Physics 202 - galileo.phys.virginia.edugalileo.phys.virginia.edu/~pqh/202_23n.pdfPhysics 202 – p....
Transcript of Physics 202 - galileo.phys.virginia.edugalileo.phys.virginia.edu/~pqh/202_23n.pdfPhysics 202 – p....
Physics 202Professor P. Q. Hung
311B, Physics Building
Physics 202 – p. 1/41
Atomic Physics
The Hydrogen Atom: Spectrum
Pre-Bohr Observation: Emission of light byatoms in a gas. Let the light pass through anarrow-slit aperture ⇒ Discrete set of lines ofdifferent colors (or wavelengths).
For hydrogen, Balmer empirically fitted thespectrum withλ = (364.5 nm) n2
n2−4
n = 3, 4, 5, ....
Physics 202 – p. 2/41
Atomic Physics
The Hydrogen Atom: Spectrum
Pre-Bohr Observation: Emission of light byatoms in a gas. Let the light pass through anarrow-slit aperture ⇒ Discrete set of lines ofdifferent colors (or wavelengths).
For hydrogen, Balmer empirically fitted thespectrum withλ = (364.5 nm) n2
n2−4
n = 3, 4, 5, ....
Physics 202 – p. 2/41
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 3/41
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 4/41
Atomic Physics
The Hydrogen Atom: Spectrum
One can invert the previous equation to give:1λ
= R( 122 − 1
n2 )
where R = 10.97373 µm−1 is the Rydbergconstant.
More general empirical formula which alsoapplies to heavier atoms: The Rydberg-Ritzformula1λ
= R Z2( 1n2
2
− 1n2
1
)
where n1 > n2 and Z = 1 for hydrogen.
Physics 202 – p. 5/41
Atomic Physics
The Hydrogen Atom: Spectrum
One can invert the previous equation to give:1λ
= R( 122 − 1
n2 )
where R = 10.97373 µm−1 is the Rydbergconstant.
More general empirical formula which alsoapplies to heavier atoms: The Rydberg-Ritzformula1λ
= R Z2( 1n2
2
− 1n2
1
)
where n1 > n2 and Z = 1 for hydrogen.
Physics 202 – p. 5/41
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 2, n1 = n: Balmer series.This series covers the visible spectrum. Thelongest wavelength is when n = 3 ⇒λ = 656.3 nm, on the reddish side. Theshortest wavelength is when n = ∞ ⇒λ = 364.6 nm, on the ultraviolet side.
Physics 202 – p. 6/41
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 1, n1 = 2, 3, ...: Lyman series.λ = (91.13 nm) n2
n2−1
n = 2, 3, ...This is in the ultraviolet regime. The longestwavelength is when n = 2 ⇒ λ = 121.5 nm.The shortest wavelength is when n = ∞ ⇒λ = 91.13 nm.
Physics 202 – p. 7/41
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 3, n1 = 4, 5, ...: Paschen series’λ = (820.14 nm) n2
n2−9
n = 4, 5, ....This is in the infrared region. The longestwavelength is when n = 4 ⇒ λ = 1874.6 nm.The shortest wavelength is when n = ∞ ⇒λ = 820.14 nm.
Physics 202 – p. 8/41
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 9/41
Atomic Physics
The Bohr Model
Two crucial elements: The Rutherford’spicture of the atom and the Balmer formula(or more generally the Rydberg-Ritz formula).
Defect with the Rutherford’s picture:Classically, the electron in orbit around thenucleus will lose energy through radiation andwill collapse into the nucleus.Something else: Bohr’s model in 1913.
Physics 202 – p. 10/41
Atomic Physics
The Bohr Model
Two crucial elements: The Rutherford’spicture of the atom and the Balmer formula(or more generally the Rydberg-Ritz formula).
Defect with the Rutherford’s picture:Classically, the electron in orbit around thenucleus will lose energy through radiation andwill collapse into the nucleus.Something else: Bohr’s model in 1913.
Physics 202 – p. 10/41
Atomic Physics
The Bohr Model: Bohr’s 2 postulates
Electrons move in certainstationary, non-radiating, circular orbitsconsistent with Coulomb’s law and Newton’slaw and specified by the quantization ofangular momentumL = mvr = n~
n: an integer.
Radiation of frequencyf =
Ei−Ef
h
occurs when the electrom jumps from orbit iof energy Ei to orbit f of energy Ef .
Physics 202 – p. 11/41
Atomic Physics
The Bohr Model: Bohr’s 2 postulates
Electrons move in certainstationary, non-radiating, circular orbitsconsistent with Coulomb’s law and Newton’slaw and specified by the quantization ofangular momentumL = mvr = n~
n: an integer.
Radiation of frequencyf =
Ei−Ef
h
occurs when the electrom jumps from orbit iof energy Ei to orbit f of energy Ef . Physics 202 – p. 11/41
Atomic Physics
The Bohr Atom
Physics 202 – p. 12/41
Atomic Physics
The Bohr Atom
Physics 202 – p. 13/41
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2postulates
Circular orbit:Take an atom with a positively chargednucleus of charge Ze at the center and oneelectron going around it. The potential energyis U = −kZe2/r. The total energy isE = 1
2mv2 + U . For a circular orbit,kZe2/r2 = mv2/r ⇒ 1
2mv2 = kZe2/2r ⇒E = −kZe2/2r.But how do we know what r is for variousorbits? Physics 202 – p. 14/41
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2postulates
Orbital angular momentum is quantized:From kZe2/r2 = mv2/r, one findsrn = n2 ~
2
mkZe2 = n2 a0
Z(1)
a0 = ~2
mke2 = 0.0529 nmis the Bohr’s radius.
Physics 202 – p. 15/41
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2postulates
How did Bohr derive the Rydberg-Ritzformula?Using f =
Ei−Ef
hand E = −kZe2/2r, one finds
f = (kZe2
2hr)( 1
rf− 1
ri). Using (1) and f = c/λ,
one finds the Rydberg-Ritz formula withR = mk2e4
4πc~3
In very good agreement with the empiricaldetermination of the Rydberg constant.
Physics 202 – p. 16/41
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2postulates
How does one write the total energy of anelectron in an orbit rn?From E = −kZe2/2r and Eq. (1), one findsEn = −Z2 E0
n2 (2)whereE0 = k2e4m
2~2 ≈ 13.6 eV
is the ionization or binding energy of theelectron in the ground state of the hydrogenatom. It is the minimum energy to remove theelectron from the hydrogen atom.
Physics 202 – p. 17/41
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2postulates
For the hydrogen atom, one hasEn = −13.6 eV
n2
E1 = −13.6 eV is the ground state energy ofthe hydrogen atom.Caveat: Bohr’s model only applies to the casewhere there is one electron. It fails for atomshaving several electrons!
Physics 202 – p. 18/41
Atomic Physics
The Bohr Atom: Consequences
Physics 202 – p. 19/41
Atomic Physics
The Bohr Atom: Example 1The Lithium atom Li has Z = 3. Since it isneutral, it also has 3 electrons. Bohr’s model isnot applicable. If two electrons are stripped away,one ends up with the ion Li2+. What is theionization energy of Li2+?Solution:Since one has only one electron left, the Bohr’smodel can be used. The ionization energy isfound by putting n = 1 and Z = 3 in Eq. (2).E1 = −32(13.6 eV ) = −122 eV .So the minimum ionization energy needed toremove the remaining electron is 122 eV .
Physics 202 – p. 20/41
Atomic Physics
The Bohr Atom: Example 2Use the Bohr’s model to calculate the quatumnumber n of the Earth in its orbit about the Sun.Solution:1) Replace ke2 by GMEMS in Eq. (1) with Z andusing rn = 6 × 106 m.2) rn = n2 ~
2
GM2
EMS⇒
3) n = ME
√rnGMS
~= 2 × 1072. A large number!
Lesson: For a very large quantum number n, weget back the results of classical physics!
Physics 202 – p. 21/41
Atomic Physics
Origin of angular momentum quantization
Crucial ingredient of Bohr’s model: thequantization of angular momentum,L = mvr = n~. Origin?De Broglie’s wave-particle duality
Electrons behave like waves. On an orbit,standing wave phenomena ⇒ condition forstanding wave:2πr = nλλ = h
p
p = mv⇒ L = mvr = n h
2π .⇒ Wave nature of particles is crucial inatomic physics.
Physics 202 – p. 22/41
Atomic Physics
Origin of angular momentum quantization
Crucial ingredient of Bohr’s model: thequantization of angular momentum,L = mvr = n~. Origin?De Broglie’s wave-particle duality
Electrons behave like waves. On an orbit,standing wave phenomena ⇒ condition forstanding wave:2πr = nλλ = h
p
p = mv⇒ L = mvr = n h
2π .⇒ Wave nature of particles is crucial inatomic physics.
Physics 202 – p. 22/41
Atomic Physics
Origin of angular momentum quantization
Physics 202 – p. 23/41
Atomic Physics
Origin of angular momentum quantizationShortcomings of Bohr’s model:
No prediction for relative intensities ofspectral lines.
Cannot describe complex atoms.
Lack of theoretical foundation.
Physics 202 – p. 24/41
Atomic Physics
Origin of angular momentum quantizationShortcomings of Bohr’s model:
No prediction for relative intensities ofspectral lines.
Cannot describe complex atoms.
Lack of theoretical foundation.
Physics 202 – p. 24/41
Atomic Physics
Origin of angular momentum quantizationShortcomings of Bohr’s model:
No prediction for relative intensities ofspectral lines.
Cannot describe complex atoms.
Lack of theoretical foundation.
Physics 202 – p. 24/41
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantummechanics:
Propagation of matter waves is governed bythe Schroedinger equation.
Matter wave is described by a wave functionwhose absolute square represents theprobability to find the particle within a rangeof position (roughly speaking)
Physics 202 – p. 25/41
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantummechanics:
Propagation of matter waves is governed bythe Schroedinger equation.
Matter wave is described by a wave functionwhose absolute square represents theprobability to find the particle within a rangeof position (roughly speaking)
Physics 202 – p. 25/41
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantummechanics:
Propagation of matter waves is governed bythe Schroedinger equation.
Matter wave is described by a wave functionwhose absolute square represents theprobability to find the particle within a rangeof position (roughly speaking)
Physics 202 – p. 25/41
Atomic Physics
Quantum MechanicsIn Bohr’s model, only one quantum number: nIn quantum mechanics:
Principal quantum number n.
Orbital quantum number l.l = 0, 1, 2, .., (n − 1).Magnitude of angular momentum:L =
√
l(l + 1)~.
Physics 202 – p. 26/41
Atomic Physics
Quantum MechanicsIn Bohr’s model, only one quantum number: nIn quantum mechanics:
Principal quantum number n.
Orbital quantum number l.l = 0, 1, 2, .., (n − 1).Magnitude of angular momentum:L =
√
l(l + 1)~.
Physics 202 – p. 26/41
Atomic Physics
Quantum Mechanics
Magnetic quantum number ml.ml = −l, ..,−1, 0, 1, .., l.Lz = ml~.
Spin quantum number ms. ms = ±12.
Physics 202 – p. 27/41
Atomic Physics
Quantum Mechanics
Magnetic quantum number ml.ml = −l, ..,−1, 0, 1, .., l.Lz = ml~.
Spin quantum number ms. ms = ±12.
Physics 202 – p. 27/41
Atomic Physics
Quantum mechanics: Hydrogen atom
Physics 202 – p. 28/41
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:No two electrons can have the same set ofvalues for n, l, ml, ms.
An atom is characterized by energy levels,e.g. n = 1, l = 0 is the lowest energy level orground state.
Ground state of an atom: the electrons are inthe lowest available energy levels consistentwith the Pauli exclusion principle.
Physics 202 – p. 29/41
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:No two electrons can have the same set ofvalues for n, l, ml, ms.
An atom is characterized by energy levels,e.g. n = 1, l = 0 is the lowest energy level orground state.
Ground state of an atom: the electrons are inthe lowest available energy levels consistentwith the Pauli exclusion principle.
Physics 202 – p. 29/41
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:No two electrons can have the same set ofvalues for n, l, ml, ms.
An atom is characterized by energy levels,e.g. n = 1, l = 0 is the lowest energy level orground state.
Ground state of an atom: the electrons are inthe lowest available energy levels consistentwith the Pauli exclusion principle.
Physics 202 – p. 29/41
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Subshell: For a given shell, levels withdifferent l, ml, ms. For l, one usually call: s, p,d, f, g, h,...
The Pauli Exclusion principle allows only amaximum number of electrons that can fit intoan energy level or subshell.
Physics 202 – p. 30/41
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Subshell: For a given shell, levels withdifferent l, ml, ms. For l, one usually call: s, p,d, f, g, h,...
The Pauli Exclusion principle allows only amaximum number of electrons that can fit intoan energy level or subshell.
Physics 202 – p. 30/41
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Subshell: For a given shell, levels withdifferent l, ml, ms. For l, one usually call: s, p,d, f, g, h,...
The Pauli Exclusion principle allows only amaximum number of electrons that can fit intoan energy level or subshell.
Physics 202 – p. 30/41
Atomic Physics
Pauli Exclusion Principle
For each l, there can be 2(2l + 1) electronsthat can fit. (This is because of the number ofcombinations of ml and ms.)
Notation for the ground state of an atom:n lnumber of electrons.Hydrogen(H): 1 electron,1 s1.Helium (He): 2 electrons, 1 s2.Lithium (Li): 3 electrons, 1 s2 2 s1.Beryllium (Be): 4 electrons, 1 s2 2 s2
etc..
Physics 202 – p. 31/41
Atomic Physics
Pauli Exclusion Principle
For each l, there can be 2(2l + 1) electronsthat can fit. (This is because of the number ofcombinations of ml and ms.)
Notation for the ground state of an atom:n lnumber of electrons.Hydrogen(H): 1 electron,1 s1.Helium (He): 2 electrons, 1 s2.Lithium (Li): 3 electrons, 1 s2 2 s1.Beryllium (Be): 4 electrons, 1 s2 2 s2
etc..Physics 202 – p. 31/41
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 32/41
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 33/41
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 34/41
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 35/41
Atomic Physics
Atomic RadiationTwo types of X-rays: bremsstrahlung andcharacteristic X-rays
bremsstrahlung: radiation from energeticelectrons which decelerate upon hitting atarget. Continuous spectrum.
characteristic X-rays: An incoming electronknocks off say a K-shell electron. That void isfilled immediately by electrons from outershells ⇒ X-rays. To do that the incomingelectron must have at leastEK = −13.6 eV (Z−1)2
12 .
Physics 202 – p. 36/41
Atomic Physics
Atomic RadiationTwo types of X-rays: bremsstrahlung andcharacteristic X-rays
bremsstrahlung: radiation from energeticelectrons which decelerate upon hitting atarget. Continuous spectrum.
characteristic X-rays: An incoming electronknocks off say a K-shell electron. That void isfilled immediately by electrons from outershells ⇒ X-rays. To do that the incomingelectron must have at leastEK = −13.6 eV (Z−1)2
12 .Physics 202 – p. 36/41
Atomic Physics
Atomic Radiation
Notice that the energy of the electron in theother shells is En = −13.6 eV (Z−1)2
n2 .
Characteristic peaks, e.g. Kα,Kβ.Read the book concerning lasers, etc...
Physics 202 – p. 37/41
Atomic Physics
Atomic Radiation
Notice that the energy of the electron in theother shells is En = −13.6 eV (Z−1)2
n2 .
Characteristic peaks, e.g. Kα,Kβ.Read the book concerning lasers, etc...
Physics 202 – p. 37/41
Atomic Physics
Atomic Radiation
Physics 202 – p. 38/41
Atomic Physics
Atomic Radiation
Physics 202 – p. 39/41
Atomic Physics
Atomic Radiation
Physics 202 – p. 40/41
Atomic Physics
Atomic Radiation
Physics 202 – p. 41/41