[Schrödinger]

Hey guys, these questions i am about to ask might sound absolutely absurd, but please help me, as i have zero insight into quantum theory and the likes....

First of all, am i correct in saying that the electrons can be considered as standing waves?
Just like how we consider a string fixed at both ends and oscillating to and fro...

[renge ishyo]

Well, the behavior of an electron in a potential field (created say by a positively charged nucleus where the electron is trapped in one of its orbitals) is *analogous* to the behavior of other confined waveforms such as a vibrating guitar string. In this instance, the distinct different standing wave patterns that can form with a guitar string are similar *in concept* to the different energy levels that an electron can exist at within the field surrounding the atom (meaning that the energy levels only occur at specific levels and not just any energy value just as the standing wave patterns only appear at certain vibrational frequencies).

However, you can't take this analogy too far...because if you ionize the electron from the atom so that it is floating around in free space the "standing wave" pattern of energy values disappears and the electron can now have any energy value it chooses as if it was a pure particle. So the analogy either works or fails depending on where the electron is located (this is another part of that pesky quantum wave particle duality rearing its ugly head).

[Schrödinger]

Ok, now that i understand the anology part, here is my next question.

Now that we can consider the electron trapped in a potential field as a standing wave, what does the wave equation actually represent?

For instance, for a wave equation for a guitar string, the wave-function , given by y=f(x,t) represents the displacement of a particle situated at distance x from the origin (say one end of the string is the origin) at any time t. This will be in the form of a sine or cosine function.

Similarly, what details about the electron is given by Schrodinger's wave equation?

[MrTeo]

Similarly, what details about the electron is given by Schrodinger's wave equation?

You should know that better than any other (sorry for the bad joke, but I couldn't resist seeing Schrödinger asking explanation about his own equation)

What the equation describes is really a wave, the stationary wave of the electron, just like that of the guitar... but due to quantum mechanics it's not so easy to represent this fact: here's an example of what I'm trying to say, even if you should consider it only a visual aid (there's no real wave in the atom and this can only show you why the stationary states are multiples of the wavelenght, according to DeBroglie's relations).



Anyway the solutions of the wave equation express the negative charge density distribution caused by the electron cloud (being more precise only some solutions are acceptable while other must be excluded, just like stationary states, the only possible for electrons). That's why the square value of gives us the probability of the presence of the electron in that point and from this fact we get the shape of the orbitals (considering only areas with 90% or more probability).

[Schrödinger]

What the equation describes is really a wave, the stationary wave of the electron, just like that of the guitar...

Ok, so the wave equation basically represents the negative charge density produced by the electron cloud.

Now, if we were to consider the electrons as standing waves, then i have another question.

I hope we are all familiar with the pictorial description of a standing wave, say, a giutar string vibrating (a sine or cosine function).

Now, if you represent any sinusoidal wave function on the x-y graph, then there will be point of time where all particles of the medium are lying on the x-axis with zero displacement from their mean positions. And there will be certain points whose displacement from the mean position is always zero, and we call them 'nodes'.

What does this correspond to in the case of electrons and orbitals? What do nodes refer to in the context of quantum theory???

[renge ishyo]

In the context of quantum theory, "nodes" appear in places where the probability of finding an electron is vanishingly small or zero. If you compare a 1s orbital to a 2p orbital for instance you will see that the 2p orbital has a node near the center that splits the electron distribution into two separate areas. The 1S orbital lacks nodes which is why the electrons are distributed all around the area in a spherical manner with no separate areas for the electrons.

[Yggdrasil]

First a few clarifications.  When we say electrons can be described as acting as waves, we really mean that the math dealing for dealing with the quantum behavior of electrons is the mathematics of waves.  The analogy should not suggest that the electrons are oscillating around inside of the orbitals or as they are traveling in free space (often a point of confusion for many).

Now, what do we mean by "the mathematics of waves"?  Well, for normal particles if you add one particle to another particle, you always get two particles.  However, there are cases where one electron plus another electron will give zero electrons.  How does this happen?  Waves have a property called phase, and waves that are exactly out of phase with each other can interfere destructively and cancel each other out.  This property of having a phase and being able to interfere either constructively or destructively is why we treat electrons (and other quantum mechanical objects) as waves.

The actual wavefunction (Ψ) of a system of electrons does not really have a physical meaning.  It is primarily a good mathematical representation of the electron that gives it the required properties (e.g. because it has positive and negative peaks and troughs, it can interfere with other wavefunctions).  However, the wavefunction is related to a very important physical parameter, the probability density distribution (|Ψ|²).  "Squaring" the wave function gives you a function that tells you the relative probability of observing the electron at each point in space.

Now your point about the nodes is both correct and very important in quantum mechanics.  Because Ψ(x) = 0 when x is the position of a node, |Ψ(x)|² is zero, meaning that the electron has exactly zero probability of being found at that position.  One example is the p-orbital.  The plane perpendicular to the p-orbital that goes through the nucleus is a node.  That means the electron can never be found on the plane between the two lobes of the two lobes of the p-orbital.  Yet, the electron can be found in either lobe of the p-orbital and traverse between the two lobes.  This is a simple example of "tunneling": the ability of quantum mechanical objects to move through forbidden regions (without ever actually being in the forbidden region).

[Schrödinger]

Yeah, that was good.

But my question is thus :
When you consider a guitar string vibrating (a standing node), there will be a point of time where the string is perfectly horizontal, or all particles of the medium have zero displacement from their mean positions. i.e, at this instant y=f(x,t)=0, since y represents the displacement of a particle situated at 'x' from the origin.

What does this correspond to in quantum theory? Does it have any sort of connection with nodes?

Is at such an instant?

[renge ishyo]

"When you consider a guitar string vibrating (a standing node), there will be a point of time where the string is perfectly horizontal, or all particles of the medium have zero displacement from their mean positions. i.e, at this instant y=f(x,t)=0, since y represents the displacement of a particle situated at 'x' from the origin."

There is no useful meaning for this in the quantum theory as far as I can see (except maybe that the electron in question is not actually in the atom where this could be interpreted as being zero everywhere because no electron is actually there). A standing wave pattern is not formed on a still string, and the nodes refer to points of zero displacement within a *vibrating string*. Like Yggs said above, don't try to understand too much about the quantum theory from the string analogy. The similarities between the two ideas are mathematical only, and you can actually do all of quantum mechanics using matrices while shedding the wave analogy all together, and it still works. The displacement of the wavefunction doesn't actually refer to physical displacements of the electrons as mentioned in Yggs post, if you square the function you get the probability of an electron being at a certain position. That's all. If anything the wavefunction can be interpreted as a probability distribution function that relates to the various possible positions of the electron...it is not a description of the electron *itself*.

Hope this helped.

[Schrödinger]

Yeah, it really helped. Thank you very much guys