# Particle in a Box situations

1. Jan 10, 2005

### FUNKER

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.

2. Jan 10, 2005

### vincentchan

you use 1-D shrondinger eq in 1D problem,which displacement should be in 1d. if you wanna do a 2D problem, you should use a 2D Shrondinger eq..... hope this answer your question

3. Jan 10, 2005

### FUNKER

it has helped thanks

4. Jan 10, 2005

### vincentchan

you are welcome

5. Jan 10, 2005

### dextercioby

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... Could u be a little more specific,and what kind of Box are u talking about??1D,2D,3D???

Daniel.

6. Jan 12, 2005

### FUNKER

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

7. Jan 12, 2005

### dextercioby

Solutions of the spactral equation for the Hamilton operator
$$\hat{H}|\psi\rangle = E|\psi\rangle$$

,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?

8. Jan 14, 2005

### FUNKER

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?

9. Jan 14, 2005

### dextercioby

All possible values for the energy of the Q system.

What do u mean by bounded??

Daniel.

10. Jan 18, 2005

### FUNKER

i.e. an electrons potential energy is negative whilst "belonging" to an atom

11. Jan 18, 2005

### dextercioby

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. Jan 18, 2005

### Norman

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. Jan 18, 2005

### dextercioby

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

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...??

Daniel.

14. Jan 18, 2005

### Norman

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?

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. Jan 18, 2005

### dextercioby

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

Yes,u got the picture...

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

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. Jan 18, 2005

### Norman

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