What is Wave packet: Definition and 116 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.
I have typed up the main problem in latex (see photo below)
It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.
I am unsure if ##h(x,t)## really is a wave packet, but it looks like one, hence the title. Anyway, so I'd like to determine ##\hat{h}(k,t=0)##. My attempt so far is recognizing that, without the real part in the integral, i.e.
##g(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} a(k)e^{i(kx-\omega...
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...
since, in order to view the shape changes in our wave packet we are presented with the taylor expansion of the frequency
ω(k) = ω(k0) + (k − k0)dω/dk + 1/2*(k − k0)^2 (d^2ω/dk^2)
we are told that only the third term that is the
1/2*(k − k0)^2 (d^2ω/dk^2)
contributes to change in shape of the...
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 $$\psi(x) = \frac {N \pi...
I tried plugging Psi into the right of the Schrodinger equation but can't get anything close to the solution or anything that is usable. How should I solve this?
It's been a long time since my last exam on QM, so now I'm struggling with some basic concept that clearly I didn't understand very well.
1) The Sch. Eq for a free particle is ##-\frac {\hbar}{2m} \frac {\partial ^2 \psi}{\partial x^2} = E \psi## and the solutions are plane waves of the form...
Consider a gaussian wave packet whose wave function at a particular instant of time is
Its time dependence is implicit in the "constants" A, a, <x> and <p>, which may all be functions of time.
But regardless of what functions of time they may be, these constants will take on some values at...
I've been troubled by this problem for some time now and have received several answers to it none of which I find compelling, so I am posing it again in hopes of getting something more convincing.
Here's the problem. Consider one had a large optical interferometer with two siderostats place...
I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet.
First I'm decomposing a Gaussian wave packet
$$\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...
Can anyone tell me what a Gaussian Wave Packet is?
What happens to the atoms inside a Gaussian Wave Packet?
Can more than one Gaussian Wave Packet Exist in the same place?
Thank you,
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) =...
Hi,
to describe electronic transport and for example bloch oscillations, one uses a wave-packet build of bloch waves (with a band index n and an effective mass m*).
Do these wave-packets of blochwaves also spread (disperse) over time?
Hi, it was suggested previously on PF by others that a way to solve a ODE where the domain of the operator in Hilbert space allowed a real solution, is through the construction of wavepackets.
The conditions for real solutions are according to Kreyszig's Functional Analysis that E, in the...
Homework Statement
I know that for a dispersive wave packet, the group velocity equals the phase velocity, which is given by v=w/k. But how do I calculate the group velocity of a non-dispersive wave packet? I'm supposed to be giving an example with any functional form.
Homework Equations...
Let’s suppose we have an electron with a Gaussian eigenstate, as the time runs, the wave spreads in space without changing its energy, however, the induced EM field caused by the particle decreases its energy. I assert this from the classical electromagnetism result in which the more...
Homework Statement
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Homework EquationsThe Attempt at a Solution...
Homework Statement
I want to plot using Mathematica a wave packet entering a delta potential ##V(x) = s\delta(x) ## (s is the strength) but I need to get the physics right first and I'm having trouble with a a few parts. I need to compute the integral ## \int_{0}^{\infty} e^{-i\omega...
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)...
Hi,
what kind if integration used on equation 1 so it turned into equation 2? this does not look like integration by parts. and where (x-x0) appeared from instead of (k-k0) ?
thanks for your help.
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...
How to add up some Fourier component for example (cosinus function with different phase) to form a wave packet on GLE?
acctually, I don't know there is a such command in GLE to add up Fourier components or not. If there is not, so are there other aplications which can be used to do that, such as...
In some texts of fundamental quantum mechanics, it introduces the wave packet by Fourier transformation of a momentum wave into a spatial version. This is easy to understand because, analogy to the optical wave, a typical beam could compose waves of more than one frequencies. The general form is...
Homework Statement
Consider a wave packet satisfying the relation ## \Delta x \Delta p_x \approx \hbar ##
Show that if the packet is not to spread appreciably while it passes through a fixed position, the condition ## \Delta p_x << p_x ## must hold.
Homework Equations
## p_x = m \ddot{x} ##...
Homework Statement
The text states:
"Let us consider a wave packet whose Fourier inverse ##\phi (\vec{k})## is appreciably different from zero only in a limited range ##\Delta \vec{k}## near the mean wave vector ##\hbar \vec{\bar{k}} ##. In coordinate space, the wave packet ##\psi(\vec{r}, t)##...
The text states:
"Let us consider a wave packet whose Fourier inverse $\phi (\vec{k})$ is appreciably different from zero only in a limited range $\Delta \vec{k}$ near the mean wave vector $\hbar \vec{\bar{k}} $. In coordinate space, the wave packet $\psi(\vec{r}, t)$ must move approximately...
Homework Statement
Assume ## \phi(k_x ) = \sqrt2 {\pi}## for ## \bar{k}_x - \delta \le k_x \le \bar{k}_x + \delta##, and ##= 0## for all other values of ##k_x##. Calculate ##\psi(x, 0)##, and show that ## \Delta x \Delta k_x \approx 1 ## holds if ## \Delta x## is taken as the width at half...
Hello,
In David Bohm,s Quantum Theory, in chapter 3 section 2(motion of pulse of light)
There is this equation, I could not understand how, integrating LHS produced a sin.
Please help.
note:
LHS: k0 +Δk is super script. I did not know how to put it closer to integration sign.
Ex(x)= ∫k0-Δk...
I have a few general questions about how wave packets relate to particle creation and motion.
1) In the following video, Steve presents us with a simple case of a free particle at rest, where we have zero momentum and thus know the momentum exactly, but as a consequence we have no idea what...
Homework Statement
can liquid helium at 4k (.1 nanometers interatomic spacing) be modeled by non spreading wave packets?
Homework Equations
1/(√2π)∫∞-∞ei(kx-wt)Φ(k)dk
W(k) was Taylor expanded to a quadratic term Δpx2/(2mħ)
The book then sas for a non expanding wave packet:
|t| << mħ/Δpx2...
Hi all,
The Gaussian wave packet is widely discussed in the text. I got the following expression for wave packet in momentum space
##\psi(p, 0) = A \exp\left[-(p-p_0)^2/ (2\sigma_p^2)\right]##
with ##A=\sqrt{2\sigma_p/\sqrt{2\pi}}##
As my understanding, the corresponding wave packet in...
The problem stated below is from Liboff "Introductory Quantum Mechanics" (2nd Edition), exercise 5.4.
Homework Statement
A pulse ## 1m ## long contains ##1000 \alpha ## particles. At ## t=0## each ##\alpha## particle is in the state:
\psi (x,0)=\frac{1}{10}\exp (ik_ox)
for |x|\leq 50cm and...
Hi everyone,
I have a dispersive wave packet of the form:
##\frac{1}{\sqrt{D^2 + 2i \frac{ct}{k_0}}} e^{-y^2/(D^2+2i\frac{ct}{k_0})}##
The textbook says that the enlargement of the package, on the y direction, is:
##L=\frac{1}{D}\sqrt{D^4+4\left(\frac{ct}{k_0}\right)^2} ##
However I have some...
Hey everyone!
1. Homework Statement
I've been giving the equation for a gaussian wave packet and from that I have to derive this formula:
T_{Kepler}=2\pi \bar n ^3 by doing a first order taylor series approximation at \bar n of the phase:
f(x)=f(\bar n)+\frac{df}{dx}|_{\bar n}(x-\bar...
In solid state physics, I learned that the velocity of a bloch electron is ##\frac{\partial E(k)}{\partial k}##, where ##E(k)## is the energy dispersion. This formula is derived on the basis of the assumption that electrons is a wave packet of bloch state in solids.
However, I have a question...
Homework Statement
So, I start off with initial condition:
\psi(x,0) = e^{\frac{-(x-x_0)^2}{4a^2}}e^{ilx}
This wavepacket is going to move from x_0 = -5pm towards a potential barrier which is 1 MeV from x = 0 to x = 0.25 pm, and 0 everywhere else.
a = 1 pm
l = 2 pm-1
Homework Equations
I...
Homework Statement
The free particle wave packet in question is $$\psi=ce^{-(r/r_0)^2}$$
Homework EquationsThe Attempt at a Solution
I've been going through books and class notes but I really have no idea where this came from. I'm thinking that if I can decompose this in plane waves I could...
Homework Statement
Following gaussian wave packet: ## \psi (x)= \frac{1}{\sqrt{\sqrt{\pi a^2}}} e^{-\frac{x^2}{2a^2}}##
Prove that this function is normalized.
Homework Equations
## \int_{- \infty}^{\infty} |\psi (x)|^2 dx = 1##
The Attempt at a Solution
Is ## \frac{1}{\sqrt{\sqrt{\pi a^2}}}...
In my course there's a chapter with the mathematical explanation to find the real expression and localisation of a free particle with the superposited wave function. The same is used to explain the movemement of a wave packet (which is a free particle). I've worked out almost all the math behind...
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 are the expansion coefficients of a wavepacket \Psi (x) = \sqrt{\frac{2}{L}}sin \frac{\pi x}{L} in the basis Ψn(x) of a particle in a periodic box of size L?
Homework Equations
\Psi (r,t) = {\sum_{n}^{}} a_{n}(t) \Psi _{n}(r)
The Attempt at a Solution
\left \langle...
If you solve TISE with V=0, you get a plane wave.
This is not normalizable, so it's not a physically achievable state.
But a linear combination does (?) I know why it does from the mathematics, a linear combination of plane waves is normalizable, but what does it really mean? A free particle is...
Homework Statement
Wave function of an electron:
##\frac{1}{N}∫_a^b[e^{ik_0x}(1+\frac{1}{2}e^{i\frac{0.1}{2}x}+\frac{1}{2}e^{-i\frac{0.1}{2}x})]dx##
The integrand becomes zero both to the left and right of x = 0 . Let a be the first time it hits zero to the left and b the first time it...
I am trying to understand how a gaussian packet varies in time.
Suppose we have a Gaussian wave packet that is displaced form the origin by an amount x0 and given initial momentum p0. So the wave function in coordinate space is
ψ(x,0) =\frac{√β}{√√\pi}exp(-β2(x-x0)2/2)*exp(ip0x/ħ)
where...
Hi :)
I'm reading a didactic paper and the author defined the initial state ket as
|\Phi_{in}> = {\int}dq\phi_{in}(q)|q>
where q is a coordinate and
\phi_{in}(q) = <q|\Phi_{in}> = exp\left[\frac{-q^{2}}{4\Delta^{2}}\right]
I don't know if I'm missing something but isn't this definition a...
Hi, I'm trying to get my head around Schrödinger's equation and quantum wave theory. I'll try to shortly state how I understand it so you may see where I'm wrong and better answer the question.
In classical mechanics if you solve a linear differential equation, the sum of the solutions is also...
Generally speaking, what is the difference between these two? What I mean by that is: in what kind of different processes are these produced (and used in physics)?