What is Wave functions: Definition and 153 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
  1. zaramahdi

    A Can particles appear and disappear "with" a cause?

    The first thing we need for this is to define what a particle is... It is an object that has specific intrinsic properties and is described by a wave sign How to measure it? This is done by the interaction of the particle to be measured with the measurement system. When measuring, the wave...
  2. J

    A LCAO graphene orbitals wave functions

    Hello, My name is Josip Jakovac, i am a student of the theoretical solid state physics phd studies. First I want to apologize if my question has already been answered somewhere here, I googled around a lot, and found nothing similar. My problem is that I need to apply TBA to Graphene. I went...
  3. Physics Slayer

    B What proof do we have of wave functions?

    How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##. I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...
  4. raz

    A Bloch momentum-space wave functions

    Hello, I wonder if it is possible to write Bloch wave functions in momentum space. To be more specific, it would calculate something like (using Sakurai's notation): $$ \phi(\vec k) = \langle \vec k | \alpha \rangle$$ Moving forward in a few steps: Expanding: $$ \phi(\vec k) = \int d^3\vec r...
  5. Hallucinogen

    I Common features of set theory and wave functions?

    I would like to know if any of you think there's any sort of connection, analogy, or common features between, sets in set theory and wave functions in QT? Wave functions lack trajectories, so do sets. Wave functions also distribute over areas, as sets can do. To my understanding, wave...
  6. S

    I Eigenfunctions and wave functions

    I saw this statement from the textbook "Quantum physics of atoms, molecules, solids, nuclei, and particles" second edition pg 166. According to the text, is the author saying the solution to the TISE is the eigenfunction and when you multiply the time dependent part, you get the wave function? I...
  7. joneall

    I Where do wave functions come from?

    In classical mechanics, we have either Newton’s laws or a Lagrangian in terms of coordinates and their derivatives (or momenta) and we can solve them for the behavior of the system in terms of these variables, which are what we observe (measure). In QM, we quantize classical mechanics by making...
  8. Riccardo Marinelli

    Initial condition of Wave functions with Yukawa Potential

    Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...
  9. A

    I Single ket for a product of two wave functions

    Hello, I would like to write a product of two wave functions with a single ket. Although it looks simple, I do not remember seeing this in any textbook on quantum mechanics. Assume we have the following: ##\chi(x) = \psi(x)\phi(x) = \langle x | \psi \rangle \langle x | \phi \rangle## I would...
  10. DuckAmuck

    B Question about how the nabla interacts with wave functions

    Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.
  11. F

    A Wave functions for positrons and electrons

    Is the wave function for the positron the complex conjugate of the wave function for the electron? I've tried to google this, but I can't seem to get a definite answer from a reliable source. It seems that antimatter is derived in quantum field theory which does not concentrate on wave...
  12. L

    Help with finding the expectation value of x^2

    The question is as follows: A particle of mass m has the wave function psi(x, t) = A * e^( -a ( ( m*x^2 / hbar) +i*t ) ) where A and a are positive real constants. i don't know how to format my stuff on this website, so it may be a bit harder to read. Generally when i write "int" i mean the...
  13. F

    I Thought experiment about wave functions

    Suppose we have a particle, let's say an electron, in a box of size D. And we add another one next to it at some distance L center to center. How do we solve for the wavefunctions of the electron. Can it be solved in normal QM or do we need QFT. Thanks.
  14. S

    Recurrence relation for harmonic oscillator wave functions

    1. Homework Statement I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is : where Rnl are radial harmonic oscillator wave functions of form: The problem is that I can't prove the relation above with the form of Rnl given by the author(above). I've...
  15. J

    I Wave Functions of Definite Momentum

    Hi, Apologies if this questions is really easy but it is something quite subtle which is annoying me. In my book of quantum physics it gives a wave function of definite momentum: ψ = Aeipx/ħ It goes on to say that since there is a momentum 'p' in the exponential then the momentum is known...
  16. K

    Show that the Hydrogen wave functions are normalized

    Homework Statement Show that the (1,0,0) and (2,0,0) wave functions are properly normalized. We know that: Ψ(1,0,0) = (2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi)) where: R(r) = (2/(a0^(3/2))*e^(-r/a0) Θ(θ) = (1/sqrt(2)) Φ(φ) = (1/sqrt(2*pi)) Homework Equations (1) ∫|Ψ|^2 dx = 1 (2)...
  17. SherLOCKed

    A Operation of Hamiltonian roots on wave functions

    How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.
  18. N

    A Do higher dimensional branes have wave functions?

    Do higher dimensional branes, like the super membrane (which is a 2D brane) or the NS5/M5 brane, have wave functions? I know that they become unstable once they are quantized, but does that mean that they do not have wave functions? You will never here about any thing regarding an M2 wave...
  19. T

    A Are all wave functions with a continuum basis non-normalizable?

    For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...
  20. J

    I Explaining Music Notes Consonance with Wave Functions

    Hello all, First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be...
  21. RicardoMP

    I Square integrable wave functions vanishing at infinity

    Hi! For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...
  22. J

    Physical difference between various wave functions

    Homework Statement Is there a physical difference between the following wave functions? If yes, why? If no, why not? \Psi(x,0) =5e^{-ax^2} \Psi(x,0) =\frac{1+i}{\sqrt{3}}e^{-ax^2} \Psi(x,0) =e^{i\pi/7}e^{-ax^2} Homework Equations - The Attempt at a Solution They only differ in the...
  23. K

    Properties of Wave Functions and their Derivatives

    Homework Statement I am unsure if the first statement below is true. Homework Equations \frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2}=\frac{\partial^2 \psi}{\partial x^2}\frac{\partial \psi^*}{\partial x} Assuming this was true, I showed that \int \frac{\partial...
  24. Y

    Wave Functions With Same Energies Are the Same (only differ by a complex phase)

    Homework Statement Assume a particle with a wave function ##\psi(x)## such that ##-\infty < x < \infty##, that move under some potential ##V(x)##. Show that: a) two wave functions with same energies can only differ by a complex phase; b) if the potential is real, then you can choose the wave...
  25. S

    I Triplet States and Wave Functions

    Why is the triplet state space wave function ΨT1=[1σ*(r1)1σ(r2)-1σ(r1)1σ*(r2)] (ie. subtractive)? How does it relate to its antisymmetric nature? Also, why is this opposite for the spin wave function α(1)β(2)+β(1)α(2) (ie. additive)? And why is this one symmetric even though it describes the...
  26. amjad-sh

    I Wave-particle duality and localization

    I have read recently that the motion of an electron of momentum p must be described by the means of a plane waves :\psi(\vec r,t)=Ae^{i(\vec k \cdot \vec r -wt)}=Ae^{i(\vec p\cdot \vec r -Et)/\hbar} de Broglie hypothesis states that every particle of momentum p has a wavelength lamda. I will...
  27. Negatratoron

    Wave Functions: What Are They?

    What's the type of wave functions? Is it just: function from a point in spacetime to Z; takes a location and returns an amplitude in discrete units? (bonus question: according to your favorite theory, what is the type of points in spacetime (that is, topology of spacetime)? is it like r^n for...
  28. PhysicsKid0123

    Asymmetric wave functions

    Can the general solution to the Schrodinger equation be asymmetric (has neither even or odd solutions)? Question (1): I saw somewhere that you cannot have a solution that is both-- it must be either odd or even, and I was wondering: why? I was working on a problem where the potential function...
  29. R

    Position Vector in Wave Functions

    Hello, I need to create a 2-D electron energy density plot in Mathematica to compare with my STM experimental results in my lab class. This would be done by plotting the superposition of the symmetric and anti-symmetric wave functions, $$\Psi_s(\textbf{r}) =...
  30. W

    Tight Binding Wave Functions

    Dear all, Could somebody please, indicate me some tutorial, in order to generate a 3D grid to plot the wave function using the Hamiltonian eigenvalues and the slater type orbitals ? Thanks in advance, Wellery
  31. C

    Do wave functions go to zero at ~ct?

    Suppose you have a free election and you make a measurement of its position r_0 at time t = 0. You then wait some time t required for the wave function to evolve out of its collapsed eigenstate. The electron now supposedly has a wave function expanding to infinity in all directions, albeit with...
  32. K

    Where does band index come from in block wave functions?

    Can you show me as explicitly as possible through equations?
  33. D

    Potential Step and Wave Functions

    Homework Statement Homework EquationsThe Attempt at a Solution For x>b, Ψ(x) = Ae-ikx + Beikx , where k = (√2mE)/hbar a<x<b Ψ(x) = Ce-ik'x + Deik'x , where k = (√2m(U2 - E)/hbar This is the problem part 0<x<a Ψ(x) = Fsink''x...
  34. B

    Wave function with a certain wavelength

    I have a number of questions about the wave function - 1. Do photons have wave functions like the one in Schrodinger equation? 2. If they do, when you send out a wave function with a certain wavelength, then because you know the momentum with no uncertainty the uncertainty of the position...
  35. A

    Physical interpretation of Neumann-Dirichlet conditions

    I am working on a PDE problem like this: Consider the wave equation with homogeneous Neumann-Dirichlet boundary conditions: ##\begin{align} u_{tt} &= c^2U_{xx}, &&0<x<\mathscr l, t > 0\\ u_x(0, t) &=u(\mathscr l, t) = 0, &&t > 0\\ u(x, 0) &=f(x), &&0<x< \mathscr l\\ u_t(x, 0) &=g(x), &&0<x<...
  36. J

    Solve Wave Equation: e^(-x^2), x*e^(-x^2), -infinity<x<infinity

    Homework Statement So it says solve this wave equation : [y][/tt] - 4 [y][/xx] = 0 on the domain -infinity<x<infinity with initial conditions y(x,0) = e^(-x^2), yt(x,0) = x*(e^(-x^2)) Homework Equations I used the D Alembert's solution which is 1/2(f(x+ct)+f(x-ct)) + 1/2c ∫ g(z) dz The...
  37. ghost313

    Systems with more than 1 wave function

    Hello,I am new to quantum mechanics.I just want to clear this equation: ψ(x) = ∑n anψn(x) What does this actually mean?Is this equation telling us that the system is moving as a wave? Or,as I think,for example let's suppouse we have 2 electrons in a system,and the wave function becomes this...
  38. Khaleesi

    Normalizing Wave Functions

    Hi, so I'm having a bit of trouble understanding the normalization of radial waves. I understand that the equation is the integral of ((R^2)r^2)=1 but I'm not understanding how the process works. I need the normalization constant on R32. I got the function to come out to be...
  39. P

    Orthogonality of wave functions

    Homework Statement Homework EquationsThe Attempt at a Solution http://i.imgur.com/tktQBsp.jpg [/B] I assume that you need to prove that the integral of psi1*psi0 is 0, so I have written out the integral and attempted to solve using integration by parts, but whichever way I write out the...
  40. 5

    Normalizing wave functions / superposition

    A remote control shot a single photon at a window that has a 50% chance of transmitting and 50% chance of reflecting photons. Once the photon wavepacket hits the glass (without an observer present), it propagates as a superposition of states, one in which it was transmitted and one in which it...
  41. Ascendant78

    Help graphing wave functions and probability densities.

    Homework Statement Homework Equations The Attempt at a Solution Well, I felt like the first part wasn't too bad and graphed the potential like so: However, to be honest, I'm not even sure if I did that right, as I wasn't sure whether he wanted it as a 2-D or 3-D graph (and I can't...
  42. T

    Finding constants for 3 wave functions

    Homework Statement The ground, 1st excited state and 2nd excited state wave functions for a vibrating molecules are: Ψ0(x) = a e-α2/2 Ψ1(x) = b (x+d) e-α2/2 Ψ2(x) = c (x2 + ex + f) e-α2/2 respectively, with constants: a, b, c, d, e and f. Use the fact that these wave functions are part of an...
  43. L

    <U|V> overlap integral of two many-electron determinant wave functions

    Hello, If we let U and V be two single determinant wave functions built up of spin orbitlas ui and vj respectively, will the overlap between them be as follows: <U|V> = Det{<ui|vi>} Thank you
  44. VoidChimera

    Question about collapsing wave functions

    So, from my understanding, some particles like electrons exist as a particle and a wave/probability field. What I was wondering, was that when the wave function collapses, is its location determined on the actual location of the particle, which we just can't measure and so represent it as a...
  45. T

    Query about quantum superposition and wave functions

    Hi Everyone I have four questions about the nature of quantum superposition and wave functions: 1. If a particle is quantumly superpositioned in more than one location then as soon as the slightest evidence of the particle's existence in one of the locations is detected by a "measurement"...
  46. N

    Are all wave functions energy-eigenstates?

    So I was reading this http://www.oberlin.edu/physics/dstyer/TeachQM/misconnzz.pdf, a list of common misconceptions students have after an intro course in QM. I'm aware that energy eigenstates are the wave functions at "time = zero" and thus do not completely describe the system. However, it is...
  47. gfxroad

    Hydrogen atom wave functions

    Homework Statement I solved the Schrödinger equation, obtaining a wave function in terms of Radial and the spherical harmonics as follows: $$Ψ(r,0)= AR_{10} Y_{00} + \sqrt{\frac23} R_{21} Y_{10} + \sqrt{\frac23} R_{21} Y_{11} - \sqrt{\frac23} R_{21} Y_{1,-1}$$ Homework Equations...
  48. gfxroad

    Show that the two wave functions are eigenfunction

    Homework Statement Consider the dimensionless harmonic oscillator Hamiltonian H=½ P2+½ X2, P=-i d/dx. Show that the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively. Find the value of the coefficient a such that...
  49. 0

    Orthogonality of wave functions to negative momentum states

    This is a question I have about the textbook discussion, so I'll do away with the standard format. The author of my QM book (Shankar, Principles of Quantum Mechanics) used the term "negative momentum states," all of a sudden, and I've never heard of it before. He has a little note saying that...
  50. S

    Can second quantization on strings create states out of the vacuum?

    Does anything connect the discrete wave functions? I thought they were suppose to be connected.
Back
Top