Understanding Wavepacket Spreading in QM

  • Thread starter Saketh
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In summary, the wavepacket spreads as time goes on because the uncertainty in the particle's position and momentum increases with time.
  • #1
Saketh
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I'm working through the first few chapters of my QM textbook, so I am not yet familiar with the Schrodinger equation.

Consider a free particle, say an electron, moving through free space. I have done the calculations, and concluded that the wavepacket must spread -- that is, get wider. However, this does not make sense to me. How can the wavepacket spread with time? That is, why doesn't the wavepacket just translate?

Then I realized that if the wavepacket just translated, we could determine the particle's momentum from its translation.
My questions:
  1. What does the spreading of the wavepacket with time represent?
  2. Would spreading appear to both an observer at rest and a non-relativistic observer in motion?
  3. Why does the "hump" in the packet flatten out as time goes on? (I know the mathematics, but I'm not sure what it represents.)
  4. Am I right in assuming that the momentum wave function [tex]\phi(k)[/tex] is constant with respect to time?

Thanks for your assistance.
 
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  • #2
1. If you measured the particle's position, then there was some resolution to your postion-measuring apparatus. Due to the uncertainty relation, you will have a minimum uncertainty in the particle's momentum, determined by the precise form of your initial wavefunction and the resolution of your device, and this uncertainty will translate to a spreading of your uncertainty in its location, as time progresses.
2. Spreading would appear to both, though the exact amount would differ, in some way. However, in order to treat this properly you will need a relativistic theory, such as a quantum field theory, not just bare non-relativistic quantum mechanics. The exact field theory will depend on the particle. If it is an electron, then that will be quantum electrodynamics...
3. The flattening out represents an uncertainty in both the variable that you measured and in the variable conjugate to it. These uncertainties will grow with time according to your initial measurement precision.
4. The momentum-space wavefunction is not constant. Based on what I said regarding the statistical spreading and the uncertainty principle, think about why this should be so.
 
Last edited:
  • #3
degrees of knowledge

My suggestion is to think about quantum mechanics as a theory that describes what you know and how well you can know it, rather than some reality that is "out there", independent of experiments.
 
  • #4
So the more accurately you know position or momentum at the beginning, the less accurately you know it later on.

That makes sense...thanks for your explanation!
 

1. What is a wavepacket in quantum mechanics?

A wavepacket in quantum mechanics is a mathematical representation of a particle in motion. It is a localized wave that describes the probability of finding a particle at a certain position and time.

2. How does a wavepacket spread in quantum mechanics?

In quantum mechanics, a wavepacket spreads due to its inherent wave-like nature. This spreading is known as wavepacket spreading or wavepacket dispersion. It is caused by the uncertainty principle, which states that it is impossible to know the exact position and momentum of a particle at the same time. As the wavepacket spreads, the uncertainty in position and momentum increases.

3. What factors affect the rate of wavepacket spreading?

The rate of wavepacket spreading is affected by several factors, including the initial width of the wavepacket, the shape of the potential well, and the energy of the particle. Higher initial widths, deeper potential wells, and lower energy particles will result in slower spreading of the wavepacket.

4. How is wavepacket spreading related to the Heisenberg uncertainty principle?

Wavepacket spreading is a direct result of the Heisenberg uncertainty principle. As the wavepacket spreads, the uncertainty in position and momentum increases, which is in line with the principle's assertion that it is impossible to know both quantities with absolute certainty.

5. What is the significance of understanding wavepacket spreading in quantum mechanics?

Understanding wavepacket spreading is crucial for accurately predicting the behavior of particles in quantum systems. It allows scientists to make precise calculations and predictions about the position and momentum of particles, which is essential for many applications in fields such as material science, chemistry, and quantum computing.

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