Wave packet Definition and 116 Threads

In particular, i am solving part b. I pulled off a couple of formulas from a textbook, but I'm quite sure they are incorrect to apply here. Can anyone guide me? Below is my attempt.

Homework Statement A laser with λ = 1 μm is pulsed (turned on and then back off) with a duration of 100 fs. What is the resulting frequency spread Δf in output of the pulsed laser? Assume that the pulse has a Gaussian envelope. Homework Equations Not really sure, but possibly ψ(x,0)...

Homework Statement Suppose a Gaussian wave packet ψ(x,0) is built out of plane waves according to the amplitude distribution function A_{k} = \frac{Ca}{\sqrt{\pi}}e^{(-a^2(k-k_{0} )^2)} Calculate ψ(x,t) for this packet and describe its evolution. Homework Equations ψ(x,t) =...

When electromagnetic waves of different frequencies interact, they give rise to secondary wave structures called envelopes in which individual waveforms form at the rear and die out at the front. These envelopes are called groups and they travel with a velocity called group velocity and the...

I was wondering if there has been a study done on what happens to a photon wave packet when half out of a device while the device emitting it is moved. If never done it would make an interesting scientific experiment to see where the influence of moving the device ends. I'd like to see...

Homework Statement Any wavepacket can be obtained by the superposition of an infinite number of plane waves using the so-called Fourier integral or Fourier transform f(x,t) = \frac{1}{\sqrt{2\pi}} _{-\infty}\int^\infty A(k)e^{i(kx-\omega t)} dk Find at t=0 the representation of the...

Homework Statement Hi , I am reading a little on introductory QM , initial chapters on waves. They have given an integral for a wavepacket , assuming at t= 0. Which is: ψ(x,0) = \int A cosk'x dk' (I don't know how to define limits to the integral in Latex upper = k+Δk , lower limit =...

A wave packet is formed when a multitude of sine waves of different periods are linearly added. This localizes the pdf, and hence creates certainty in a particle's position. However, basic sinusoidal functions are cyclical, hence the additton of two sinusoidal functions are always...

Anyone know how “long” a photon wave packet envelope is? I get the impression from Purcell that it might be only about 1 wavelength long. But then I think I saw on this forum someone mention that it could be “thousands” of wavelengths long. How would or could one determine that? I am...

Hello Colleagues, I am curious about a problem in Quantum Mechanics that incorporates the evolution of a Gaussian Wave Packet under the Gravitational Potential. What I am interested in is equation (3) in the following paper: "On the quantum analogue of Galileo's leaning tower...

for a 1D free particle with initial wave function \phi(x') square shaped(e.g. \phi(x')=1,x'\in [a,b],otherwise it vanishes), my question is: how does it evolve with time t? if we deal with it in P basis, it is easily solved, using the propagator U(t)=∫|p'><p'|e^{-\frac{ip'^2...

A wave packet of mean energy E0 is incident on a potential square barrier. The figure below shows stills from a film showing 9 successive stages in interaction. a) True or false? The incident packet at time t = 0 describes many particles, each in a different momentum state. If false, tell...

Homework Statement This is problem 2.22 from D.J. Griffiths Introduction to Quantum Mechanics A free particle has the initial wave function: \Psi(x,0)=Ae^{-ax^{2}} Find \Psi(x,t). Hint Integrals of the form: \int_{-\infty}^{\infty}e^{-(ax^{2}+bx)}dx can be handled by completing the square...

Homework Statement Consider a one-dimensional wave packet with \phi(p) = A \Theta ((h/2\pi) / d -|p-p0|) Determine the constant A and find the spatial wave function ψ(x). Ignore temporal evolution. Homework Equations The Attempt at a Solution Honestly this is a little embarrassing, but the...

Why do wave packets expand when they evolve in time? Is there any physical reason behind this? Is it a conservation law of some quantities that makes it expanded?

Homework Statement Consider a force-free particle of mass m described, at an instant of time t = 0, by the following wave packet: \begin{array}{l} 0 \ \mathrm{for} \ |x| > a + \epsilon \\ A \ \mathrm{for} \ |x| ≤ a \\ -\frac{A}{\epsilon} (x − a − \epsilon) \ \mathrm{for} \ a < x ≤ a + \epsilon...

Hello, (for those in a hurry: the last paragraph contains the essence) I'm in my last year of bachelor of physics and following a QM class and as is standard we calculated the reflection R and transmission T coefficients for a plane wave in some potential well/barrier situation. Our...

What's the kinetic energy uncertainty for gaussian wave packet ψ(x)=((α/pi)^(1/4))exp(-αx^2/2)?

Homework Statement Calculate <x> for the Gaussian wave packet \psi(x)=Ne-(x-x0)/2k2 Homework Equations \left\langle x \right\rangle = \int dx x|\psi(x)|2 The Attempt at a Solution So I've been reviewing for the up-coming midterm and I've had the painful realization that I'm...

Homework Statement I have been given the function \varphi_{G}(z,t) = \frac{2}{\Delta k_0\sqrt{\pi}} \int_{-\infty}^{\infty}dk\ e^{-\frac{4(k-k_0)^2}{\Delta k_0^2}}e^{i(kz-\omega t)} and been told to do the integration and then to specify the phase and group velocity of the wave package. I...

Homework Statement start with the wave function \Psi(x,0) = Ae^{-cx^{2}}e^{ikx} where A,c, and k are real constants (and c is positive) i) Normalize \Psi(x,0) ii) Determine \Psi(x,t) and |\Psi(x,t)|^{2} Homework Equations The Attempt at a Solution I normalized it to get...

I am doing my extended essay on the maths involved in GPS, and I am trying to understand the concept of the group velocity of a wave packet, http://www.mathpages.com/home/kmath210/kmath210.htm" they explain it more or less, but I still don't get how do they differenciate dw/dk to get...

Homework Statement A wave packet is described by the momentum-space wave function A(p)=C when 0<p<p0, and A(p)=0 for all other values of p. Here C is a constant. i) Normalize this wave function by solving for C in terms of p0. ii) Calculate the expectation values <p> and <p2>. From...

It says on my homework that: "A wave packet is like a periodic function with a repeat length of _______." I am confused by "repeat length" because I understand a wave packet to NOT repeat anywhere. It has a spread x but the peaks are all different heights, so I do not really understand what...

Homework Statement It can be shown that the position probability density for the one-dimensional, free-particle Gaussian wave packet can be expressed as shown below, where m is the mass of the particle and L is the position uncertainty at time t=0, vgr = hbar *k0/m, k0 is the average wave...

Homework Statement A free particle has wave function (i have no idea how to get these come out correctly, so I can't use LaTeX. :frown: Edit on the LaTeX. I'm getting closer! Ok...I can't get rid of the space but that's another subject.:-p...

Homework Statement Consider the wave packet \varphi(k) = \int B(k)cos(kx) dk from 0 to infinity and B(k) = exp(^{-a^{2}k^{2}}). Find \varphi(k) Homework Equations The Attempt at a Solution After looking up integral tables, i got an expression involving error function (erf) and...

In a QM introductory book , I have read that the wave packet is not a solution of the Schrodinger equation, is this true in some context or is it just an mistake of the author?

Homework Statement Consider this wave packet: \Psi(x)=A exp \left[\frac{i(p + \Delta p)x}{\hbar}\right] + A exp \left[\frac{i(p - \Delta p)x}{\hbar}\right] (from a previous problem) This time-dependent form of the packet is: \Psi(x)=A exp \left[\frac{i(p_{1}x-E_{1}t)}{\hbar}\right] + A exp...

This question is about normalizing a wave packet, this is actually the solution and I couldn't understand 2 points. - I cannot see how the red encircled part do not diverge to infinity. - And I cannot understand how the very last line is derived from the 2nd last one. For the second one...

Hi, everyone! I'm thinking of a problem how a wave packet moves in a constant magnetic field. I want to solve it by numerical calculation.First,I initiate it as a Gauss wave packet. What I understand is it will circle in the magnetic field as time goes by,and it will diffuse at the same time...

We finally started modern quantum mechanics. I'm definitely feeling a bit lost or fuzzy, especially in the more rigorous book Quantum Physics by Gasiorowicz. At any rate, we have two more problem sets before the last exam. As for the problem, I'm not sure what to. Is it asking to calculate...

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...

Homework Statement So I already finished most of this problem, but I'm having trouble figuring out the very last part second part. The last part of the problem is: "Finally, take one additional term in the Taylor series expression of \omega(k) and show that \sigma is now replaced by a...

Why Schrodinger postulated that a material particle in motion is equivalent to a wave packet rather than a single wave train?

The idea of a wave packet is confusing a bit. First let me tell you what I have understood about it. A wave packet is obtained by the superposition of harmonic waves of nearby frequencies which represents a matter wave. It is well known fact that the width of a moving Gaussian wave packet in...

Homework Statement Hi, The problems asks to calculate multiple things for a Gaussian wave packet. Steady state function: psi(x,0)=A*exp(-ax^2).My problem is that I'm stuck at calculating <p^2>.Homework Equations <p^2>=Int(|psi|^2*(-1*h^2*d^2/dx^2))dx or...

Homework Statement For a free particle, Given psi(x,0) = Aexp(-ax^2), find psi(x,t) Homework Equations phi(k) = 1/(sqrt(2pi)) times integral from -inf to +inf (psi(x,0)exp(-ikx))dx psi(x,t) = 1/(sqrt(2pi)) times integral from -inf to +inf (phi(k)exp(i(kx - (hk^2)t/2m)))dk my apologies...

Show that wave packet is an eigenstate to operator [SOLVED] Homework Statement For a harmonic oscillator we can define the step up and down operators \hat{a} and \hat{a}^{\dagger} and their action as \hat{a}=\sqrt{\frac{m\omega}{2\hbar}}(\hat{x}+\frac{\imath}{m\omega}\hat{p}) \quad...

Sound waves of 499 Hz and 506 Hz are superimposed at a temperature where the speed of sound in air is 340 m/s, Now the question is what's the lenth delta(x) of the wave packet in meters?

Hello, I have a question about basic Quantum Mechanics. In free particle study of Quantum Mechanics, why do we use the Gaussian wave packet always? I know that the Fourier Transform of it is also a Gaussian wave packet, but I was thinking, is that what we have selected? or is that what...

hi this is abhimanyu. i m a new comer in the world of quantum physics. pls tell me what is wave packet?

What is the relationship wavelength and group velocity in wave packet ? If a wave packet has big wavelength its group velocity bigger or not ?

How particle delocalization can be experimentally verified? Thanks

Hello. I am reading about the wave packet in my physics textbook, and I understood that it describes the probability of a particle having a given position and a momentum in a particular state. However, I just cannot understand where those waves that consist the wave packet come from. I will...

OK, I think I understand the problem, I'm just a little confused on some pieces in the middle. Problem: Consider a wave packet formed by the superposition of two waves psi1=cos(1.00x-2.00t) <--1.00=k, 2.00=w psi2=cos(1.01x-2.03t) <--1.01=k, 2.03=w where x...

Homework Statement A radar transmitter used to measure the speed of pitched baseballs emits pulses of 2.0cm wavelength that are .25micros 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...

1.in the attached file see the fig 8.5 What is the the physical reason,for which ahe amplitude of the wave function is larger in between x=0 & x= a ? 2.what is the reason for choosing a wave packet to describe a particle ?

Hi, I'm teaching myself quantum mechanics (so this isn't homework). I came across the following question: \Phi(p) = A\Theta\left[\frac{\hbar}{d}-|p-p_{0}|\right] I have to find the constant of normalization, \psi(x), and the coordinate space wave function \psi(x,t) in the limit...

Hello. I started reading this little book by Heisenberg. It starts giving a mathematical relation of width of wave packet and range of wavelengths necessary to mathematically construct it, and then drops in wavalength-momentum relations to give a quick insight on the nature of the uncertainty...