PHY206: Atomic Spectra

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PHY206 Atomic Spectra

• Lecturer Dr Stathes Paganis
• Office D29, Hicks Building
• Phone [phone]
• Email e.paganis_at_shef.ac.uk
• Text Arthur Beiser, Concepts of Modern Physics
• http//phy206.group.shef.ac.uk/
• Marks Final 70, Homework 2x10, Problems Class
  10

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Course Outline (1)

• Lecture 0 Introduction , Waves - Particles
• Lecture 1 Bohr Theory
• Introduction
• Bohr Theory (the first QM picture of the atom)
• Quantum Mechanics
• Lecture 2 Angular Momentum (1)
• Orbital Angular Momentum (1)
• Magnetic Moments
• Lecture 3 Angular Momentum (2)
• Stern-Gerlach experiment the Spin
• Examples
• Orbital Angular Momentum (2)
• Operators of orbital angular momentum
• Lecture 4 Angular Momentum (3)
• Orbital Angular Momentum (3)
• Angular Shapes of particle Wavefunctions
• Spherical Harmonics
• Examples

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Course Outline (2)

• Lecture 5 The Hydrogen Atom (1)
• Central Potentials
• Classical and QM central potentials
• QM of the Hydrogen Atom (1)
• The Schrodinger Equation for the Coulomb
  Potential
• Lecture 6 The Hydrogen Atom (2)
• QM of the Hydrogen Atom (2)
• Energy levels and Eigenfunctions
• Sizes and Shapes of the H-atom Quantum States
• Lecture 7 The Hydrogen Atom (3)
• The Reduced Mass Effect
• Relativistic Effects

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Course Outline (3)

• Lecture 8 Identical Particles (1)
• Particle Exchange Symmetry and its Physical
  Consequences
• Lecture 9 Identical Particles (2)
• Exchange Symmetry with Spin
• Bosons and Fermions
• Lecture 10 Atomic Spectra (1)
• Atomic Quantum States
• Central Field Approximation and Corrections
• Lecture 11 Atomic Spectra (2)
• The Periodic Table
• Lecture 12 Review Lecture

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Atoms, Protons, Quarks and Gluons
Atomic Nucleus
Atom
Proton
Proton
gluons
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Atomic Structure
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Empty space is it really empty?
Inside the proton
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De Broglie Waves
Moving bodies (particles) have a wave nature
Momentum is proportional to wave-number. Proportio
nality constant
De Broglie wavelength
Energy is proportional to wave-frequency
Need to understand waves
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Describing a wave
Consider a stretched along X-axis oscillating
along Y (simple harmonic oscillation)
Notice that this is only a function of time. To
get the x dependence, imagine that at t0 we
shake the string at x0 so that a wave will start
propagating across the string.
velocity of wave
distance travelled after time t
The displacement of y at distance x at ANY time
t, is the same as the value of y at the earlier
time
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Waves - Wavefunctions
At each point x oscillation with amplitude A and
frequency
Sinusoidal waves
At each time t photo of a wave with amplitude A
and wavelength
Wave moves in increasing x direction with velocity
Most general sinusoidal wave with wave number k
and ang. frequency w
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Complex Wavefunctions
In classical physics this is mathematically
convenient. However in QM this is a necessity
wavefunctions ARE complex
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Complex Number 1min Review
Complex numbers points on the complex plane.
They can be represented as
or
Z
Complex conjugate
Magnitude
In Classical Physics, its a matter of
convenience (physical solution can be either real
or imaginary part of a complex function)
In Quantum Physics, its a necessity.
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What does complex bring?
We have the presence of a phase
space
time
This phase depends on space and on time.
Evolution of the real part of c-wave
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Travelling waves, Plane Waves
Travelling waves
The waves we presented are all travelling (c is
the velocity)
These are solutions of the Classic Wave
Equation
Plane waves special kind of c-valued travelling
waves.
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Superposition of harmonic modes
At a certain time t the wave-group can be
expressed as an infinite sum of harmonic waves
with varying amplitudes (the Fourier Integral).
Fourier transform of Y
Sum these harmonics
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Standing Waves
20 nodes
21 nodes
22 nodes
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Phase and group velocities
- no limitations on the phase velocity, (phase of
a plane wave does not carry any information)
This phase velocity
The observable is the velocity of propagation of
a wave packet or wave group. Consider the
superposition of two harmonic waves with slightly
different frequencies (?gtgt??, kgtgt?k)
The velocity of propagation of the wave packet
fast oscillations within the wave group
envelope wave group
-the group velocity
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Group Velocity of de Broglie waves
- the group velocity of de Broglie waves coincide
with the particles velocity
Periodic processes discrete spectrum (Fourier
series).
Aperiodic processes continuous spectrum
(represented as Fourier integral)
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Dispersive waves
A non dispersive wavepacket propagates without
dispersion
Non-dispersion happens when the phase and group
velocities are equal.
Dispersion happens when
For a dispersive wavepacket the frequency is not
anymore proportional to wavenumber
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Uncertainty principle
A wave of finite length L (wavepacket) is
localized within L
An infinitely long wave is de-localized (we
cannot tell its position x)
L
A confined wavepacket does not have a definite
wavelength but a range of them.
This wave has definite wavelength
To produce the wavepacket we need to sum a large
number of harmonics (with varying wavelengths).
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Fourier Transform, Uncertainty principle
Fourier transform of Y
small
small
(Figure from Beiser)
To confine a wave in a region of space, we need a
large range of wavelengths
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Earlier (incorrect) Interpretation of de Broglie
Waves
Earlier ideas (Schrödinger) particle the wave
group. In favour the group velocity of de
Broglie waves coincide with the particles
velocity. However, the wave packet wouldnt live
for a long time because of the dispersion of de
Broglie waves in vacuum
In general
- no dispersion (c?c(k))
for light in vacuum
Deformation of a 1D wave group, mme (1
Bohr0.053nm, time units h3/mee42.4?10-17s)
Thus, a particle IS NOT the group of de Broglie
waves!
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Particle-Wave Duality
Matter exhibits wavelike aspects
De Broglies equations
apply to particles.
The wave-like character of an object becomes more
apparent at low kinetic energies as its de
Broglie wavelength increases.
Experimentally
- particle behavior dominates
- wave behavior dominates
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Postulates of Quantum Mechanics

• Every QM state is vector in a complex vector
  space.
• A system that can be in 2 possible quantum states
  (up,down).
• It has two dimensions (discrete function)
• A Wavefunction is one such vector indexed by x
  and t.
• It has infinite dimensions (continuous function)
• Observables (things that you measure in an
  experiment) correspond to Hermitian operators.
• The possible results of a measurement (eg the
  actual value of the energy of a particle) are the
  eigenvalues of H.
• Notice that it is always a real number
• The states for which the observable H is definite
  are the eigenstates (or eigenvectors).
• Example if you have prepared a system to be in a
  particular QM state and the observable you are
  measuring is not subject to stat fluctuation
  (i.e. it is definite) then this is an eigenstate.
  The spin of an electron is a good example (later).

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Postulates (continued)

• If we start (prepare) an arbitrary Quantum State
  then the probability to measure one of the
  possible observables (eigenvalues) is the square
  of the amplitude (component) in the eigenvector
  basis

state
eigenvector
component on first axis
The components are in general complex. The
probability is their c-square
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What is the meaning of a wavefunction ?
Notice that the wavefunction is a continuous
complex vector of x. We can show that the
eigenvalues of x is x, and the eigenvectors are
the Dirac delta functions (why?).
Eigenvector
component on first axis
This means that the wavefunction at a particular
point x10cm (say), is the component of the
wavefunction complex vector at this particular
axis (eigendirection). It also means that its
square is the probability to find the quantum
state at x10cm !!!
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Probability
The probability density for a particle at a
location is proportional to the square of the
wavefunction at that point
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Things to remember

• What we observe are observables NOT quantum
  states.
• For a wavefunction , the value of
  the wavefunction at a particular x5m, t10sec,
  is a separate quantum state
• Observables (experimentally) are things like
  position, momentum, energy, electric field,

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Early Models of the Atom

• Rutherfords model
• Planetary model
• Based on results of thin foil experiments (1907)
• Positive charge is concentrated in the center of
  the atom, called the nucleus
• Electrons orbit the nucleus like planets orbit
  the sun

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atoms should collapse
Classical Physics
Classical Electrodynamics charged particles
radiate EM energy (photons) when their velocity
vector changes (e.g. they accelerate).
This means an electron should fall into the
nucleus.
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Light the big puzzle in the 1800s
Light from the sun or a light bulb has a
continuous frequency spectrum
Light from Hydrogen gas has a discrete frequency
spectrum
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Emission lines of some elements (all quantized!)
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Emission spectrum of Hydrogen

• Quantized spectrum

• Continuous spectrum

DE
DE
Any DE is possible
Only certain DE are allowed

• Relaxation from one energy level to another by
  emitting a photon, with DE hc/l
• If l 440 nm, DE 4.5 x 10-19 J

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Emission spectrum of Hydrogen
The goal use the emission spectrum to determine
the energy levels for the hydrogen atom
(H-atomic spectrum)
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Balmer model (1885)

• Joseph Balmer (1885) first noticed that the
  frequency of visible lines in the H atom spectrum
  could be reproduced by

n 3, 4, 5, ..

• The above equation predicts that as n increases,
  the frequencies become more closely spaced.

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Rydberg Model

• Johann Rydberg extended the Balmer model by
  finding more emission lines outside the visible
  region of the spectrum

n1 1, 2, 3, ..
n2 n11, n12,
Ry 3.29 x 1015 1/s

• In this model the energy levels of the H atom are
  proportional to 1/n2

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Electron standing-waves on an atom

• Electron wave extends around circumference of
  orbit.
• Only integer number of wavelengths around orbit
  allowed.

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The Bohr Model (1)

• Bohrs Postulates (1913)
• Bohr set down postulates to account for (1) the
  stability of the hydrogen atom and (2) the line
  spectrum of the atom.

• Energy level postulate An electron can have only
  specific energy levels in an atom.
• Electrons move in orbits restricted by the
  requirement that the angular momentum be an
  integral multiple of h/2p, which means that for
  circular orbits of radius r the z component of
  the angular momentum L is quantized
• 2. Transitions between energy levels An electron
  in an atom can change energy levels by undergoing
  a transition from one energy level to another.

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The Bohr Model (2)

• Bohr derived the following formula for the energy
  levels of the electron in the hydrogen atom.
• Bohr model for the H atom is capable of
  reproducing the energy levels given by the
  empirical formulas of Balmer and Rydberg.

Energy in Joules Z atomic number (1 for H) n is
an integer (1, 2, .)
The Bohr constant is the same as the Rydberg
multiplied by Plancks constant!
Ry x h -2.178 x 10-18 J
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The Bohr Model (3)
Energy levels get closer together as n
increases
at n infinity, E 0
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Prediction of energy spectra
We can use the Bohr model to predict what DE
is for any two energy levels
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Example calculation (1)
Example At what wavelength will an emission
from n 4 to n 1 for the H atom be
observed?
1
4
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Example calculation (2)
Example What is the longest wavelength of
light that will result in removal of the e- from
H?
?
1
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Bohr model extedned to higher Z
The Bohr model can be extended to any single
electron system.must keep track of Z (atomic
number).
Z atomic number
n integer (1, 2, .)
Examples He (Z 2), Li2 (Z 3), etc.
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Example calculation (3)
Example At what wavelength will emission
from n 4 to n 1 for the He atom be
observed?
2
1
4
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Problems with the Bohr model

• Why electrons do not collapse to the nucleus?
• How is it possible to have only certain fixed
  orbits available for the electrons?
• Where is the wave-like nature of the electrons?

First clue towards the correct theory De Broglie
relation (1923)
Einstein
De Broglie relation particles with certain
momentum, oscillate with frequency hv.
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Quantum Mechanics

• Particles in quantum mechanics are expressed by
  wavefunctions
• Wavefunctions are defined in spacetime (x,t)
• They could extend to infinity (electrons)
• They could occupy a region in space
  (quarks/gluons inside proton)
• In QM we are talking about the probability to
  find a particle inside a volume at (x,t)
• So the wavefunction modulus is a Probability
  Density (probablity per unit volume)
• In QM, quantities (like Energy) become
  eigenvalues of operators acting on the
  wavefunctions

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QM we can only talk about the probability to
find the electron around the atom there is no
orbit!