What is Wavefunction: Definition and 575 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. K

    I When does the separation of variables work

    When studying the hydrogen atom, given that the potential depends only on the distance and not an any angle, we can do a separation of variables of the wavefunction as the product between a function depending only on the distance between particles (protons and electrons) and a spherical...
  2. S

    B Does the wavefunction of an unbound electron collapse while it's captured by a proton?

    And then, if the wavefunction does collapse as the electron binds to the proton, is that collapse temporary, or will the wavefunction remain collapsed until the electron escapes and is free again? Or, will the wavefunction of a freshly bound electron eventually return to a max superposition...
  3. C

    B Wavefunction and Lorentz Invariance

    What are the implications that the wavefunction is not Lorentz invariant?
  4. ergospherical

    <x|P|x'> in Dirac formalism

    What is ##<x|P|x'>##? (for particle in 1d, and ##\hbar = 1##)?\begin{align*} <x|P|x'> &= \int dp' <x|P|p'><p'|x'> \\ &= \int dp' \ p' <x|p'> <p'|x'> \\ &= \int dp' \ p' \frac{1}{\sqrt{2\pi}} e^{ip'x} \frac{1}{\sqrt{2\pi}} e^{-ip'x'} \\ &= \frac{1}{2\pi} \int dp' \ p' e^{ip'(x-x')} \end{align*}
  5. George Wu

    A What is a spatial wavefunction in QFT?

    My understanding is: $$\phi (\mathbf{k})=\int{d^3}\mathbf{x}\phi (\mathbf{x})e^{-i\mathbf{k}\cdot \mathbf{x}}$$ But what is ##\phi (\mathbf{x})## in Qft? In quantum mechanics, $$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> =\int{d^3}\mathbf{k}\phi...
  6. S

    B The Difference Between Wavefunction & Superposition: Exploring Photon Behaviour

    (For me to understand, please be mindful to avoid a bunch of jargon) I'm not sure if the proper word is wavefunction or superposition, and didn't find anything in a search of the difference between the two. So will elaborate on the question in my own words. To begin, as far as I undestand...
  7. Leureka

    I Quantum wavefunction is real?

    In this 2011 paper, Lundeen & colleagues used weak measurement to map both imaginary and real components of a wavefunction directly, without destroying the state. It says: “with weak measurements, it’s possible to learn something about the wavefunction without completely destroying it”. And...
  8. bhobba

    I Interestingly Bohr Did Not Believe in Wavefunction Collapse

    I was reading an article 'What Einstein Really Thought about Quantum Mechanics' in Scientific American. There they mentioned something I didn't know - Bohr did not believe in Wave Function Collapse as an issue (I don't either - but that's just my opinion, so means nothing). I found this...
  9. James1238765

    I Time evolution of the electromagnetic wavefunction on a lattice

    The Maxwell wavefunction of a photon is given in [here] as follows: Because the curl operation mixes 3 different components, this wavefunction only works for a minimum of 3 space dimensions, with each grid point having 6 component numbers ##{E^1, E^2, E^3, B^1, B^2, B^3}##, and with the...
  10. K

    Normalize function - quantum chemistry

    Normalize function f(r) = Nexp{-alpha*r} Where alpha is positive const and r is a vector I was just wondering if the fact that we have a vector value in our equation changes anything about the solution
  11. A

    Calculate the probablity density and current density of a wavefunction

    i have use time evolution operator to get the wavefunction at any time "t" as Ψ(x,t) = U(t,t1) * Ψ(x,t1) but i don't know how to calculate next part of the question
  12. 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...
  13. Dario56

    I Density Operators of Pure States

    Quantum states are most often described by the wavefunction ,##\Psi##. Variable ,##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)## defines probability density function of the system. Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is...
  14. J

    I The Wavefunction when Mass is Much Greater than Planck's Constant

    I understand that the uncertainty is low when you're dealing with a "macro" scale area that is much bigger than Planck's constant. But what's confusing to me is when you know with extreme precision the location, but there's so many particles involved that there is little uncertainty since the...
  15. R

    I Interpreting ##A^{\mu}(x)|0\rangle## and ##\psi (x) |0\rangle##

    I can understand how ##\phi (x)|0\rangle## represents the wavefunction of a single boson localised near ##x##.I don't understand how the same logic appies to ##A^{\mu}(x)|0\rangle## and ##\psi |0\rangle##. Both of these operators return a four component wavefunction when operated on the vaccuum...
  16. Dario56

    I Electron Indistinguishability and Repulsion

    If we had a system of ##N## non – interacting electrons than a wavefunction of such a system is a product of one-electron wavefunctions otherwise known as a Hartree product: $$ \Psi(x_1,x_2,...,x_N) = \prod_{n=1}^N \psi(x_n) $$ This means that in such a hypothetical system , it is possible to...
  17. P

    I Wavefunction of ionized hydrogen electron

    At what point does the electron become ionized in the hydrogen atom solution
  18. Salmone

    I Wavefunction properties tunneling effect

    I am considering tunnel effect with a potential barrier of a certain height that is ##\neq 0## only for ##0 \le x \le a## . I write the Hamiltonian eigenfunctions outside the barrier as:## \psi_E(x)=\begin{cases} e^{ikx}+Ae^{-ikx} \quad \quad x \le0 \\ Ce^{ikx} \quad \quad x\ge a \\...
  19. S

    Wavefunction normalization

    I ran into this question in my problem sheet. Does anybody know how to work it out?
  20. D

    Normalizing Wavefunction: Hard QM Question)

    So I have come up with my solution(attempt) which is: where ( $$\psi_ 1 \triangleq Asin(kx),0<x<L$$ $$\psi_ 2 \triangleq Be^{-sx},x>L$$ $$k \triangleq \sqrt{\frac{2mE}{\hbar^2}} $$ $$s \triangleq \sqrt{\frac{2m(V-E)}{\hbar^2}} $$) But this has a serious problem about boundary: I think...
  21. Mr_Allod

    Position expectation value of 2D harmonic oscillator in magnetic field

    Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...
  22. K

    I Wavefunction in polar coordinates and its bra ket notation

    The wavefunction of ##|\psi\rangle## is given by the bra ket ##\psi (x,y,z)= \langle r| \psi\rangle## I can convert the wavefunction from Cartesian to polar and have the wavefunction as ## \psi (r,\theta,\phi)## What bra should act on the ket ##|\psi\rangle## to give me the wavefunction as ##...
  23. K

    I How to find the wavefunction in this case?

    We've a two interacting particle system, with Hamiltonian as: ##H_{s y s}=\frac{\mathbf{p}_{1}^{2}}{2 m_{1}}+\frac{\mathbf{p}_{2}^{2}}{2 m_{2}}+V\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right)## we reduce it to two non interacting fictitious particles,one moving freely other in a central field...
  24. K

    I Wavefunction of a free particle has carrier and envelope parts

    If ##\psi(x, t)=\left(\frac{1}{2 \pi \alpha^{2}}\right)^{1 / 4} \frac{1}{\sqrt{\gamma}} e^{i p_{0}\left(x-p_{0} t / 2 m\right) / \hbar} e^{-\left(x-p_{0} t / m\right)^{2} / 4 \alpha^{2} \gamma}##where * ##\gamma=1+\frac{i t} {\tau}##( a complex number) * ##\tau=\frac{m h}{2...
  25. K

    I Finding Momentum Mean & Variance from Wavefunction

    I've a Gaussian momentum space wavefunction as ##\phi(p)=\left(\frac{1}{2 \pi \beta^{2}}\right)^{1 / 4} e^{-\left(p-p_{0}\right)^{2} / 4 \beta^{2}}## So that ##|\phi(p)|^{2}=\frac{e^{-\left(p-p_{0}\right)^{2} / 2 \beta^{2}}}{\beta \sqrt{2 \pi}}## Also then ##\psi(x, t)=\frac{1}{\sqrt{2 \pi...
  26. K

    I Collapse of wavefunction into a forbidden eigenstate for a free particle

    For the free particle in QM, the energy and momentum eigenstates are not physically realizable since they are not square integrable. So in that sense a particle cannot have a definite energy or momentum. What happens during measurement of say momentum or energy ? So we measure some...
  27. dabafsdf

    I Spin and wavefunction of excitons

    My Bachelor thesis is all around Excitons (specifially transitions between excitons of different energies). During my work I often had trouble with the spin and the wavefunction of them. Is there maybe some good (free) literature about the theory of excitons ? I found some books in the internet...
  28. J

    A Concept of wavefunction and particle within Quantum Field Theory

    -1st: Could someone give me some insight on what a ket-state refers to when dealing with a field? To my understand it tells us the probability amplitude of having each excitation at any spacetime point, but I don't know if this is accurate. Also, we solve the free field equation not for this...
  29. I

    How can I plot the function g(x) = sin(πn/L) x and its corresponding g²(x)?

    Summary:: We are currently studying basics of quantum mechanics. I'm getting the theory part but it's hard to visualise everything and understand. We are given this question to plot the function so if someone could help me in this. Plot the following function and the corresponding g²(x) g(x)...
  30. redtree

    B Difference between a continuously differentiable function and a wave

    What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?
  31. F

    I Determining Momentum from Wavefunction

    The goal I am trying to achieve is to determine the momentum (2D) in a quantum system from the wavefunction values and the eigenergies. How would I go about this in a general manner? Any pointers to resources would be helpfull.
  32. H

    Electron wavefunction in hydrogen

    Hi. I would love if someone could check my solution since me and the answer sheet I found online don't agree. The probability is given by the triple integral \begin{align*} \int_0^{r_b} \int_0^{2\pi} \int_0^\pi |\psi (r)|^2 r^2 \sin{\theta} \,d\theta \,d\phi \,dr &= \frac{1}{\pi...
  33. J

    I Is the Wavefunction a Contravariant Component?

    I've heard that the wavefunction as a function of x has units of square root of inverse distance, but I haven't heard an intuitive description of why this is aside from that the math works out when you integrate to get the probability. But aside from the math working out, I'm hoping to get a...
  34. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  35. shankk

    I Issue with wavenumber in the free particle wavefunction

    To me, the ##K## obtained by solving the Schrodinger equation and the de broglie wavelength seem two completely unrelated quantities. Can someone explain why have we equated ##K## and ##\frac{2\pi}{\lambda}##. Also, isn't writing ##p = \hbar K## implying that eigenstate of energy is also an...
  36. Gotbread

    I Possible to derive geodesics from the wavefunction?

    Background While watching Does time cause gravity? from PBS Spacetime, i wondered if its possible to "derive" the geodesic equation not from GR alone, but by assuming each particle is described by an extended wave function and the time evolution of this wave is not constant but the rate varies...
  37. tanaygupta2000

    Normalize the Gaussian wave packet

    For normalizing this wave function, I began by finding the complex conjugate of psi and then multiplied it with the original psi. Now what I am getting is A^2 integral exp(2cx^2-4ax) dx = 1 Now I am not getting how to solve this exponential term. I tried by completing the square method but it is...
  38. T

    Normalization of a wavefunction

    I tried writing the function as: Ѱ = c1Φ1 + C2𝚽2 + C3𝚽3 in order to then find mod C1^2... But ɸ = √2/a sin(ᴨx/a) and not sin(ᴨx/a) I cannot understand how the factor of "√2/a " comes
  39. P

    AdS/CFT electromagnetic wavefunction emergence

    If an electomagnetic wave like blue light, for example, exists in 3 dimensions, then how does/can the AdS/CFT conjecture explain it's emergence? Are the electric and the magnetic components of the blue wave both in 2 dimensions in CFT, and if so how would they combine and emerge into AdS to form...
  40. S

    A Wavefunction of the Universe considering all possible boundaries?

    The Hawking-Hartle no boundary condition is well known. The authors considered a many worlds/histories model considering a sum over all compact euclidean metrics. But are there any models or theories that consider a sum over all possible metrics or boundaries? And finally, if all possible...
  41. Mayan Fung

    I Discreteness of bound vs unbound states

    I observe that all bound states have discrete energy levels, eg. particle in a box, hydrogen atoms. But unbound states always have a continuous energy spectrum. For example, for the case of a finite potential well, when ##E<V_0##, we have discrete energy for the bound states. When ##E>V_0##, the...
  42. BohmianRealist

    I A non-branching interpretation of the universal wavefunction

    Would it make sense to say that the entire evolutionary history of the universal wavefunction could simply be a single, continuous moment of self-measurement? In other words, that the universe exists for no other reason than to be the apparatus that is always in the process of measuring its own...
  43. 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...
  44. mjmnr3

    Why does a symmetric wavefunction imply the angular momentum is even?

    I looked in the instructor solutions, which are given by: But I don't quite understand the solution, so I hope you can help me understand it. First. Why do we even know we are working with wavefunctions with the quantum numbers n,l,m? Don't we only get these quantum numbers if the particles...
  45. A

    Exponential Wavefunction for Infinite Potential Well Problem

    Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...

    I A question about the Collapse of a Wavefunction

    I’m new in QM. I have a simple question: when one says that the wavefunction collapses, is it the same as saying that the variance of an observable is 0? Thanks.
  47. X

    Normalizing wavefunction obtained from Lorentzian wave packet

    Part a: Using the above equation. I got $$\psi(x) = \int_{-\infty}^{\infty} \frac{Ne^{ikx}}{k^2 + \alpha^2}dk $$ So basically I needed to solve above integral to get the wave function. To solve it, I used Jordan's Lemma & Cauchy Residue Theorem. And obtained $$\psi(x) = \frac {N \pi...
  48. allisrelative

    I Does Decoherence get rid of all Quantumness?

    The answer is no and even when decoherence occurs for Wigner's Friend in the lab, quantum coherence remains. Let's start with the paper that illustrates this. Assisted Macroscopic Quantumness CONT. https://arxiv.org/abs/1711.10498 Wow, I recently read this paper and the results are simply...
  49. 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?
  50. 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...