# Does a Gaussian wave packet remain Gaussian?

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In summary: How do you work on a(t)?Hint: It depends on the Hamiltonian! Just think about the question, which class of Hamiltonians ##H(x,p,t)## have a chance to preserve the Gaussian shape (of course with time-dependent parameters, which for a Gaussian are just the average and the standard deviation, as in your ansatz). The hint about Ehrenfest's theorem is also very valuable :-)).
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Consider a gaussian wave packet whose wave function at a particular instant of time is

Its time dependence is implicit in the "constants" A, a, <x> and <p>, which may all be functions of time.
But regardless of what functions of time they may be, these constants will take on some values at another instant of time and remain independent of x. So the wave function (at this new time) is still gaussian.
So a gaussian wave packet remains gaussian. True or false?

I think it's false. But what's wrong with the deduction?

At the moment you simply assume that the wave function will always have that shape. You have to show that there are functions A(t), a(t), <x>(t), <p>(t) such that the resulting time-dependent wave function solves the Schroedinger equation. The latter two should be quite easy (Ehrenfest theorem), then you can work on a(t).

mfb said:
At the moment you simply assume that the wave function will always have that shape. You have to show that there are functions A(t), a(t), <x>(t), <p>(t) such that the resulting time-dependent wave function solves the Schroedinger equation. The latter two should be quite easy (Ehrenfest theorem), then you can work on a(t).

If the wave packet hits an obstacle, like a potential barrier or a potential well or collide with another wave packet, then it no longer remains gaussian.

But if it is free to move ahead on its own, then it remains gaussian.

True?

Don't guess, calculate it.

mfb said:
Don't guess, calculate it.
How do you work on a(t)?

Hint: It depends on the Hamiltonian! Just think about the question, which class of Hamiltonians ##H(x,p,t)## have a chance to preserve the Gaussian shape (of course with time-dependent parameters, which for a Gaussian are just the average and the standard deviation, as in your ansatz). The hint about Ehrenfest's theorem is also very valuable :-)).

## 1. What is a Gaussian wave packet?

A Gaussian wave packet is a type of wave function that describes the probability amplitude of a particle in quantum mechanics. It is characterized by a bell-shaped curve and is used to model the behavior of particles in many physical systems.

## 2. Does a Gaussian wave packet remain Gaussian over time?

In most cases, yes. A Gaussian wave packet will typically remain Gaussian as it evolves over time, as long as it is not subjected to any external influences or interactions. However, there are certain scenarios where the wave packet may become distorted or spread out over time.

## 3. What factors can cause a Gaussian wave packet to deviate from its original shape?

External forces or interactions, such as collisions or measurements, can cause a Gaussian wave packet to deviate from its original shape. Additionally, the uncertainty principle in quantum mechanics can also contribute to the spreading out of a wave packet over time.

## 4. How does the width of a Gaussian wave packet affect its behavior?

The width of a Gaussian wave packet is directly related to the uncertainty in the position and momentum of a particle. A narrower wave packet corresponds to a smaller uncertainty in position and a larger uncertainty in momentum, and vice versa. This can affect the behavior of the particle, as a narrower wave packet will have a more defined position but a less defined momentum.

## 5. Are there any real-world applications of Gaussian wave packets?

Yes, Gaussian wave packets are commonly used in various fields of physics, such as quantum mechanics, optics, and acoustics. They are also used in signal processing and data analysis, as they can accurately model and predict the behavior of physical systems.

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