2006-7 Quantum Theory Slides Lecture 3

Quantum theory and atomic spectroscopy Lecture 3 Angular momentum

The story so far…. • Atoms contain electrons in discrete energy levels • The energy levels can be divided into shells (n) and subshells (l) • The number of subshells available is determined by n. • The maximum number of electrons in a subshell is given by 2 x (2l + 1) • Now read on …

Today’s question(s) is (are)…… • Why is the maximum number of electrons in a subshell dependent on l? • Why do we call l the orbital angular momentum quantum number?

Orbital angular momentum

• Because it will “orbit” the nucleus, each electron has an orbital angular momentum • In quantum theory, angular momentum is quantised!! • The direction of the angular momentum vector depends on whether the electron is moving clockwise or anti-clockwise • This orbital angular momentum of the electron is given by l

Optical selection rules • Angular momentum plays in a role in spectroscopy • The most common transitions are called electric dipole transitions • The light adds or subtracts 1h/2π (a single quantum) of angular momentum to the atom • Hence there is a selection rule is ∆l = ± 1 • In hydrogen we have an extra rule, ∆n ≠ 0

Total orbital angular momentum • We can add the a.m. of the individual electrons to get a TOTAL orbital angular momentum for the atom • If a subshell is closed (full) the total for that subshell is 0 • If the subshell is partially filled, we determine its total angular momentum, usually given the value L, by a VECTOR sum • Therefore, when we add two non-zero values of l we get two or more values of L

Vector addition

• When adding two vectors, we can get more than one result • Only discrete values allowed i.e. for orbital angular momentum, the result must be an integer • Total is: L = l1 + l2, l1 + l2 – 1,….. (l1 – l2) + 1, l1 – l2

Zeeman effect • Become a scientist and get your face on a stamp! • Zeeman wondered whether a magnetic field would affect atomic emission lines • He had no idea what the result would be, nor did he know anything about quantum numbers!

Zeeman’s experiment

• Looked at light emission as a function of frequency • With and without magnetic field Fabry FabryPerot Perot––frequency frequencyscanner scanner (bit (bitlike likeaaspectrometer) spectrometer)

Zeeman’s observation • The crucial experiment in cadmium • The transition splits into three lines • The magnitude of the splitting depends on the magnetic field

Lorentz’s explanation • Lorentz “explained” the Zeeman effect in terms of an electric current in an atom (no mention of quantised angular momentum at all) • Lots of champagne, 1902 Nobel prize for Hendrick (and Zeeman)! • Great, so long as you always see three lines in a magnetic field • But….. eBz ν 0 and ν 0 ± 4πme c

Anomalous Zeeman effect • Thomas Preston had already demonstrated in 1897 that “splittings” could be observed that did not fit Zeeman’s pattern.

Zeeman effect in helium

• The 2p subshell is in fact into 3 sublevels which have different energies in a magnetic field. • New quantum number introduced: ml • For any l there are (2l + 1) ml sublevels with values ml = -l, -l + 1,….. l – 1, l Maximum of 2 electrons in each sublevel!

Calculating the Zeeman shift • The modern form is very similar to Zeeman’s formula, but for completely different reasons!

∆E M L

ehBz ML = 4πme c

Old formula – complete pants!

New formula- quantum

ν0

and

eB z ν0 ± 4πme c

What do these sublevels mean? • Arnold Sommerfeld suggested Space Quantization was the answer • Evidence? None!

Enter Stern and Gerlach • Stern and Gerlach believed space quantization was literally true • Set up an experiment to prove it!

Stern – Gerlach experiment • The interaction between the electron angular momentum and the magnetic field depends on their relative orientations • Classically, the angular momentum can point in any direction in the field: quantum says space quantisation takes place • Use the magnetic interaction to move the atoms in space and look on a screen

The Stern – Gerlach result • Classically, expect a diffuse blob! Quantum theory predicts separate spots • Using silver atoms, they observed TWO separate spots: space quantization!!!

1-0 1-0to to quantum quantum theory! theory!But But there thereisis something something aabit bitodd odd here…. here….

Answers to the earlier questions.. • Different subshells hold different numbers of electrons because they have different numbers of sublevels! • The number of sublevels is determined by l • Each sublevel holds a maximum of two electrons

Hang on…… • Why do we see two spots? We only have an ODD number of sublevels for any value of l or L! • Why can a sublevel hold just two electrons? • Next time, we will see that electron SPIN is the key!