Download Pauli Exclusion Principle

Pauli Exclusion Principle
No more than one electron may occupy a given
quantum state specified by a particular set of singleparticle quantum numbers n, l, ml ms.
ground state
Ground State of Atoms
H : 1s1
He: 1s2 (filled shell n=1)
Na: 1s2 2s2 2p6 3s1
Mg: 1s2 2s2 2p6 3s2
Li: 1s2 2s1
Al: 1s2 2s2 2p6 3s2 3p1
Be: 1s2 2s2
Si : 1s2 2s2 2p6 3s2 3p2
B : 1s2 2s2 2p1
P: 1s2 2s2 2p6 3s2 3p3
C : 1s2 2s2 2p2
S: 1s2 2s2 2p6 3s2 3p4
N: 1s2 2s2 2p3
O: 1s2 2s2 2p4
F: 1s2 2s2 2p5
Ne: 1s2 2s2 2p6 (filled shell n=2)
Cl: 1s2 2s2 2p6 3s2 3p5
Ar: 1s2 2s2 2p6 3s2 3p6
K: 1s2 2s2 2p6 3s2 3p6 4s1
7-46. Write the ground-state electron configuration of (a) carbon, (b) oxygen, and (c)
argon.
C : 1s2 2s2 2p2
O: 1s2 2s2 2p4
Ar: 1s2 2s2 2p6 3s2 3p6
7-51. What elements have these ground-state electron configurations? (a) 1s22s22p63s23p2
and (b) 1s22s22p63s23p64s2?
a)
7-49. If the 3s electron in sodium did not penetrate the inner core, its energy would be
-13.6 eV/32 = -1.51 eV. Because it does penetrate, it sees a higher effective Z and its
energy is lower. Use the measured ionization potential of 5.14 V to calculate Zeff for the 3s
electron in sodium.
7-72 (a) Show that the function
is a solution of Equation 7-9, where A is a constant and a0 is the Bohr radius. (b) Find the
constant A.
7-72 (a) Show that the function
is a solution of Equation 7-9, where A is a constant and a0 is the Bohr radius. (b) Find the
constant A.
7-72 (a) Show that the function
is a solution of Equation 7-9, where A is a constant and a0 is the Bohr radius. (b) Find the
constant A.
7-72 (a) Show that the function
is a solution of Equation 7-9, where A is a constant and a0 is the Bohr radius. (b) Find the
constant A.
7-65 Show that the expectation value of r for the electron in the ground state of a one
electron atom is <r > = 3/2 <a0>/Z.
Change of variable
Excited States and Spectra of Alkali Atoms
- An excited state of the atom usually involves a change in the state of one of the electrons
or, more rarely, two or even more electrons.
-Even in the case of the excitation of only one electron, the change in state of this electron
changes the energies of the others.
-This effect is negligible, and the energy levels can be calculated accurately from a
relatively simple model of one electron plus a stable core for the alkali metals:
Li, Na, K, Rb, and Cs
-Similar to H
Na: Ne 3s1
Na: 1s2 2s2 2p6 3s1
outermost electron 3s
2p electrons
1s electrons
Transitions between occupied levels are prohibited by Pauli principle
Only outermost electrons are involved
Excited States and Spectra of Alkali Atoms
Na: 1s2 2s2 2p6 3s1
Ground state
Excited states
Na: Ne 3s1
7-31. The optical spectra of atoms with two electrons in the same outer shell are similar,
but they are quite different from the spectra of atoms with just one outer electron
because of the interaction of the two electrons. Separate the following elements into two
groups such that those in each group have similar spectra: lithium, beryllium, sodium,
magnesium, potassium, calcium, chromium, nickel, cesium, and barium.
lithium, sodium, potassium, chromium, cesium, (one
electron in the outermost shell)
7-55. Which of the following elements should have optical spectra similar to that of
hydrogen and which should have optical spectra similar to that of helium: Li, Ca, Ti, Rb,
Ag, Cd, Ba, Hg, Fr, Ra?
Li, Rb, Ag, Fr (similar to H)
Spin-orbit coupling
Atomic states with the same n and l ,but different j have slightly different energies
7-49. Which of the following atoms would you expect to have its ground state split by the
spin-orbit interaction: Li, B, Na, Al, K, Ag, Cu, Ga? (Hint: Use Appendix C to see which
elements have / = 0 in their ground state and which do not.)
Li: 1s2 2s1
Na: 1s2 2s2 2p6 3s1
B : 1s2 2s2 2p1
Al:
1s2
2s2
2p6
3s2 3p1
Ga: 1s2 2s2 2p6 3s2 3p6 3d104s2 4p1
K: 1s2 2s2 2p6 3s2 3p6 4s1
Ag: 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s1
Cu: 1s2 2s2 2p6 3s2 3p6 3d10 4s1
7-59. Since the P states and the D states of sodium are all doublets, there are four possible
energies for transitions between these states. Indicate which three transitions are allowed
and which one is not allowed by the selection rule of Equation 7-65.
7-62 Show that the change in wavelength Dl of a transition due to a small change in
energy is
(Hint: Differentiate E = hc/λ)
7-69 The wavelengths of the photons emitted by potassium corresponding to transitions
from the 4P3/2 and 4P1/2 states to the ground state are 766.41 nm and 769.90 nm. (a)
Calculate the energies of these photons in electron volts. (b) The difference in energies of
these photons equals the difference in energy ΔE between the 4P3/2 and 4P1/2 states in
potassium. Calculate ΔE . (c) Estimate the magnetic field that the 4p electron in potassium
experiences.
a)
b)
c)
7-74 If relativistic effects are ignored, the n = 3 level for one-electron atoms consists of
the 32S1/2, 32P1/2, 32P3/2, 32D3/2, and 32D5/2 states. Compute the spin-orbit-effect splittings of
3P and 3D states for hydrogen.
Thomas precession
7-74 If relativistic effects are ignored, the n = 3 level for one-electron atoms consists of
the 32S1/2, 32P1/2, 32P3/2, 32D3/2, and 32D5/2 states. Compute the spin-orbit-effect splittings of
3P and 3D states for hydrogen.
7-74 If relativistic effects are ignored, the n = 3 level for one-electron atoms consists of
the 32S1/2, 32P1/2, 32P3/2, 32D3/2, and 32D5/2 states. Compute the spin-orbit-effect splittings of
3P and 3D states for hydrogen.
Zeeman effect
When an atom is placed in an external magnetic field B, the total angular momentum
J is quantized in space relative to the direction of B and the energy of the atomic
state characterized by the angular momentum quantum number j is split into 2j + 1
energy levels corresponding to the 2j +1 possible values of the z component of J and
therefore to the 2j +1 possible values of the z component of the total magnetic
moment.
Lamb shift
energy level fluctuations of the vacuum
7-41. The Lamb shift energy difference between the 22S1/2 and 22P1/2 levels in atomic
hydrogen is 4.372 x 10-6 eV. (a) What is the frequency of the photon emitted in this
transition? (b) What is the photon’s wavelength? (c) In what part of the electromagnetic
spectrum does this transition lie?
a)
b)