Wave-packet in configuration space

In summary, Wigner discusses the elements of quantum mechanics in his book "Group theory and its Applications to the Quantum Mechanics of atomic spectra." He explains that in a many-dimensional space with position coordinates, the motion of a system can be described by a point in configuration space. This point's movement corresponds to the classical motion of the system and the motion of a wave packet in configuration space, assuming a specific index of refraction. However, the meaning and interpretation of this refractive index is not clear.
  • #1
Pradyuman
6
0
In the book "Group theory and it's Applications to the Quantum Mechanics of atomic spectra " by Eugene P. Wigner

in chapter 4 The elements of quantum mechanics it is written

Consider a many dimensional space with as many coordinates as the system considered as position coordinates. Every arrangement of the positions of the particles of the system corresponds to a point in this multidimensional configuration space. This point will move in the course of time tracing out a curve by which the motion of the system can be completely described classically. There exists a fundamental correspondence between the classical motion of this point, the system point in configuration space, and the motion of a wave packet also considered in configuration space, if only we assume that the index of refraction for these waves is ##\sqrt{2m(E-V)}\over E##, where ##E## is the total energy of the system;##V## is the potential energy as a function in the configuration space.
What does the wave-packet and the refractive index implies here.How to interpret this?
 
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  • #2
I do not know the index of refraction in this context. According to the formula you quote, it has physical dimension of ##L^{-1}T##, inverse of velocity, if he does not apply some convention of unit that you have not quoted there.
 

FAQ: Wave-packet in configuration space

What is a wave-packet in configuration space?

A wave-packet in configuration space is a localized wave function that represents a particle's probability amplitude in a given position space. It is a superposition of multiple wave functions with different wavelengths and momenta, allowing the particle to be described with a finite spread in both position and momentum.

How is a wave-packet formed?

A wave-packet is formed by the interference of multiple plane waves with different wavelengths and momenta. Mathematically, it is represented by the Fourier transform of a range of wave functions. The constructive and destructive interference of these waves creates a localized packet that can evolve over time.

What is the significance of a wave-packet in quantum mechanics?

In quantum mechanics, a wave-packet provides a more realistic description of a particle compared to a single plane wave. It allows for the localization of a particle in space, which is crucial for understanding phenomena such as quantum tunneling and the Heisenberg uncertainty principle. The wave-packet approach also helps in analyzing the time evolution and spreading of particle states.

How does a wave-packet evolve over time?

The time evolution of a wave-packet is governed by the Schrödinger equation. In free space, a wave-packet generally spreads out over time due to the different phase velocities of its constituent waves. This spreading can be described by the dispersion relation, and the rate of spreading depends on the initial conditions and the properties of the medium in which the wave-packet propagates.

What is the relationship between a wave-packet and the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the product of the uncertainties in position and momentum of a particle cannot be smaller than a certain value (ħ/2). A wave-packet inherently satisfies this principle because it has a finite spread in both position and momentum. The more localized the wave-packet is in position space, the broader its spread in momentum space, and vice versa, illustrating the trade-off described by the uncertainty principle.

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