Wave Functions and Orbitals

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Take the electron of a hydrogen atom, in the one s orbital. The chance of finding it in a tiny volume around a point is greatest right at the nucleus. So, at what distance from the nucleus is this electron most likely to be? Zero, right? Hmm. No, no, not so fast. Today we find out where this electron really likes to be. And look, JEE does test this idea. So, let's begin.

0:33The wave function ψ

So, everything starts with the Schrödinger equation. Solve it for an electron in an atom, and the double arrow on screen shows what comes out: the wave function, psi, this Greek letter that looks like a trident. Okay? Now, at each point, psi can be positive, or negative. So tell me, can psi itself be the chance of finding the electron? Can a chance be negative? No, never, right? Chance cannot go below zero. So psi by itself has no physical meaning. Write that down. Then what has meaning? Mod psi, whole squared. The two straight bars mean modulus, and the small two means squared. Mod psi squared is the probability density of finding the electron at a point, and it is never negative. Got it?

1:20So what is an orbital?

Alright, now the big word of this chapter: orbital. An atomic orbital is a one electron wave function. We picture it as the region round the nucleus where the probability of finding the electron is high. The orbital shapes you see in books are boundary surfaces. Such a surface usually encloses about ninety percent of the probability. Ninety, not one hundred, okay? And every orbital is fixed by three quantum numbers: n, l, and m l, the m with a small l below. It holds at most two electrons, with opposite spins. Clear?

1:57Orbit or orbital? The classic trap

Now, a classic JEE trap. Bohr's model had orbits. Today we talk about orbitals. Only two extra letters, so both words mean the same thing, right? No, no, careful! In Bohr's model, an orbit is a definite circular path, with a fixed radius and a fixed speed. Like a train on a fixed track, okay? An orbital is a probability distribution, described by psi. No track at all. It only tells you where the electron is likely to be found. Now think back to the last chapter. Which principle says the electron in an atom cannot follow such a definite path? Yes, exactly, the uncertainty principle! For a definite path, you need the exact position and the exact momentum at the same time, and that is impossible. So the uncertainty principle forbids orbits, not orbitals. One more difference that examiners love. An orbit holds up to two n squared electrons. But an orbital holds at most two. That's it, just two. Got it?

3:05Radial probability: the thin-shell idea

Now, back to our puzzle. Instead of one single point, imagine a thin spherical shell around the nucleus, like one layer of an onion, with radius r and a tiny thickness d r. With me? The probability of finding the electron in this shell is four pi r squared, psi squared, d r; pi is the Greek letter from circles. Four pi r squared d r is the shell's volume, and psi squared tells how densely the probability is packed there. The graph plots four pi r squared psi squared against r over a naught; more on a naught soon. Now, for one s, psi squared is largest right at the nucleus, our hook fact. But put r equals zero in four pi r squared psi squared. What do you get? Zero, yes! Because of that r squared. At r equals zero, the shell shrinks to a point, with no volume at all. So the radial probability is zero right at the nucleus, even though psi squared is biggest there! Interesting, right? Then, as r grows, the shell gets bigger but psi squared gets smaller. So the radial probability rises, peaks at r equals a naught, and then falls. Here a naught, the a with a small zero below, is zero point five two nine angstrom: the capital A with a little ring on top. See the blue curve's peak at a naught. Look, puzzle solved! The most likely distance is not zero, it is a naught.

4:34Counting maxima and radial nodes

Alright, now look at the other two curves on the graph, two s and two p. Here is a very simple rule for counting the peaks: every radial probability curve has n minus l maxima. And between those peaks sit the radial nodes, n minus l minus one of them. At a radial node, the radial probability drops to zero, okay? Quick check on the graph. For two s, n is two and l is zero; for two p, l is one. So how many maxima, and how many nodes? Two s: two maxima, and one node. See the orange curve: a small hump, a dip to zero where the arrow points, then a big hump. Two p: one maximum, and no node, the green curve's one smooth hump. Perfect match!

5:23Example 1: radial probability maxima of 3s, 3p and 3d

Okay, your turn now. How many maxima does the radial probability curve of a three s orbital have? And of three p, and three d? Pause the video, and try it first. Done? Here is what many students say. All three have n equals three, so all three curves have three maxima, right? No, no, that is the trap! The rule is n minus l, not just n. And l is zero for s, one for p, and two for d, okay? So three s: three minus zero, three maxima. Three p: three minus one, two maxima. Three d: three minus two, only one maximum. Three, two, one. Really simple, right?

6:11Example 2: positions of the two radial nodes of 3s

Now a harder one, JEE level. For hydrogen, psi three s, that is psi with three s written small below, is proportional to, this curvy sign, twenty seven minus eighteen sigma plus two sigma squared, times e to the power minus sigma by three. Sigma, the Greek letter, is r over a naught. Find the radii of the radial nodes. Pause and try. Hint: count the nodes first. Done? Okay, let's check. First, the count: for three s, n minus l minus one is three minus zero minus one, two radial nodes. So we must get two answers. Now, at a node, psi is zero. And psi is the bracket times e to the power minus sigma by three. Which of these two factors can actually become zero? Only the bracket, yes! The exponential part is never zero; the equals sign with a slash means not equal. So the nodes come from two sigma squared minus eighteen sigma plus twenty seven equals zero, the same bracket rearranged. Now the quadratic formula. Eighteen squared is three hundred twenty four, and four times two times twenty seven is two hundred sixteen. So sigma is eighteen, plus or minus root of three hundred twenty four minus two hundred sixteen, all over four. That root, the tick shaped sign, is about ten point three nine. So, eighteen plus or minus ten point three nine, over four. So sigma is about one point nine zero, or seven point one zero; the wavy equals means approximately. Sigma is r over a naught, so the double arrow gives the nodes: r about one point nine zero a naught, and seven point one zero a naught. Two nodes, as predicted. That quick count is your safety check.

8:07Quick revision: psi squared, orbitals, maxima and nodes

Clear? So, quick revision. Psi has no physical meaning; mod psi squared is the probability density. An orbital is a probability region, not a path, holding at most two electrons. Radial probability is four pi r squared psi squared d r, with n minus l maxima, and n minus l minus one nodes. Now go and try the practice problems on this page. You will find them easy. All the best!