square root of probability functions. I have also read that some wavefunctions are complex which means that they involve the value i which is the square root of minus one. These two things...
  50. M

    Quantum Mechanics: Complex Wavefunctions Explained

    Why do wavefunctions in quantum mechanics need to be complex? What are the drawbacks of using real valued wavefunctions like: Asin(kx+ωt+ø) etc...or a standing wave equation: Asin(kx)cos(ωt)? I'm an undergraduate student and recently passed 12th grade...So any answer of my level would be...