Momentum information from position space wave function

In summary, the conversation revolves around developing a graphical simulation of a wave function in position space, with the added feature of being able to collapse the wave function by measuring its position or momentum. The issue at hand is calculating the momentum space wave function from the position space wave function using a Fourier transformation. The example given involves a 1D Gaussian wave packet and the necessary normalization factor and standard deviation calculations for both the position and momentum space wave functions are discussed. The conversation also touches on the Heisenberg-Robertson uncertainty relation and the unique properties of Gaussian wave packets.
  • #1
jmd_dk
10
0
I am trying to develop a graphical, interactive simulation of a wave function in position space, given an arbitrary potential. It works great, but as a final touch, I would like the user to be able to collapse the wave function, either by measuring its position or its momentum.

My problem is then, how do I calculate the momentum space wave function, given the position space wave function? I know that a simple Fourier transformation should do the job, but when I calculate it I always end up with some even momentum space wave function.

Example: Make a 1D Gaussian wave packet in position space, traveling with speed v. Its momentum space wave function will also be Gaussian, centered on the origin. But it should be centered on v! How do you capture the information of movement from the position wave function alone, at a single instant?

Thank you.
 
Physics news on Phys.org
  • #2
The Gaussian wave packet centered around [itex]p_0[/itex] is ([itex]\hbar=1[/itex])
[tex]\tilde{\psi}(p)=N \exp \left (-\frac{(p-p_0)^2}{4 \Delta p^2} \right),[/tex]
The normalization factor should be chosen such that
[tex]\int_{\mathbb{R}} \mathrm{d} p \tilde{\psi}(p)=1 \; \Rightarrow \; N=\frac{1}{(2 \pi \Delta p^2)^{1/4}}.[/tex]
Further [itex]\Delta p[/itex] is the standard deviation of [itex]p[/itex].

The same state in position space is
[tex]\psi(x)=\frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} \mathrm{d} p \tilde{\psi}(p)=\left (\frac{2 \Delta p^2}{\pi} \right)^{1/4} \exp \left (-\Delta p^2 x^2+\mathrm{i} p_0 x \right).[/tex]
Note that from this we read off
[tex]2 \Delta x^2=\frac{1}{2 \Delta p^2} \; \Rightarrow \; \Delta x=\frac{1}{2 \Delta p},[/tex]
which implies that
[tex]\Delta x \Delta p=\frac{1}{2}.[/tex]
This shows that the Gaussian wave packet has the minimal product of [itex]\Delta x \Delta p[/itex] according to the Heisenberg-Robertson uncertainty relation. It's a coherent state, and one can show that the Gaussian wave packets are the only ones with that minimum-uncertainty property.
 

Question 1: What is momentum information from position space wave function?

Momentum information from position space wave function is a way to analyze the momentum of a particle based on its position. This is done by using the wave function, which represents the probability of finding a particle in a certain position in space.

Question 2: How is momentum information obtained from position space wave function?

Momentum information is obtained from position space wave function through mathematical operations such as taking the derivative or Fourier transforming the wave function. These operations allow us to extract information about the momentum of the particle from its position in space.

Question 3: Why is momentum information important in quantum mechanics?

Momentum information is important in quantum mechanics because it is one of the fundamental properties of particles. It helps us understand the behavior and interactions of particles, and is crucial in many quantum mechanical equations and principles.

Question 4: Can momentum information from position space wave function be used to predict the future behavior of a particle?

No, momentum information from position space wave function cannot be used to predict the future behavior of a particle with certainty. According to the Heisenberg uncertainty principle, there is always a limit to how accurately we can know both the position and momentum of a particle at the same time.

Question 5: How does momentum information from position space wave function relate to classical mechanics?

Momentum information from position space wave function relates to classical mechanics through the concept of wave-particle duality. In classical mechanics, particles are described as having a definite position and momentum, whereas in quantum mechanics, these properties are represented by the wave function and are subject to uncertainty. However, in certain cases, the momentum information obtained from position space wave function can approach the classical momentum of a particle.

Similar threads

  • Quantum Physics
Replies
24
Views
590
  • Quantum Physics
Replies
1
Views
595
Replies
32
Views
2K
  • Quantum Physics
Replies
27
Views
2K
  • Quantum Physics
Replies
1
Views
686
Replies
8
Views
1K
  • Quantum Physics
Replies
19
Views
2K
Replies
1
Views
633
  • Quantum Physics
Replies
5
Views
848
Back
Top