1. CHAPTER 7 The Hydrogen Atom
Orbital Angular momentum
Application of the Schrödinger Equation to the Hydrogen Atom
Solution of the Schrödinger Equation for Hydrogen
Homework due next Friday Oct 23rd:
Read Chapter 7: problems 1, 4, 5, 6, 7, 8, 10, 12, 14, 15
Werner Heisenberg
( )
The atom of modern physics can be symbolized only through a partial differential equation in an abstract space of many dimensions. All its qualities are inferential; no material properties can be directly attributed to it. An understanding of the atomic world in that primary sensuous fashion…is impossible.
- Werner Heisenberg

2. Intrinsic Spin
In 1925, grad students, Samuel Goudsmit and George Uhlenbeck, in Holland proposed that the electron must have an intrinsic angular momentum and therefore a magnetic moment. Paul Ehrenfest showed that, if so, the surface of the spinning electron should be moving faster than the speed of light! In order to explain experimental data, Goudsmit and Uhlenbeck proposed that the electron must have an intrinsic spin quantum number s = ½.

3. Intrinsic Spin
As with orbital angular momentum, the total spin angular momentum is:
The z-component of the spinning electron is also analogous to that of the orbiting electron:
So Sz = ms ħ
Because the magnetic spin quantum number ms has only two values, ±½, the electron’s spin is either “up” (ms = +½) or “down” (ms = -½).

4. What about Sx and Sy?
As with Lx and Ly, quantum mechanics says that, no matter how hard we try, we can’t also measure them! If we did, we’d measure ±½ ħ, just as we’d find for Sz. But then this measurement would perturb Sz, which would then become unknown!
The total spin is , so it’d be tempting to conclude that every component of the electron’s spin is either “up” (+½ ħ) or “down” (ms = -½ ħ). But this is not the case! Instead, they’re undetermined. We’ll see next that the uncertainty in each unmeasured component is equal to their maximum possible magnitude (½ ħ)!

5. Generalized Uncertainty Principle
Define the Commutator of two operators, A and B:
If this quantity is zero, we say A and B commute.
Then the uncertainty relation between the two corresponding observables will be:
So if A and B commute, the two observables can be measured simultaneously. If not, they can’t.
Example:
So:
and

6. Uncertainty in angular momentum and spin
We’ve seen that the total and z-components of angular momentum and spin are knowable precisely. And the x and y-components aren’t. Here’s why. It turns out that:
and
Using:
We find:
So there’s an uncertainty relation between the x and y components of orbital angular momentum. And the same for spin.
Measurement of one perturbs the other.

7. Two Types of Uncertainty in Quantum Mechanics
1. Some quantities (e.g., energy levels) can, at least in principle, be computed precisely, but some cannot (e.g., Lx and Lz simultaneously).
Even if a quantity can, in principle, be computed precisely, the accuracy of its measured value can still be limited by the Uncertainty Principle. For example, energies can only be measured to an accuracy of ħ /Dt, where Dt is how long we spend doing the measurement.
2. And there is another type of uncertainty: we often simply don’t know which state an atom is in.
It can be in a superposition of states

8. Superposition of states
Energy
Ground level, E1
Excited level, E2
DE = hn
Stationary states are stationary. But an atom can be in a superposition of two stationary states, and this state moves.
where |ai|2 is the probability that the atom is in state i.
Interestingly, this lack of knowledge means that the atom is vibrating:
Vibrations occur at the frequency difference between the two levels.

9. Operators and Measured Values
In any measurement of the observable associated with an operator A, the only values that can ever be observed are the eigenvalues. Eigenvalues are the possible values of a in the Eigenvalue Equation: ˆ
where a is a constant and the value that is measured.
For operators that involve only multiplication, like position and potential energy, all values are possible.
But for others, like energy and momentum, which involve operators like differentiation, only certain values can be the results of measurements.

10. Writing H atom states in the bra-ket notation
The bra-ket notation provides a convenient short-hand notation for H states. Since n, ℓ, mℓ, and ms determine the state, we can write a state as a ket:
There’s no need to write the value of s, since it’s always ½ for electrons.
The specific mathematical functions involved are well known, so everyone knows what this means.
And when relevant, we can write the bra form for the complex conjugate, as well:

11. CHAPTER 8 Many-electron atoms
Dimitri Mendeleev
What distinguished Mendeleev was not only genius, but a passion for the elements. They became his personal friends; he knew every quirk and detail of their behavior.
- J. Bronowski

12. Chapter 8: Atomic Structure and the Periodic Table
What if there’s more than one electron?
Helium: a nucleus with charge +2e and two electrons,
the two electrons repelling one another.
Cannot solve problems exactly with the Schrödinger equation because of the complex potential interactions.
Can understand experimental results without computing the wave functions of many-electron atoms by applying the boundary conditions and selection rules.

13. Multi-electron atoms
When more than one electron is involved, the potential and the wave function are functions of more than one position:
Solving the Schrodinger Equation in this case can be very hard. But we can approximate the solution as the product of single-particle wave functions:
And it turns out that we’ll be able to approximate each Yi with a Hydrogen wave function.

14. Pauli Exclusion Principle
To understand atomic spectroscopic data, Pauli proposed his exclusion principle: No two electrons in an atom may have the same set of quantum numbers (n, ℓ, mℓ, ms). It applies to all particles of half-integer spin, which are called fermions, and particles in the nucleus are also fermions. The periodic table can be understood by two rules: The electrons in an atom tend to occupy the lowest energy levels available to them. The Pauli exclusion principle.

15. Atomic Structure
Hydrogen: (n, ℓ, mℓ, ms) = (1, 0, 0, ±½) in ground state. In the absence of a magnetic field, the state ms = ½ is degenerate with the ms = −½ state. Helium: (1, 0, 0, ½) for the first electron. (1, 0, 0, −½) for the second electron. Electrons have anti-aligned (ms = +½ and ms = −½) spins. The principle quantum number also has letter codes. n = Letter = K L M N… n = shells (eg: K shell, L shell, etc.) nℓ = subshells (e.g.: 1s, 2p, 3d)
Electrons for H and He atoms are in the K shell.
H: 1s
He: 1s2

16. Atomic Structure
How many electrons may be in each subshell? Recall: ℓ = … letter = s p d f g h … ℓ = 0, (s state) can have two electrons. ℓ = 1, (p state) can have six electrons, and so on.
Total
For each mℓ: two values of ms 2
For each ℓ: (2ℓ + 1) values of mℓ
2(2ℓ + 1)
Electrons with higher ℓ values are more shielded from the nuclear charge.
Electrons with higher ℓ values lie higher in energy than those with lower ℓ values.
4s fills before 3d.

17. The Periodic Table

18. The Periodic Table
Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies Electrical conductivity is relatively good Alkaline Earths: Two s electrons in outer subshell Largest atomic radii High electrical conductivity

19. The Periodic Table
Halogens: Need one more electron to fill outermost subshell Form strong ionic bonds with the alkalis More stable configurations occur as the p subshell is filled Transition Metals: Three rows of elements in which the 3d, 4d, and 5d are being filled Properties primarily determined by the s electrons, rather than by the d subshell being filled Have d-shell electrons with unpaired spins As the d subshell is filled, the magnetic moments, and the tendency for neighboring atoms to align spins are reduced

20. The Periodic Table
Lanthanides (rare earths): Have the outside 6s2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have unpaired electrons that align themselves The large orbital angular momentum contributes to the large ferromagnetic effects Actinides: Inner subshells are being filled while the 7s2 subshell is complete Difficult to obtain chemical data because they are all radioactive

21. 8.2: Total Angular Momentum
Orbital angular momentum
Spin angular momentum
Total angular momentum
L, Lz, S, Sz, J, and Jz are quantized.

22. Total Angular Momentum
If j and mj are quantum numbers for the single-electron hydrogen atom: Quantization of the magnitudes: The total angular momentum quantum number for the single electron can only have the values

23. Spin-Orbit Coupling
An effect of the spins of the electron and the orbital angular momentum interaction is called spin-orbit coupling. is the magnetic field due to the electron’s orbital motion. where a is the angle between .
The dipole potential energy
The spin magnetic moment µ ●

24. Total Angular Momentum
Now the selection rules for a single-electron atom become Δn = anything Δℓ = ±1 Δmj = 0, ±1 Δj = 0, ±1 Hydrogen energy-level diagram for n = 2 and n = 3 with spin-orbit splitting.

25. Many-Electron Atoms
Hund’s rules: The total spin angular momentum S should be maximized to the extent possible without violating the Pauli exclusion principle. Insofar as rule 1 is not violated, L should also be maximized. For atoms having subshells less than half full, J should be minimized. For a two-electron atom There are LS coupling and jj coupling to combine four angular momenta J.