What's the kinetic energy uncertainty for gaussian wave packet?

In summary, the uncertainty in the kinetic energy of a gaussian wave packet ψ(x)=((α/pi)^(1/4))exp(-αx^2/2) can be computed by taking the expectation value of the kinetic energy operator K=p^2/2m and using the formula σK=(p/m)σp. However, due to some subtleties, this may not always work and an alternative method involving the 4th moment of the Gaussian may be necessary.
  • #1
AlonsoMcLaren
90
2
What's the kinetic energy uncertainty for gaussian wave packet ψ(x)=((α/pi)^(1/4))exp(-αx^2/2)?
 
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  • #2
Can you compute the uncertainty in p? If you know the uncertainty in p and you know that K=p^2/2m can you compute the uncertainty in K?
 
  • #3
I guess you can get the EXPECTATION VALUE of K, not the uncertainty in K, by squaring the uncertainty in p and dividing it by 2m
 
  • #4
[tex]K=\frac{p^2}{2m}[/tex]

[tex]<K>=<\frac{p^2}{2m}>=\frac{<p^2>}{2m}[/tex]

[tex]\sigma_K=\frac{\partial K}{\partial p}\sigma_p[/tex]

Does that look reasonable?
 
  • #5
σK=(p/m)σp

what is p? does p= <p>?
 
  • #6
Yes, as usual, you put the expectation values in there.
 
  • #7
Then <p> should be zero..
 
  • #8
Ah, well I think there may be something subtle happening here that prevents my formula from working. In the mean time then, you can compute this by brute force:

[tex]\sigma_K=\sqrt{\langle \psi | K^2|\psi\rangle-(\langle \psi | K|\psi\rangle)^2}=\frac{1}{m}\sqrt{\langle \psi |\frac{p^4}{4}|\psi\rangle-(\langle \psi | \frac{p^2}{2}|\psi\rangle)^2}[/tex]

This involves the 4th moment of the Gaussian though...which may be kind of hard...maybe someone more enlightened can fill us in then...XD
 

1. What is a gaussian wave packet?

A gaussian wave packet is a mathematical representation of a particle's wave function in quantum mechanics. It describes the probability of finding the particle at a certain position and time.

2. How is kinetic energy uncertainty calculated for a gaussian wave packet?

Kinetic energy uncertainty for a gaussian wave packet is calculated using the uncertainty principle, which states that the product of the uncertainties in position and momentum must be greater than or equal to h/4π, where h is Planck's constant. The specific formula for calculating the kinetic energy uncertainty involves the width and spread of the wave packet.

3. Why is kinetic energy uncertainty important in quantum mechanics?

Kinetic energy uncertainty is important in quantum mechanics because it is a fundamental limit to how well we can know the position and momentum of a particle at the same time. This uncertainty is a fundamental aspect of quantum mechanics and is necessary for understanding the behavior of particles at the subatomic level.

4. How does the uncertainty in kinetic energy affect the behavior of a particle?

The uncertainty in kinetic energy affects the behavior of a particle by limiting our ability to know both its position and momentum accurately. This can result in a particle appearing to behave in a wave-like manner, rather than a purely particle-like manner, as its exact position and momentum cannot be simultaneously determined.

5. Can the kinetic energy uncertainty for a gaussian wave packet be reduced?

The kinetic energy uncertainty for a gaussian wave packet is a fundamental aspect of quantum mechanics and cannot be reduced. However, by decreasing the spread and width of the wave packet, the uncertainty can be minimized, resulting in a more accurate determination of the particle's position and momentum.

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