Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation)

In summary, the conversation discusses the process of numerically solving a relativistic version of an infinite square well with an oscillating wall using the Klein-Gordon equation. The speaker shares that they transformed the spatial coordinate and used NDSolve to solve the system. However, when they plotted the result, the wavefunction blew up when the frequency of the oscillation was high. The speaker then questions the reason for this and whether it has any connection to particle production. After locating their error, it is revealed that the velocity of the wall was not allowed in the units being used. The solution to this was to adjust the value of L(t) to avoid this issue.
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Numerical solution of the wavefunction blows up when the frequency of oscillation of the wall is high. What does this mean?
I am trying to numerically solve (with Mathematica) a relativistic version of infinite square well with an oscillating wall using Klein-Gordon equation. Firstly, I transform my spatial coordinate ## x \to y = \frac{x}{L[t]} ## to make the wall look static (this transformation is used a lot in solving non-static boundary condition in the non-relativistic case), which brings Klein-Gordon equation to :
Input :
1644917229990.png

Output :
1644917258275.png

All constants have been set to 1
1644917266102.png

I tried to solve this system where ##L(t)=2+sin(1000 t)## using NDSolve :
1644917349008.png

Then I plot my result as a function of ##(y,t)## :
1644917373977.png

The wavefunction ##\psi## blows up. This doesn't happen when I tune the frequency down to 1, ##L(t)=2+sin(t)##
1644917399498.png

My question is why does ##\psi## blow up when the frequency of the oscillation is high and what does it mean? Does it have anything to do with particle production? Or did I just mess up my code?
 

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I have located my error. It was the fact that when I set ##\omega## to 1000, the velocity of the wall, ##\dot{L} = \omega cos(\omega t) \ge 1##, which is not allowed in the units that I am working with ##(c=1)##.
The solution to this is just to set ##L(t) = \frac{1}{\omega} (2+sin(\omega t))##, and this disastrous result doesn't happen anymore.
 

Related to Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation)

1. What is the Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation)?

The Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation) is a mathematical model used to describe the behavior of a particle confined in a one-dimensional box with an oscillating boundary. It is based on the Klein-Gordon equation, which is a relativistic wave equation that describes the behavior of particles with spin zero.

2. How does the oscillating wall affect the particle's behavior in the Infinite Square Well?

The oscillating wall in the Infinite Square Well creates a time-dependent potential that causes the particle to experience a force, leading to changes in its energy and momentum. This results in the particle exhibiting wave-like behavior, with its probability distribution changing over time.

3. What are the implications of the Klein-Gordon Equation in the context of the Infinite Square Well?

The Klein-Gordon Equation allows us to study the behavior of particles with spin zero in the Infinite Square Well, taking into account relativistic effects. This is important in understanding the behavior of subatomic particles, such as mesons, which have zero spin.

4. Can the Infinite Square Well with an Oscillating Wall be solved analytically?

No, the Infinite Square Well with an Oscillating Wall cannot be solved analytically. The Klein-Gordon equation is a partial differential equation, and the time-dependent potential adds another level of complexity. Therefore, numerical methods must be used to solve this equation and obtain solutions.

5. What are the practical applications of studying the Infinite Square Well with an Oscillating Wall?

The Infinite Square Well with an Oscillating Wall has applications in various fields, including quantum mechanics, solid-state physics, and particle physics. It can help us understand the behavior of particles in confined spaces and provide insights into the properties of subatomic particles. Additionally, it can be used to model systems with time-dependent potentials, such as quantum dots and Josephson junctions.

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