What Do the Nodes in a Quantum Particle's Wave Function Represent?

[Niles]

Homework Statement

Hi all.

Please take a look at: http://en.wikipedia.org/wiki/Particle_in_a_box

My problem is: I do not know how to interpret the eigenfunction of a particle in an infinite square well. We have that the wave function is a function of sine, and Psi_1 has no nodes, Psi_2 has one node, Psi_3 has two nodes and so on, i.e. they look like standing waves on a string.

What are these nodes an expression for? What do they mean? And does the wave function mean that the particle in the infinite square well oscillates around the energy-levels?

 

[muppet]

First things first. Do you know what a wave function actually is?

 

[Niles]

Yeah, I do. From it we can find any observable (or at least find the probability) at a given time t.

 

[muppet]

OK. Now you're on the right lines when you talk about the different energy levels. But if you have a wavefunction that is a sum of psi_1, psi_2, ... then the electron doesn't really "oscillate" between them- it's in all of them at once! Have you heard the term "superposition of states?"

I'm not sure what you're really asking when you ask what the nodes are "an expression for"- as they aren't an equation. They're points at which the wavefunction is constantly zero. (For future reference, it's usually a good idea to express homework problems word-for-word for this reason.) So you could really be asking two questions:
1)What does a point at which the wavefunction is constantly zero mean in terms of the probability density?
2)What does the number of nodes have to do with the physical state of the system?
(HINT: Work out what it tells you about the mathematical wave. What are the {Psi} eigenstates of? Then make the connection.)

 

[Niles]

OK. Now you're on the right lines when you talk about the different energy levels. But if you have a wavefunction that is a sum of psi_1, psi_2, ... then the electron doesn't really "oscillate" between them- it's in all of them at once! Have you heard the term "superposition of states?"

Is this just that we write the wavefunction as a linear combination of each possible state, and the square of each constant in front of the possible state is the probability of a particle being in that state?

1)What does a point at which the wavefunction is constantly zero mean in terms of the probability density?
2)What does the number of nodes have to do with the physical state of the system?
(HINT: Work out what it tells you about the mathematical wave. What are the {Psi} eigenstates of? Then make the connection.)

It's not homework - I am trying to get a better grasp on the infinite square well:

1) This is easy. This just means that the probability of the particle being in this state is zero. But in my book (Griffiths, under the "Infinite Square Well") there are some graphs with position (x) along the x-axis and Psi up the y-axis. Then there is a node for Psi_2 (i.e. n = 2). Does this just mean that in the infinite square well, we particle cannot be found at that exact point?

2) This is just an indication of the energy of the particle?

Thanks in advance! These questions/answers are very helpful.

 

[muppet]

Is this just that we write the wavefunction as a linear combination of each possible state, and the square of each constant in front of the possible state is the probability of a particle being in that state?

Yes
There's one caveat I have to add, however. When you say "each possible state", that should really be "eigenstate of a particular observable." You can't expand a wavefunction as a sum of eigenfunctions of different operators; you can't add a dirac delta function and a compex exponential, for example. Have you taken any linear algebra courses? The key mathematical idea behind advanced QM is that the one "state" of a system $$\Psi$$, as a point in a specific kind of vector space, can be represented in a basis of eigenfunctions $$\psi$$ of one particular observable. (If you haven't taken any courses on linear algebra, don't worry if that sounds like double dutch!)

It's not homework - I am trying to get a better grasp on the infinite square well:

It didn't sound like a homework problem! The best place to discuss conceptual stuff like this isn't the homework forum, which is really for help in problem solving. If you take a problem like this to the Quantum Mechanics subforum, you'll find that 1)more people check it 2)The answers you get are much more straightforward as people will quite happily explain anything to you, wheras they don't want to do your homework for you.

1) This is easy. This just means that the probability of the particle being in this state is zero. But in my book (Griffiths, under the "Infinite Square Well") there are some graphs with position (x) along the x-axis and Psi up the y-axis. Then there is a node for Psi_2 (i.e. n = 2). Does this just mean that in the infinite square well, we particle cannot be found at that exact point?

If by "state" you mean "at that point" then that second sentence is correct. Actually, because $$|\Psi|^{2}$$ is a probability density function, the probability that you'll find a particle at any single point is zero. (Gold has a huge mass density- but how much does zero cm^3 of gold weigh?) But a node means that the probability density associated with that point is zero, so that the chance of finding it in a small region centred around that point is zero.

2) This is just an indication of the energy of the particle?

Pretty much. The specific reason for it: in quantum mechanics the kinetic energy operator is obtained by applying the substitiution $$p\rightarrow -i\hbar\frac{d}{dx}$$ (in one dimension) to the classical formula for kinetic energy $$\frac{p^2}{2m}$$. The momentum is linked to the wavefunction (think de Broglie) and hence so is the energy. The more nodes you have, the shorter your wavelength is. As you can only fit an integer number of half-wavelengths into an in infinite potential well, there's only certain allowed wavelengths, and hence allowed energies; this is the reason for the quantisation of energy in this problem, and the case of atomic energy levels is similar (only with much more horrible maths )

 

[Niles]

Great, this really helped me! Thanks for taking the time to answer me - good job!

 

[muppet]

You're welcome