Quantum mechanics of a particle

[sarabellum02]

I have a couple things I don't understand:

1. Why is it that the more you conifne a particle, the higher its energy is?
2. Why is it that the more nodes there are in the wavefunction the higher the energy is?
3. What causes the energy of a particle to be quantized?

thanks!

 

[dextercioby]

1.What do you mean by "conifne a particle"...?If you mean "confine a particle",i have to ask you what do you mean by this...?
2.What are nodes of a wave function...?I've never heard that a function would have nodes...
3.Principles of QM...Essentially the second.

Daniel.

 

[marlon]

I have a couple things I don't understand:

1. Why is it that the more you conifne a particle, the higher its energy is?

Are you referring to quarkconfinemnt, then your answer is asymptotic freedom...

2. Why is it that the more nodes there are in the wavefunction the higher the energy is?

Are you referring to nodal planes of orbitals?then your answer is the magnetic quantumnumber...(at least to some extent)

3. What causes the energy of a particle to be quantized?

thanks!

Mother Nature

marlon

 

[masudr]

It's not true that the more you confine something the more energy it has -- what is true is that the more you confine something, the variance in it's momentum will increase.

As for the node thing; I think sarabellum means that in a "particle in a box", the number of nodes refers to the number of points where the wavefunction $\psi(x)=0$. There is no reason as to why that is -- it's just how the solutions to Schrödinger's equation works out.

And for the third point, as marlon says, that's just how nature is. If energy wasn't quantised, one could ask "Why is energy continuous?" and so on.

Masud.

 

[marlon]

what is true is that the more you confine something, the variance in it's momentum will increase.

This is not entirely correct. Just look at quarks...

Besides, what exactly do you mean by this and how does it apply to QM?

marlon

 

[Edgardo]

I have a couple things I don't understand:
1. Why is it that the more you conifne a particle, the higher its energy is?

sarabellum probably means, if you make the well smaller, the energies become higher.
Click on this link http://www.quantum-physics.polytechnique.fr/en/pages/p0203.html, and change the width of the well, by dragging the left bottom corner of the well.

2. Why is it that the more nodes there are in the wavefunction the higher the energy is?

sarabellum probably means for example the harmonic oscillator eigenstates.
The higher the energy, the more zeros (sarabellum calls them nodes) the wavefunction has http://encyclopedia.laborlawtalk.com/Quantum_harmonic_oscillator

I think that's just the result of QM calculations.

3. What causes the energy of a particle to be quantized?

Hmm...as marlon said, it's nature.

 

[masudr]

...how does it apply to QM?

It's the celebrated Uncertainty Principle.

 

[BlackBaron]

1. Why is it that the more you conifne a particle, the higher its energy is?

I think I know what you mean, bur truth is, I'm not sure how exactly to give you a good answer.

2. Why is it that the more nodes there are in the wavefunction the higher the energy is?

As some mentioned before, nodes are the points where the wavefuncion equals 0 (like the nodes of a vibrating string).
The fact that the more nodes there are, the higher is the energy actually happens with any kind of wave. Again, think in a vibrating string, the stronger you make it vibrate (i.e. the more energy you give to it), the bigger is the number of nodes.
The explanation on why that is so, is pretty much like this: the higher the number of nodes per unit distance, the more oscilations there are, which means the bigger is the wavenumber, and thus the higher is the energy (remember that in the case of the wavefunction, the momentum of the particle it's proportional to the wavenumber).
In fact, in more than one QM problem (I'm not sure if there's a general rule, maybe not), the number of nodes is actually closely related to the quantum number you use for the energy.

3. What causes the energy of a particle to be quantized?

More or less, this comes as a result of Schrödinger equation, energy is quantized whenever there are bound states (i.e. the potential confine the wavefunction to be in a certain place), when that happens, you have to force the wavefunction to be zero at certain point in space (like infinite, or the extremun of a infinite square well) and those boundary conditions can only be met by certain values of the wavenumber (and thus, the energy).
There are good discussions on this point in some standard Quantum Mechanics textbooks, like Eisberg & Resnick and Cohen-Tanoudji.

 

[sarabellum02]

Thanks for your help everyone!

 

[kanato]

2. Why is it that the more nodes there are in the wavefunction the higher the energy is?

The kinetic energy of a wave function comes from:
$$\frac{-\hbar^2}{2m} \frac{d^2 \Psi}{dx^2}$$
so it's related to the derivative of the wavefunction. If the wave function wiggles faster, then it has a larger derivative and will have more KE.

I have a couple things I don't understand:

1. Why is it that the more you conifne a particle, the higher its energy is?

If you're looking at the square well, when you make the well size smaller, the energies get larger. If you look at the ground state wavefunction, making the well size smaller, with the restriction that it has to be zero at the ends and it has to be normalized makes it so the wavefunction has to go to zero faster as it approaches the edges of the well. This makes it so its derivatives are larger, and hence larger kinetic energy.