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 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...
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...
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...
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...
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}}}...
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...