4k helium wave packet model

[chris2020]

Homework Statement

can liquid helium at 4k (.1 nanometers interatomic spacing) be modeled by non spreading wave packets?

Homework Equations

1/(√2π)∫^∞_-∞e^i(kx-wt)Φ(k)dk

W(k) was Taylor expanded to a quadratic term Δp_x²/(2mħ)

The book then sas for a non expanding wave packet:

|t| << mħ/Δp_x²

Another relation:

|t| Δv_x =|t| Δp_x/m << ħ/Δp_x = Δx

The Attempt at a Solution

I'm not sure where to begin. I figure... Δx is 1×10^-10 meters

Δp>ħ/Δx
Then I could say

|t| << mΔx²/ħ

Which tell me t must be very small since we are talking about ≈ (10^-10)^2

Other then that I don't know where to go from here.

 

[mark726]

I appreciate your curiosity and interest in this topic. To answer your question, yes, it is possible to model liquid helium at 4K with non-spreading wave packets. However, it is important to note that the behavior of liquid helium at very low temperatures is quite complex and cannot be fully described by simple wave packet models.

In order to model liquid helium at 4K, we must first understand the properties of this material. At this temperature, helium exists in a state called a Bose-Einstein condensate, where a large number of individual helium atoms behave as a single quantum mechanical entity. In this state, the helium atoms are very closely packed together, with an interatomic spacing of approximately 0.1 nanometers.

To model this system, we can use the Schrodinger equation, which describes the quantum mechanical behavior of particles. In this case, we can use the wave function Ψ(x,t) to describe the behavior of the helium atoms. This wave function can be written as a superposition of non-spreading wave packets, which represent the individual helium atoms.

The equations that you have provided are part of the mathematical framework for describing wave packets. The first equation is the Fourier transform of a wave packet, which allows us to represent the wave packet in terms of its momentum components. The second equation is a Taylor expansion of the momentum components, which is used to approximate the wave packet behavior. The third equation relates the time and position uncertainties of the wave packet.

To fully model liquid helium at 4K, we must take into account the interactions between the helium atoms and any external forces acting on the system. This can be done by including potential energy terms in the Schrodinger equation. Furthermore, we must also consider the effects of quantum statistics, as helium atoms are bosons and follow different rules than fermions.

In conclusion, while it is possible to model liquid helium at 4K using non-spreading wave packets, it is important to keep in mind that this is a simplified approach and may not fully capture the complex behavior of this system.