What is Wave packets: Definition and 36 Discussions
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.
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 $$\psi(x) = \frac {N \pi...
Show that ##v_{av}=\frac{\hbar k_2 + \hbar k_1}{2m}## is equal to ##v_{av}=\frac{\omega_2 - \omega_1}{k_2-k_1}##. Which of the identities listed above (if any) would make the sign change between ##k_2## and ##k_1##?
One can attain a "wave packet" by superposing two or more sinusoidal waves...
Hello,
I am a high school physics teacher, and I have been thinking about a way to model quantum mechanics in an intuitive way in order to teach it better, but I don't want to lead my students down the wrong path. I am certainly no expert in quantum theory. In looking at the guidelines, I...
Has anyone ever tried to measure the number of waves in photon wave packets? It seems like that would be an important feature and would be equal to the number of fringes in the double slit experiment (on one side), unless it is a huge number. Also, the decrease in the intensity of fringes as...
I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet.
First I'm decomposing a Gaussian wave packet
$$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0...
Homework Statement
Find the wave packet Ψ(x, t) if φ(k) = A for k0 − ∆k ≤ k ≤ k0 + ∆k and φ(k) = 0 for all other k. The system’s dispersion relation is ω = vk, where v is a constant. What is the wave packet’s width?
Homework Equations
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I solved for Ψ(x, t):
$$\Psi(x,t) =...
Given a source of electrons, like from an electron gun. Physicists call these freely traveling particles and often use a Gaussian wave packet to represent them with the group velocity being precisely defined as the velocity of the center of the packets. But if we do not measure the position of...
Homework Statement
Assume a wave packet is has contributions from various frequencies, give by g(ω)=C for |ω|<ω0, and g(ω) =0 for elsewhere.
a)What is the signal strength as a function of time, i.e., V(t)=?
b) Sketch g(ω) and V(t); You can use fooplots.com, for example, or python.
c)...
I'm reading Gasiorowicz's Quantum Physics and at the beggining of chapter 2, SG introduces the concept of "wave packet" and gaussian functions associated to them. The first attached image is the 28th page of the book's 1st edition I suppose, and my question is about the paragraph inside the red...
My layman's intuition tells me that wave packets normally spread out in space and disperse, except in special circumstances. Photons don't behave like that.
In The Principles of Quantum Mechanics, pp 124-125, Dirac discusses the equations of motions of a photon wave packet. He says...
Homework Statement
This is the problem sheet that I am solving at the moment:
2. The attempt at a solution
I have already solved 17.
Here is my solution to 17:
Now I am working on 18.
I am trying to show that the probability density function is conserved. I.e the integral of the...
Hi, I'm fairly new to quantum physics so go easy on me! I've just come across the uncertainty principle and have some confusion over wave packets. If we superimpose multiple waves then as I understand it we get a wave packet with a finite length which lowers the uncertainty in position. If your...
I've found the wave-packet picture quite useful as I work my way through the very basics of quantum mechanics. But I'm having trouble finding a wave-mechanical picture of operators. For example, at least in terms of a free particle, using the wave mechanics treatment (as opposed to the matrix...
I'm reading an article about the photo induced ligand loss of metal carbonyl complexes at the moment and here's a bit I'm having trouble getting my head around:
Its the wavepacket part that I'm confused about. All I know is that a wavepacket is what you get when you combine multiple sine...
I have difficulty understanding the exact concept of group velocity. Consider a wave packet as a linear combination of a number of eigenstates of a 1-D particle in box. The dispersion curve(\omegaversus k) is composed of discrete points located on a parabola. Well, for each point one can...
PROBLEM:
Laser pulses of femptosecond duration can be produced, but for such brief pulses it
makes no sense to speak of the ‘color’ of the laser. To demonstrate this, compute the time duration of a laser pulse whose range of frequencies covers the entire visible spectrum (4.0*10^14 Hz to...
Homework Statement A radar transmitter used to measure the speed of pitched baseballs emits pulses of 2.0 cm wavelength that are .25 μs in duration. a) What is the length of the wave packet produced? b) To what frequency should the receiver be tuned? c) What must be the minimum bandwidth of the...
Do the wave functions for smaller objects disperse faster?
How long does a macroscopic wave function (for example that corresponding to a measurement device) keep its broadness (roughly speaking)?
Context: I'm reading a paper where they rely on the claim
and I would have no clue about...
Homework Statement
I: A telephone line can transmit a range of frequencies \Delta f = 2500 Hz. Roughly what is the duration of the shortest pulse that can be sent over this line?
II: A space probe sends a picture containing 500 by 500 elements, each containing a brightness scale with 256...
Uncertainty Principle - Wave Packets
Homework Statement
If a phone line is capable of transmitting a range of frequencies delta(f) = 5,431 Hz, what is the approximate duration of the shortest pulse that can be transmitted over the line? Give your answer in millseconds to 4 significant figures...
I am having trouble understanding (not for homework) what a wave packet is in terms of the correspondence of the idea of a wave packet to a "point like" particle. I'd like to focus on the 1d wave packet ultimately, but in order to describe my consternation -- let me detour to a well defined...
Homework Statement
The phase velocity of each wavelength of white light moving through or-
dinary glass is a function of the wavelength i.e. glass is a dispersive medium. What is the general dependence of phase velocity on wavelength in glass? Is dvp/dlamda positive or
negative? Why...
Forgetting about spin and polarity for the moment, do individual identical particles of the same type (electrons, photons, etc.) really have their own individual wave functions (Gaussian packets)? The mathematical definition of probability (relative frequency, etc.) has meaning only at the...
I'm reading up on Quantum Mechanics and I don't follow an integration they use.
They start with this:
\psi(x,t) = \int^{\infty}_{-\infty} dk A(k) e^{i(kx-\omega t)}
They begin by considering the wave packet at time t=0:
\psi(x,0) = \int^{\infty}_{-\infty} dk A(k) e^{ikx}
"and...
Hi, I'm trying to derive a wave equation for a gaussian wavepacket for both the position (x) and the momentum (k), for a wave packet of width sigma, at some initial position x0 and with an initial momentum k0.
Now I have worked out the initial wavepacket equation to be:
psi(x) =...
If you imagine a string, the first part of the string (that the gaussian looking wave peak is moving along) has thicker mass density than the latter part of the string. (so it's essentially a thick bit of string going on to a thinner bit).
What would happen when the wavepacket reaches the...
I'm trying to interpret a research paper on single photon ionization using extreme UV attosecond laser pulses, and I realized I have some very basic questions concerning electrons in an atom. Technically, the answers to these questions are found online and in the literature, but the wording...
Hi,
In some books and sites it's said to be nothing but physical wave packet for physical particles. They says a real-physical wave packet can exhibit all the features of a massive particle.
Is it true(shown with experiments?), or is it one of the interpretations?
Hello:
I need some help with a homework problem that was taken from Quantum Physics by Gasiorowicz.
The problem goes like this: You have a beam of electrons and know the size of the wave packet, and by the uncertainty principle you can estimate the dispersion in p at t=0. The problem is to...
Hi all.
Can someone explain me physically why we need to deal with wave packets in water waves?
I know the the nonlinear schrodinger equations deals with wave packets in water wave.
But why bother dealing with wave packets?
For the KdV equation, the concept of wave packets is not needed...
Are there any nice wave packets you could write as a superposition of eigenstate solutions of a one-dimensional harmonic oscillator? The question deals with a situation, where a particle feels a harmonic potential, but is far away from the center and is traveling as a wave packet, probably...
I do not know if this is the correct modern term, but it seems that particles can be thought of as wave packets? no matter what they are, what is manufacturing these packets in such a way that our universe is as we see it?
Wave packets has a group velocity of
v_{group}=\frac{d\omega}{dk}
and its phase velocity is
v_{phase}=\frac{\omega}{k}
Show that the group velocity and the phase velocity are related by:
v_{group} = v_{phase} - \lambda\frac{dv_{phase}}{d\lambda}
Can someone please tell me...
Why does a wave packet spread out in space as time passes?
What difference would it make to the universe if it did not?
And have a look at this link which shows a movie of "the time evolution of a quantum wave packet."
http://webphysics.davidson.edu/mjb/acs_transformations_qm/packet.html...