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.
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...
The Schrödinger equation I need to prove is this one
And the Gaussian wavepacket is found here
Thanks for your advice.
JorgeM
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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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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.
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...
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...
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...
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...
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"...