Wavepacket Definition and 37 Threads

In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere.
Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating.
Quantum mechanics ascribes a special significance to the wave packet; it is interpreted as a probability amplitude, its norm squared describing the probability density that a particle or particles in a particular state will be measured to have a given position or momentum. The wave equation is in this case the Schrödinger equation. It is possible to deduce the time evolution of a quantum mechanical system, similar to the process of the Hamiltonian formalism in classical mechanics. The dispersive character of solutions of the Schrödinger equation has played an important role in rejecting Schrödinger's original interpretation, and accepting the Born rule.In the coordinate representation of the wave (such as the Cartesian coordinate system), the position of the physical object's localized probability is specified by the position of the packet solution. Moreover, the narrower the spatial wave packet, and therefore the better localized the position of the wave packet, the larger the spread in the momentum of the wave. This trade-off between spread in position and spread in momentum is a characteristic feature of the Heisenberg uncertainty principle,
and will be illustrated below.

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  1. L

    I 1D wavepacket scattering simulation, momentum distribution formula

    Hello everybody at the forum I'm from Ukraine, I have Chemistry degree, and last year I began to self studying Quantum Mechanics. I'm reading this article: R. Garcia, A. Zozulya, and J. Stickney, “MATLAB codes for teaching quantum physics: Part 1,” [Online]. Available...
  2. JorgeM

    Gaussian wavepacket as a solution of the Schrödinger equation

    The Schrödinger equation I need to prove is this one And the Gaussian wavepacket is found here Thanks for your advice. JorgeM <Moderator's note: upload images to PhysicsForums. Do not use external image servers.>
  3. ilper

    I The construction of particles in QFT

    In all books about QFT I have seen I can not find anything about what a localized particle concept is. Suddenly I found this note in Zee's 'QFT in nutshell' page 4: "As usual, we can form wave packets by superposing eigenmodes. When we quantize the theory, these wave packets behave like...
  4. I

    B Uncertainty of a Gaussian wavepacket

    Hi, I know that a Gaussian wavepacket has minimum uncertainty. The issue is, some sources are telling me that σxσp=ħ and others are telling me that σxσp=ħ/2. I am really confused. I think the latter is correct due to what I have been taught about the uncertainty principle, but then I don't...
  5. K

    About wavepacket unit and probability

    Hi all, In quantum mechanics, we consider the squared modulo of a wave function has meaning of probability, so does it mean a wave packet should be unitless? I am reading some materials online about the Gaussian wave packet...
  6. blue_leaf77

    "Wavepacket of electrons" or "wavepacket of electron"?

    For simplicity let's assume Gaussian shape. I just want to verify my understanding about wavepacket. As the thread title says, when we have a wavepacket, is it understood as the superposition of many plane waves corresponding to many free electrons, or as the wavefunction of only one electron...
  7. jfizzix

    Group velocity of a wavepacket vs its mean phase velocity

    The mean velocity of a wavepacket given by the general wavefunction: \Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int dk A(k)e^{i(k x - \omega(k) t)}, can be expressed in two ways. First, we have that it's the time derivative of the mean position (i.e., its mean group velocity): \frac{d \langle...
  8. C

    Physical meaning of a wave packet w/ respect to HUP&duality

    I'm a QM noob/newb trying to understand the physical implication of a wave packet, in my mind it is something like this: On the x-axis there is displacement (vibration), probability on the y. I Imagine stretching and compressing the wave packet. When I stretch it out, the amplitude must...
  9. T

    Expectation Value of Momentum for Wavepacket

    Homework Statement What is the average momentum for a packet corresponding to this normalizable wavefunction? \Psi(x) = C \phi(x) exp(ikx) C is a normalization constant and \phi(x) is a real function. Homework Equations \hat{p}\rightarrow -i\hbar\frac{d}{dx}The Attempt at a Solution...
  10. O

    Free particle wavefunction represent a fixed profile= wavepacket?

    Why the Griffiths book says : any function of x and t that depends on these variables in the special combination (x±vt) represent a wave of fixed profile, traveling in the -+x direction at speed v... I don't really get the reason why from 2 terms of wavefunction can become only one term...
  11. I

    Wavepacket incident on a potential step

    Hello, I am writing a Fortran 95 program to model the scattering of a wavepacket by a potential step of height V0 at x=0. My wavepacket is formed by the superposition of numerous travellling waves of different k values. The wavepacket has the dispersion relation ω(k)=k2. I want the wavepacket...
  12. C

    Quantum Mechanics, a free particle prepared as a gaussian wavepacket

    Homework Statement The problem is given in the attached image. I'm currently trying to work out question one.Homework Equations \phi (k) = \dfrac{1}{2 \pi} \int_{- \infty}^{ \infty} \Psi (x,0) e^{-ikx} dxThe Attempt at a Solution Okay, so the first thing I did was to normalise it, but then I...
  13. N

    Wavepacket Evolution: Questions on Time Evolution and Plane Waves

    I had a few questions regarding the time evolution of wavepackets of the form ∫ dk*A(k)*cos(kx-wt), where w = w(k) If the group velocity is zero, i.e; dw/dk evaluated at k' = 0, where k' the central wavenumber of the narrow packet, then do we essentially see complete constructive...
  14. T

    Calculating kinetic energy density of wavepacket in deep water waves

    Assume that an orca can generate a wavepacket which is half a sine wave. Show that the kinetic energy density in this wave can be written U= ρgAλ/(32(π^2)) 2. Homework Equations mass per unit length=∫ρ dxdy KE=1/2mv^2 3. The Attempt at a Solution I'm confused on what to use...
  15. T

    Calculating kinetic energy density of wavepacket in deep water waves

    Homework Statement Assume that an orca can generate a wavepacket which is half a sine wave. Show that the kinetic energy density in this wave can be written U= ρgAλ/(32(π^2)) Homework Equations mass per unit length=∫ρ dxdy KE=1/2mv^2 The Attempt at a Solution I'm confused...
  16. S

    Gaussian wavepacket and position-momentum uncertainty

    We know that momentum is proportional to k so by adding more waves to localise our particle we are adding more waves with independent momentum values Upon measurement, the gaussian wavepacket must collapse into one eigenstate of momentum, but if we have a very localised packet there will be...
  17. B

    Photon coherence time as the wavepacket length and dephasing

    Quite often one can see descriptions saying that the coherence time of single photons corresponds to the length of the single photon wavepacket (for example Jelezko et al, PRA 67 041802(R) (2003) http://pra.aps.org/abstract/PRA/v67/i4/e041802). I find it hard to come to terms with this picture...
  18. Y

    Gaussian Wavepacket Momentum Squared

    Homework Statement I will not post a specific problem, but rather, I would like to ask a general question. Say I am given a Gaussian wavepacket (function psi(x,t) ) and asked to find the expectation value for x-squared and momentum squared. Now, x-squared is rather straightforward...
  19. S

    Gaussian Wavepacket: Position-Momentum Uncertainty

    how can the gaussian wavepacket presents a physical picture of the origin of position-momentum uncertainty?
  20. P

    Wavepacket Dispersion and how this links to particle behaviour

    Hi, I'm trying to get my head round modelling particles in free space in quantum mechanics. I appreciate that we can "build" wavepackets by superposing many plane waves with different k-numbers (i.e. with different frequencies & momentums & energies I think). The greater the number of phase...
  21. B

    What Is psi_PRIME in the Lorentz Transform of a Wavepacket?

    Homework Statement see attached .pdf. all parts of problem statement are italicized. Homework Equations see attached .pdf The Attempt at a Solution see attached .pdf Actually: my question is pretty qualitative. You can look at everything I've done with this problem so far...
  22. N

    Why Use Homogeneous Dispersion for Group Velocity in Inhomogeneous Lattice?

    Hi The group velocity of an electron wavepacket in a homogeneous lattice is vgroup(k) = ∇kEk, where Ek is the dispersion. I have just read an article, where they use this to find the group velocity of a wavepacket in an inhomogeneous lattice, but they use the homogeneous dispersion. I...
  23. S

    Wave packet of: 2 spin half non interacting indentical particles, wavepacket=?

    Homework Statement The two particles are confined to a 1D infinite potential well both have spin up. the spin part of the wavepacket is thus: |arrowup,arrowup> I need to write the wavepacket of the ground state Homework Equations The Attempt at a Solution 1. spatial part...
  24. W

    Expectation value of position of wavepacket

    Hello, this is just a general question, how is <x^2> evaluated, if <x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket) Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ? I'm only wondering how the squared works...
  25. P

    2d Wavepacket in relativistic QM

    Dear users, I wonder if there is anybody who can give me a hint on how to handle the following situation: In the 2+1 dimensional Klein-Gordon equation with coordinates (t,x,y), I use as initial condition for \Psi(x,0) a spherically symmetric Gaussian. The relativistic dispersion relation...
  26. M

    Gaussian Wavepacket Tunnel Through double barrier

    Hello, I was trying to design a movie in Matlab. A Gaussian wave packet moves toward double barrier then some wave reflect and some pass out from the barrier . i design the double barrier in Matlab. but don't know how to make movie in Matlab with Gaussian wavepacket. can you help. Do u have...
  27. T

    Wavepacket Problem: Writing |psi(x,t)|^2

    Homework Statement A wavepacket, psi(x,t), can be expressed as a linear combination of eigenstates. Assuming that only 2 eigenstates, phi0(x) and phi1(x), contribute to the linear combination write down the expression for |psi(x,t)|^2. Homework Equations [Boltzmann's constant = 1.38 x10^23 J...
  28. Y

    How we can find wavelength of a wavepacket?

    How we can find wavelength of a wavepacket? Which distance is wavelength?
  29. A

    Native language: EnglishLevel of quantum mechanics: Introductory or beginner

    Dear, I have a trouble understanding QM. What's the difference between wavepacket and wavefunction? Can we use a wavepacket for a particle in a box? Please reply to this questions. Thank you in advance.
  30. A

    Questioning Wavepacket: Seeking Clarification

    Dear everyone, I'm wondering about why we need the wavepacket? Is there anyone to clarify my stupidity? Thanks in advance.
  31. A

    Free Particle Wavepacket Spreading: Is it True?

    Is it generally true that the wavepacket of a free particle spreads out as time goes to infinity? It seems like it would, since the phase velocities of the component plane waves are different, and therefore the plane waves would get increasingly out of phase with time. A gaussian wave packet...
  32. O

    What Is the Minimum Size of a Wavepacket?

    I've read a few texts where the term "minimum sized wavepacket" is used. Can anyone explain what the "minimum size" refers to in the context of a wavepacket? Thanks.
  33. R

    How to Determine Expansion Coefficients for a Wavepacket in a Periodic Box?

    I'm trying to get my head around the idea of expansion coefficients when describing a wavefunction as \Psi(\textbf{r}, t) = \sum a_{n}(t)\psi_{n}(\textbf{r}) As I understand it, the expansion coefficients are the a_{n} s which include a time dependence and also dictate the probability of...
  34. J

    How Do Gaussian Profiles Affect Solutions to Maxwell's Equations in a Vacuum?

    Electromagnetic waves Homework Statement Find the solution of Maxwell's equations in vacuum for a continuous beam of light of frequency \omega traveling in the z direction with a gaussian profile in the x and y directions. Homework Equations Maxwell's equations in vaccuum. \nabla \cdot...
  35. H

    How Does the Envelope of a Light Wavepacket Change at a Media Interface?

    Hi, I've had trouble finding an answer to this question and was wondering if anyone could help. What happens to the envelope of a wavepacket of light when it crosses the interface between two media? I know that the field of the wavepacket will be continuous across the boundary, but does...
  36. T

    What is the relationship between wavepacket uncertainty and probability density?

    Hi, I'm puzzled by a couple of formulae in the answer sheet to a problem set I'm working on. To calc. the new uncertainty in the position of a group of electrons, initially localised to \pm1\mum, after time t, it uses the factor: \left(1+\frac{\hbar^{2}(\Delta...
  37. L

    From the Schrödinger equation to the wavepacket reduction axiom

    In the very first pages of "Quantum Mechanics" by Landau & Lifchitz, the measurement process is described as an interaction between a quantum system and a "classical" system. I like this interpretation since any further evolution of the quantum system is anyway entangled with the "classical"...
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