How Does Schrödinger's Equation Apply to a One-Dimensional Particle in a Box?

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Discussion Overview

The discussion revolves around the application of Schrödinger's equation to a one-dimensional particle in a box, exploring the nature of wave functions, displacement, and the concept of boundedness in quantum mechanics. Participants engage with theoretical aspects, definitions, and implications of the equation in both one-dimensional and multi-dimensional contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether Schrödinger's one-dimensional equation provides the wave function of a particle and how displacement can be defined in a one-dimensional context.
  • Others clarify that the one-dimensional Schrödinger equation is appropriate for one-dimensional problems, and that displacement can indeed be considered in one dimension.
  • A participant introduces the concept of the Hamiltonian's spectrum, suggesting it represents the possible energy values of the system.
  • There is a discussion about the definition of "bounded" systems, with some arguing that boundedness relates to discrete energy levels, while others challenge this notion by suggesting that real quantum systems may not be bounded.
  • Participants explore the implications of wave function behavior at infinity and how it relates to the concept of boundedness in quantum mechanics.
  • There is a request for clarification on the difference between bounded quantum states and bounded physical systems, with references to specific examples like harmonic oscillators and atoms.

Areas of Agreement / Disagreement

Participants express differing views on the definitions of boundedness and the implications of discrete energy levels. The discussion remains unresolved regarding the precise definitions and conditions under which a system is considered bounded.

Contextual Notes

Limitations include varying interpretations of displacement, the definition of boundedness, and the implications of the Hamiltonian's spectrum. The discussion does not reach a consensus on these topics.

FUNKER
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does shrodingers one dimensional equation, if solved, give you the wave function (i.e. the displacement) of a particle ?
If so, then how can this be used in a one dimensional situation since displacement is in 2 dim.?
These questions arise from Particle in a Box situations.
 
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you use 1-D shrondinger eq in 1D problem,which displacement should be in 1d. if you want to do a 2D problem, you should use a 2D Shrondinger eq... hope this answer your question
 
it has helped thanks
 
you are welcome
 
FUNKER said:
does shrodingers one dimensional equation, if solved, give you the wave function (i.e. the displacement) of a particle ?
If so, then how can this be used in a one dimensional situation since displacement is in 2 dim.?
These questions arise from Particle in a Box situations.


Schroedinger's equation,in any number of dimensions,does give not only the (generally time-dependent) wave-function,but also the Hamiltonian's spectrum,which are the only possible values for the system's energy.

I don't know what u mean by "displacement",however... :confused: Could u be a little more specific,and what kind of Box are u talking about??1D,2D,3D?

Daniel.
 
im talking bout displacement, and its a 1 d box. i said if you sovle the equation you get a wave function. please read above. what is the hamiltonian spectrum
 
Solutions of the spactral equation for the Hamilton operator
[tex]\hat{H}|\psi\rangle = E|\psi\rangle[/tex]

,which are in fact the only possible values of the energy the system can take.

Daniel.

PS.U know that "displacement" is entirely probabilistic,okay?
 
yes I do know that it is but hey thanks for your help. Just to recap the hamiltonian spectrum is just all possible values for a particle in a bound system?
 
All possible values for the energy of the Q system.

What do u mean by bounded??

Daniel.
 
  • #10
i.e. an electrons potential energy is negative whilst "belonging" to an atom
 
  • #11
Well,according to QM,the electrons in atoms can be found anywhere...So u can't call an atom as being "bounded".While the potential energy is basically unbounded in the origin.The sign is not important...

Daniel.
 
  • #12
Isn't the definition of bounded the emergence of discrete energy levels? Or is a better notion the limit of the wavefunction as position goes to infinity?
I would think the latter is more rigorous, but it is not entirely obvious to me whether the discrete energy levels are enough to define whether the system is bound. Can anyone think of a counter-example for the quantized energy levels of a non-bound system? How would you define "bounded" if an atom is not one?
 
  • #13
Norman said:
Isn't the definition of bounded the emergence of discrete energy levels?

Bounded quantum states...It's something totally different than bounded physical systems... :wink:


Norman said:
Entirely obvious to me whether the discrete energy levels are enough to define whether the system is bound.

I told u...REAL quantum systems are not "bounded".How would you define such a "boundness"??

Norman said:
Can anyone think of a counter-example for the quantized energy levels of a non-bound system? How would you define "bounded" if an atom is not one?

Hydrogen atom,bound states?LHO ?

What do you mean "atom is not one"??You mean,if the NUMBER of atoms is larger than one??Bounded quantum states of multiparticle quantum systems??
Maybe the Hamiltonian of the system should have a discrete spectrum...?? :rolleyes:

Daniel.
 
  • #14
dextercioby said:
Bounded quantum states...It's something totally different than bounded physical systems... :wink:

Ok, so which are we talking about here and what is the difference? If you mean the difference between a Harmonic Oscillator (a bounded quantum state- if I understand what you are saying) and say a Lithium atom (a real system), then I think I understand what you are saying, if not would you please elaborate?

dextercioby said:
I told u...REAL quantum systems are not "bounded".How would you define such a "boundness"??



Hydrogen atom,bound states?LHO ?

What do you mean "atom is not one"??You mean,if the NUMBER of atoms is larger than one??Bounded quantum states of multiparticle quantum systems??
Maybe the Hamiltonian of the system should have a discrete spectrum...?? :rolleyes:

Daniel.

I think you are misunderstanding me. Or I am misunderstanding you. You answered my exact question with the same question. Are we defining "boundedness" as the emergence of a discrete spectrum from the Hamiltonian or should we look at how the wavefunction behaves at infinity? I am not challenging you, so please don't become defensive, I am only trying to understand.
Cheers,
Norm
 
  • #15
Norman said:
Ok, so which are we talking about here and what is the difference?


I hope we're talking about the boundness of quantum states... :rolleyes: I said that the only way to have "bounded systems" in QM is to have infinite potential barriers.Or that thing is unattainable in reality...


Norman said:
If you mean the difference between a Harmonic Oscillator (a bounded quantum state- if I understand what you are saying) and say a Lithium atom (a real system), then I think I understand what you are saying, if not would you please elaborate?

Yes,u got the picture... :smile:



Norman said:
Are we defining "boundedness" as the emergence of a discrete spectrum from the Hamiltonian

Fot the quantum states we do...Bounded quantum states are eigenstates of the Hamiltonian...

Norman said:
or should we look at how the wavefunction behaves at infinity?

By a simple extrapolation of the first principle,we would say:bounded quantum states<=>quantum states on which the wavefunction tend to zero in the asymptotic limit (coordinate representation).


Daniel.
 
  • #16
dextercioby said:
By a simple extrapolation of the first principle,we would say:bounded quantum states<=>quantum states on which the wavefunction tend to zero in the asymptotic limit (coordinate representation).


Daniel.

Ah... very good point. I must remember to appeal to the principles of QM.

Thanks a lot for the enlightenment and the quick reply.
Cheers,
Norm
 

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