Spreading of Gaussian Wave Packet

In summary, the question is asking you to calculate the fractional change in size of the wavepacket over a time period of one second for the case when its initial size is 10^-10, and then separately calculate it for the case when its initial size is 10^-16.
  • #1
Shackleford
1,656
2
We finally started modern quantum mechanics. I'm definitely feeling a bit lost or fuzzy, especially in the more rigorous book Quantum Physics by Gasiorowicz. At any rate, we have two more problem sets before the last exam.

As for the problem, I'm not sure what to. Is it asking to calculate the fractional change in size of a wave packet when it spreads to 10^-10 m from 10^-16 m? I guess I could also call the given dimensions L, such as in delta-x = L/2 or something. Sorry for the poor scan.

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  • #2
Well to start with what is the wavefunction (as a function of space and time) of a Gaussian representing a free particle.

What could we mean by the 'size' of the wavepacket?

(I think the question is asking you to calculate the fractional change in size of the wavepacket over a time period of one second for the case when its initial size is 10^-10, and then separately calculate it for the case when its initial size is 10^-16)
 
  • #3
fantispug said:
Well to start with what is the wavefunction (as a function of space and time) of a Gaussian representing a free particle.

What could we mean by the 'size' of the wavepacket?

(I think the question is asking you to calculate the fractional change in size of the wavepacket over a time period of one second for the case when its initial size is 10^-10, and then separately calculate it for the case when its initial size is 10^-16)

The Gaussian wave function is (sorry I still haven't learned latex yet)

psi-(x,t) = integral from negative infinity to positive infinity of dk A(k)e^(kx-wt)

From remembering what's in my Modern Physics textbook, the size of the wave packet could be half of the wavelength. And there's a relation I think that applies to all wave functions generally: delta-k * delta-x = 1/2
 
Last edited:

1. What is a Gaussian wave packet?

A Gaussian wave packet is a type of wave function that describes the probability of a quantum particle being in a certain location. It is named after the mathematician Carl Friedrich Gauss and has a bell-shaped curve. It is commonly used to describe the spread of a quantum particle's position and momentum over time.

2. How does a Gaussian wave packet spread?

A Gaussian wave packet spreads due to the uncertainty principle in quantum mechanics. This means that as the particle's position becomes more certain, its momentum becomes less certain, and vice versa. As the particle moves, the wave packet spreads out in both position and momentum space, resulting in a larger uncertainty in both quantities.

3. What factors affect the spreading of a Gaussian wave packet?

The spreading of a Gaussian wave packet is affected by several factors, including the initial width of the wave packet, the mass and kinetic energy of the particle, and any external forces acting on the particle. Additionally, the time over which the packet is observed and the properties of the potential energy function also play a role in determining the rate and extent of spreading.

4. How can the spreading of a Gaussian wave packet be visualized?

The spreading of a Gaussian wave packet can be visualized using various techniques, such as plotting the wave function in position and momentum space over time or using computer simulations to show the movement of the particle and the changing shape of the wave packet. Additionally, experiments can be conducted using particles such as electrons in a potential well to observe the spreading of the wave packet.

5. Can the spreading of a Gaussian wave packet be controlled?

The spreading of a Gaussian wave packet is a natural phenomenon in quantum mechanics and cannot be directly controlled. However, by manipulating the initial conditions and external factors, such as the potential energy function, it is possible to influence the rate and extent of spreading. This is an important aspect of quantum mechanics and is utilized in various technological applications, such as in quantum computing and quantum cryptography.

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