Quantum Mechanics, one-dimensional box problem

In summary, the conversation discusses the energy eigenfunctions and eigenvalues for the one-dimensional box problem with ends at -a/2 and a/2. The solution to the problem involves finding the wavefunction, Phi(x) = Asin kx + Bcos kx, and ensuring its continuity at the boundary points. The periodicity conditions help determine the allowed values for k, which in turn give the energy spectrum. The conversation also mentions the possibility of using Psi instead of Phi and asks for a sketch of the eigenfunctions, as well as any patterns or groupings that may be observed.
  • #1
danai_pa
29
0
What are the energy eigenfunctions and eigenvalues for the one-dimensional box problem describ above if the end of the box are at -a/2 and a/2

I can find the solution of this problem Phi(x) = Asin kx + Bcos kx
and property of wavefunction is continuous at boundary
Phi(x=-a/2) = Phi(x=a/2)=0
but i don't understand to find k (wave number), please help me
 
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  • #2
danai_pa said:
What are the energy eigenfunctions and eigenvalues for the one-dimensional box problem describ above if the end of the box are at -a/2 and a/2

I can find the solution of this problem Phi(x) = Asin kx + Bcos kx
and property of wavefunction is continuous at boundary
Phi(x=-a/2) = Phi(x=a/2)=0

What is the problem "described above"?

And do you perhaps mean Psi instead of Phi ?
 
  • #3
Can you sketch your eigenfunctions?
(I'm assuming you've done the box problem with ends x=0 to x=a. Hopefully you realize that the choice of origin shouldn't change the shape of the eigenfunctions.)
See any pattern? any grouping of the eigenfunctions?
 
  • #4
The periodicity conditions shouls give you the allowed "k" values from which you can get the energy spectrum.

Daniel.
 

1. What is the one-dimensional box problem in quantum mechanics?

The one-dimensional box problem is a thought experiment used to illustrate the behavior of particles in a confined space. It involves a particle confined to a one-dimensional box, with infinite potential energy at the walls of the box.

2. How does the one-dimensional box problem relate to quantum mechanics?

The one-dimensional box problem is a fundamental concept in quantum mechanics, as it allows us to understand the behavior of particles in a confined space and the quantization of energy levels. It also helps to explain the wave-like nature of particles and the concept of wave-particle duality.

3. What is the Schrödinger equation and how is it used in the one-dimensional box problem?

The Schrödinger equation is a mathematical equation that describes the time evolution of a quantum system. It is used in the one-dimensional box problem to calculate the allowed energy levels and wavefunctions of the particle in the confined space.

4. What are the implications of the one-dimensional box problem for real-world applications?

The one-dimensional box problem has important implications for various real-world applications, such as the design of electronic devices and understanding the behavior of atoms and molecules in confined spaces. It also has implications for quantum computing and cryptography.

5. Are there any limitations or criticisms of the one-dimensional box problem?

While the one-dimensional box problem is a useful thought experiment, it does have limitations and has been criticized for oversimplifying real-world scenarios. It does not take into account factors such as interactions between particles and the effects of external forces. However, it still serves as a valuable concept in understanding the principles of quantum mechanics.

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