Wavefunction Definition and 584 Threads

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. 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...
  2. 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...
  3. 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...
  4. 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...
  5. PORFIRIO I

    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.
  6. 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...
  7. 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...
  8. P

    A Graphene wavefunction expressed in tight binding form

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  9. J

    I Correlation between Symmetry number & Total wavefunction

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  10. R

    I How do you normalize this wave function?

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  11. allisrelative

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

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  12. P

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

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  13. iymasomhumaan

    B Nuclear wavefunction and Bose-Einstein condensates

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  14. Kaguro

    Finding the potential function from the wavefunction

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  15. bob012345

    I Lithium Atom Ground State Radial Wavefunction

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  16. K

    I Understanding Nuclear Rotation: Quantum Numbers and Wavefunctions

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  17. I

    Finding meaning in the Phase of the wavefunction

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  18. A

    I Master` Equation for the Penrose-Diosi wavefunction collapse

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  19. Denselight93

    B The Virtue of Individual Perception RE: Wavefunction Collapse

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  20. E

    When does the wavefunction wavelength equal the De Broglie wavelength?

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  21. 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?
  22. DoobleD

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

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  23. Erik Ayer

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  24. 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
  25. 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?
  26. 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...
  27. dipanshum

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  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. A

    I Can Quantized Momentum Transfer Explain Double-Slit 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...
  39. 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...
  40. 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...
  41. 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
  42. D

    What is the Effect of the Number Operator on a Given 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 =...
  43. D

    I How Does the Wavefunction Evolve in Quantum Mechanics?

    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...
  44. 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...
  45. 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 $$\kappa^2 = \frac{2m(V_0 - E)}{\hbar^2}$$...
  46. 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...
  47. 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
  48. 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...
  49. 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...
  50. 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...
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