- #1
rozan977
- 9
- 0
How does energy become quantized in an infinite potential well??
Quantized energy in infinite potential well
- Context: Graduate
- Thread starter rozan977
- Start date
Click For Summary
Discussion Overview
The discussion revolves around the quantization of energy in an infinite potential well, exploring the mathematical foundations and implications of boundary conditions in quantum mechanics. Participants examine how these principles relate to other systems, such as vibrating strings.
Discussion Character
- Exploratory, Technical explanation, Conceptual clarification, Debate/contested
Main Points Raised
- Some participants suggest that energy quantization arises from the boundary conditions of the Schrödinger equation for an infinite potential well.
- One participant notes that the proper functions (eigenfunctions) are sin(πnz), with eigenvalues proportional to (πn)², indicating a mathematical basis for quantization.
- Another participant relates the concept to the frequencies of a vibrating string, proposing a similar quantization pattern.
- Some participants question the form of the solution to the Schrödinger equation, suggesting that an exponential form could also be valid, while noting that it must satisfy the same boundary conditions.
- There is a mention of historical context, referencing Pythagoras as an early figure in the study of problems related to quantum mechanics.
Areas of Agreement / Disagreement
Participants express various viewpoints on the mathematical treatment of the problem, with some agreeing on the role of boundary conditions while others explore alternative forms of the Schrödinger equation. The discussion remains unresolved regarding the implications of these different approaches.
Contextual Notes
Participants highlight the importance of boundary conditions in determining the eigenstates and eigenvalues, but there is uncertainty about the equivalence of different solution forms and their implications for understanding quantization.
- #2
alxm
Science Advisor
- 1,848
- 9
How do the harmonics of a finite string become quantized?
- #3
Feldoh
- 1,336
- 3
Honestly from a mathematical standpoint it comes about due to the boundary conditions of the Schrödinger equation for an infinite potential well.
- #4
Bob_for_short
- 1,161
- 0
rozan977 said:
The proper functions (eigenfunctions) of the Hamiltonian are sin(πnz) where z is the dimensionless length z=x/L. The proper values (eigenvalues) are proportional to (πn)2.
Any, I repeat, any wave inside the well can be decomposed in a sum of proper waves with some amplitudes. In general case the wave energy is not certain but dispersed. Only in the eigenstates the energy is certain.
- #5
Halcyon-on
- 72
- 0
rozan977 said:
As the frequencies of a string in a guitar [tex]E_n = h v_n = n h v[/tex]
- #6
Halcyon-on
- 72
- 0
That is through (periodic) boundary conditions.
- #7
Halcyon-on
- 72
- 0
For this aspect, Pythagoras was the first to study a problem of QM mechanics.
http://img523.imageshack.us/img523/5874/pitagoradagafuriotheorixk5.jpg
http://img523.imageshack.us/img523/5874/pitagoradagafuriotheorixk5.jpg
Last edited by a moderator:
- #8
rozan977
- 9
- 0
Feldoh said:
But what if the solution we assume of Schrödinger equation be in exponential form??
- #9
Halcyon-on
- 72
- 0
rozan977 said:
it is completely equivalent.
- #10
Bob_for_short
- 1,161
- 0
rozan977 said:
In order to satisfy the boundary conditions the two complex exponentials have to have certain coefficients that make their sum to be sin(pi*n*x/L).
Similar threads
Graduate
State in the infinite potential well
- · Replies 7 ·
Undergrad
Particle in a potential well
- · Replies 11 ·
- · Replies 31 ·
- · Replies 5 ·
- · Replies 15 ·
Graduate
Power series in quantum mechanics
- · Replies 1 ·
- · Replies 7 ·
- · Replies 4 ·
- · Replies 5 ·
- · Replies 10 ·