Comparison of width of a wavefunction in real space and momentum space

In summary, the speaker has solved the Schrödinger equation for a delta potential in both momentum and real space. They are now trying to find the correlation between the width of the wavefunction in both spaces and are considering using the uncertainty relation of momentum and space. They are also considering calculating the characteristic width of the wave functions in both spaces using the approximations \kappa x \approx 1 and p / \hbar \kappa \approx 1. They mention the need to calculate Δx and Δp in order to determine the correlation between the widths in both spaces.
  • #1
BasharTeg
5
0
Hello, I have a slight problem with Quantumtheory here.

Homework Statement


I have solved the schrödinger equation in the momentum space for a delta potential and also transferred it into real space. So now I have to find the correlation between the width of the wavefunction in both spaces (and then motivate it physically) and I am stuck here because I don't even know where to start.


Homework Equations


[itex]\Psi (x) = \sqrt{\kappa}e^{- \kappa |x|}[/itex]

[itex]\Psi (p) = \frac{\sqrt{2 ( \hbar \kappa)^3}}{\sqrt{\pi}(p^2 + (\hbar \kappa)^2)}[/itex]


The Attempt at a Solution


I was thinking about maybe the uncertainty relation of momentum and space would help here, but I am stuck where to start.


Hope someone can help or give a hint.
 
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  • #2
Just looking at the functions, you can approximate the characteristic width of the wave functions in position space by using [itex]\kappa x \approx 1[/itex] and in momentum space by using [itex]p / \hbar \kappa \approx 1[/itex].

If you want to be more precise, calculate [itex]\Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}[/itex] and [itex]\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}[/itex].
 
  • #3
Thanks I will look into it. I guess I have to calculate Δx and Δp since I need a correlation how the width in momentum space affects the width in real space and vice versa.
 

What is a wavefunction?

A wavefunction is a mathematical description of a particle's quantum state. It contains information about the particle's position, momentum, and other properties.

What is real space and momentum space?

Real space refers to the physical space in which particles exist, while momentum space refers to the space of all possible momentum values for a particle.

Why is the width of a wavefunction important?

The width of a wavefunction is important because it provides information about the uncertainty in a particle's position or momentum. A narrower wavefunction indicates a more precise determination of position or momentum, while a wider wavefunction indicates a greater uncertainty.

What is the relationship between the width of a wavefunction in real space and momentum space?

The width of a wavefunction in real space and momentum space are related by the Heisenberg uncertainty principle. This principle states that the more precisely a particle's position is known, the less precisely its momentum can be known, and vice versa.

How does the width of a wavefunction change over time?

The width of a wavefunction can change over time due to quantum effects, such as wavefunction collapse or wavefunction spreading. These changes can be observed in experiments and can affect the behavior of particles at the quantum level.

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