Eigenvalue Problem in Uniformly Acceleration Motion

In summary, this conversation discusses the problem of solving the eigenvalue problem for a particle in uniform acceleration motion. The force F is constant and leads to a potential function of V(x) = Fx and a Hamiltonian of H = (p^2/2m) - Fx. The conversation also delves into the comparison between classical mechanics and quantum mechanics, as well as trying to understand the concept of uniform acceleration motion for a single particle. Additionally, the possibility of using the Airy function as a solution for the Hamiltonian is mentioned.
  • #1
jshw
4
0

Homework Statement


In Uniform Acceleration Motion, the force F is constant.
then potential V(x)=Fx, and Hamiltonian H=(p^2/2m)-Fx
The problem is to solve the eigenvalue problem Hpsi(x)=Epsi(x)


Homework Equations


F=constant
V(x)=Fx
H=(p^2/2m)-Fx


The Attempt at a Solution


I have tried to compare classical mechanics and quantum mechanincs.
but, my QM textbook don't mention about uniformly acceleration motion.
I had hard time to solve it during this weekend. but I get lost the direction to solve it.
I think that this problem is mathematically messy. please give me the direction to solve this problem.
 
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  • #2
jshw,

I don't completely understand what you're asking.

I'm guessing Fx is the partial of F with respect to x, so is the potential (V(x)) Fx everywhere? in a one-dimensional setting? (i.e. is this a particle in a 1D box with constant potential across the bottom of the whole box?)

classically, I think 'uniformly accelerating' means that every part of an object is accelerating in same direction, at the same rate (as opposed to an object spinning through air, for instance) but I'm not sure what it means in the context of a single particle; perhaps it's as simple as constant acceleration...?
 
  • #3
Your Hamiltonian certainly looks like having Airy function as a possible solution.
 

1. What is the Eigenvalue Problem in Uniformly Acceleration Motion?

The Eigenvalue Problem in Uniformly Acceleration Motion is a mathematical problem that involves finding the eigenvalues and eigenvectors of a linear transformation that represents the motion of a particle under constant acceleration. It is used in various fields such as physics, engineering, and mathematics to analyze the behavior of systems undergoing uniform acceleration.

2. How is the Eigenvalue Problem in Uniformly Acceleration Motion solved?

The Eigenvalue Problem in Uniformly Acceleration Motion is typically solved using matrix methods. The linear transformation that represents the motion is written in matrix form, and the eigenvalues and eigenvectors are found by solving the characteristic equation for the matrix. These values can then be used to analyze the behavior of the system.

3. What are the applications of the Eigenvalue Problem in Uniformly Acceleration Motion?

The Eigenvalue Problem in Uniformly Acceleration Motion has many practical applications, such as predicting the behavior of a falling object under gravity, analyzing the stability of a system in motion, and determining the natural frequencies of oscillating systems. It is also used in fields like robotics, control theory, and quantum mechanics.

4. What is the significance of eigenvalues and eigenvectors in the Eigenvalue Problem in Uniformly Acceleration Motion?

Eigenvalues and eigenvectors play a crucial role in the Eigenvalue Problem in Uniformly Acceleration Motion as they represent the properties of the linear transformation that describes the motion. The eigenvalues determine the magnitude of the acceleration, while the eigenvectors represent the direction of the acceleration. They also help to identify any critical points or equilibrium states in the system.

5. Can the Eigenvalue Problem in Uniformly Acceleration Motion be applied to non-uniformly accelerating systems?

While the Eigenvalue Problem in Uniformly Acceleration Motion is specifically used for systems undergoing uniform acceleration, it can also be applied to non-uniformly accelerating systems by breaking down the motion into smaller intervals of uniform acceleration. This approach is known as the Euler method and is commonly used in numerical simulations to solve complex motion problems.

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