After reading this book (again...lol), the IB textbook and the study guide...here is how I would explain the wave function. (My head really hurt after that lesson btw)
If it clears things up...awesome...this blog is doing its job!
If this makes it more confusing, close your web browser! But I recommend reading this very slowly...skimming will make everything more confusing.
Lets begin...
1.) First, please don't be scared with the next equation...
All you need to notice is the fact that Ψ exists in the equation and will have a value. Its like having d = s x t....where t has a value when you need to solve something.
Blah Blah Blah...All you need to take away is that Ψ is a variable quantity that can take up different values.
Okay. Move on.
2.) The textbook says (Read slowly) "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".
Right. The important thing to realize here is whenever the word "wave function" pops up or the symbol Ψ appears...you can visualize the wave function as a graph.
If I said a quantity (I'll call it y) is related to the sine function, all of you can immediately picture the sine graph and the "value" that the quantity y can therefore take. So the peak of the sine graph is the maximum value of the quantity y.
However, the wave function is way more complicated than the sin graph because...
3.) The wave function looks different depending on the situation. The sine graph always looks the same. How the wave function looks like can change.
What do I mean by this? Look at this graph:
The "wave function" looks like this for the lowest energy state of an electron in the hydrogen atom.
However, in a higher energy level of the hydrogen atom...the "wave function" looks like this:
Get it get it? The "wave function" looks different depending on the situation.
When you say "wave function"...it is a collection of graphs where each one applies to a different situation.
Look at how the "value" of the wave function (ie what you would read on the y axis at a certain distance from the atom) changes according to the distance from the atom.
4.) Read slowly. "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".
Now this statement makes more sense.
If I want to find the probability of locating an electron say at 10^-11 m from the hydrogen atom, I would look for the value on the axis, go up the graph, read the y value (which is Ψ don't forget) and then square it.
There you have it...the number you have (say 0.11) is the probability of locating the electron at that point which is 10^-11 m from the hydrogen atom.
4.) Because the value of Ψ changes according to where you are from the atom...Ψ is a variable value as stated in the beginning.
In the text book you will read "At any instant in time, the wave function has different values at different points in space".
Booyah, just look at the 2 graphs...thats what the textbook is saying.
-------------Now we will link the standing waves we learnt in class with the wave function.
5.) So why are there loads of different graphs for this one symbol Ψ ?
Notice that at higher energy levels, the wave function completes more oscillations.
VS
Guess what...a standing wave in a string has more oscillations at higher harmonics.
Ahhhhh...there must is some link between the two.
This is the link:
6.) For a standing wave to set up on a string, the boundary conditions are 2 nodes on either side.
The reason why only certain frequencies can exist is because these specific frequencies satisfy the conditions necessary for a standing wave to form...2 nodes at either end.
We all agree that more than 1 frequency can thus form a standing wave as long as 2 nodes can form at either end.
7.) Notice how the graph for the wave function (the hydrogen graphs) all reach zero a long distance from the atom. This makes sense, the electron is less likely to be found very very far away from the atom.
Well, in order for Schrodinger's equation to work properly (the long one above), the wave function must rapidly tend to zero (go towards zero) at very far distances from the atom.
This is like a boundary condition. A valid wave function can only "exist" if Ψ approaches zero further away from the atom. There are only specific energies where the wave function approaches zero. These energies (or energy levels) are the ones we see in the textbook:
Other energies would make the wave function graph not reach zero, which is why that graph doesn't exist. Other energies actually make the wave function infinite as opposed to zero.
Think of this diagram:
as having a node at either end when it is reaching zero, thus a standing wave forms. If it was going to infinity (going up and and up), its like having an antinode...which will not form a standing wave.
See the link?
In the same way that certain standing waves exist on a string because only certain frequencies satisfy the boundary conditions (and there are many of these frequencies known as harmonics)...only specific energies can produce a wave function that tends to zero and thus make the Schrodinger Equation work (and there are many of these energies and hence many valid wave function graphs).
Thats why we can use the electron in a box model to describe the nature of electrons under the Schrodinger model. There are connections between standing waves and Schrodinger's model. The energy levels are still quantized like Bohr's model but are now described in terms of probability.
8.) So lets return back to "The probability of finding the particle at any point in space within the atom is given by the square of the amplitude of the wave function at that point".
We now know why there are many valid wave function graphs (in the same way there are many possible frequencies to form a standing wave on a string)
Related to the wave function is the probability (the statement above). This probability changes according to the wave function (because it is the wave function squared. duh) and exists in 3 dimensions. So the dips and peaks on the 2D graph can be mapped into a 3D electron cloud.
Aha!...look at the study guide.
A peak is the denser part of the cloud, a dip does is the less dense part of the cloud.
The shape of the cloud changes according to the wave function (think about the value of Ψ on the y axis as you move left and right), and the wave function in turn only exist for specific energies. So there are specific shapes to the electron clouds because there are specific wave functions related to them!
That should make sense...read it again =)
The shape of the cloud are the shapes we have come across in HL Chemistry when we talk about s, p and d orbitals.
s orbitals
p ortibal
d orbital
They look the way they are because they are the only possible shapes that can exist and satisfy the conditions of the wave function and Schrodinger's equation...they are solutions to Schrodinger's Equation!
COOL!
Thats why its not some random hexagon.
I love physics =D
great post Hanoi - I've learnt something new - I'll pass on to Mr Oakes to read too!
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