CHM2S1-AIntroduction to Quantum Mechanics Dr R. L. Johnston

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CHM2S1-A Introduction to Quantum Mechanics Dr
R. L. Johnston

• I Foundations of Quantum Mechanics
• 1. Classical Mechanics
• 1.1 Features of classical mechanics.
• 1.2 Some relevant equations in classical
  mechanics.
• 1.3 Example The 1-Dimensional Harmonic
  Oscillator
• 1.4 Experimental evidence for the breakdown of
  classical mechanics.
• 1.5 The Bohr model of the atom.
•  
• 2. Wave-Particle Duality
• 2.1 Waves behaving as particles.
• 2.2 Particles behaving as waves.
• 2.3 The De Broglie Relationship.

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• 3. Wavefunctions
• 3.1 Definitions.
• 3.2 Interpretation of the wavefunction..
• 3.3 Normalization of the wavefunction.
• 3.4 Quantization of the wavefunction
• 3.5 Heisenbergs Uncertainty Principle.
•  
• 4. Wave Mechanics
• 4.1 Operators and observables.
• 4.2 The Schrödinger equation.
• 4.3 Particle in a 1-dimensional box.
• 4.4 Further examples.
•  

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Learning Objectives

• To appreciate the differences between Classical
  (CM) and Quantum Mechanics (QM).
• To know the failures in CM that led to the
  development of QM.
• To know how to interpret the wavefunction and how
  to normalize it.
• To appreciate the origins and implications of
  quantization and the uncertainty principle.
• To understand wave-particle duality and know the
  relationships between momentum, frequency,
  wavelength and energy for particles and
  waves.
• To be able to write down the Schrödinger equation
  for particles in a 1-D box in 1- and 2-electron
  atoms in 1- and 2-electron molecules.
• To know the origins and allowed values of atomic
  quantum numbers and how the energies and angular
  momenta of hydrogen atomic orbitals depend on
  them.
• To be able to sketch the angular and radial nodal
  properties of atomic orbitals.
• To appreciate the origins of sheilding and its
  effect on the ordering of orbital energies in
  many-electron atoms.

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• To use the Aufbau Principle, the Pauli Principle
  and Hunds Rule to predict the lowest energy
  electron configuration for many-electron atoms.
• To appreciate how the Born-Oppenheimer
  approximation can be used to separate electronic
  and nuclear motion in molecules.
• To understand how molecular orbitals (MOs) can be
  generated as linear combinations of atomic
  orbitals and the difference between bonding and
  antibonding orbitals.
• To be able to sketch MOs and their corresponding
  electron densities.
• To construct MO diagrams for homonuclear and
  heteronuclear diatomic molecules.
• To predict the electron configurations for
  diatomic molecules, calculate bond orders and
  relate these to bond lengths, strengths and
  vibrational frequencies.

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References

• Fundamentals
• P. W. Atkins, J. de Paula, Atkins' Physical
  Chemistry (7th edn.), OUP, Oxford, 2001.
• D. O. Hayward, Quantum Mechanics for Chemists
  (RSC Tutorial Chemistry Texts 14) Royal Society
  of Chemistry, 2002.
• W. G. Richards and P. R. Scott, Energy Levels in
  Atoms and Molecules (Oxford Chemistry Primers 26)
  OUP, Oxford, 1994.
• More Advanced
• P. W. Atkins and R. S. Friedman, Molecular
  Quantum Mechanics (3rd edn.) OUP, Oxford, 1997.
• P. A. Cox, Introduction to Quantum Theory and
  Atomic Structure (Oxford Chemistry Primers 37)
  OUP, Oxford, 1996.

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1. Classical Mechanics

• Do the electrons in atoms and molecules obey
  Newtons classical laws of motion?
• We shall see that the answer to this question is
  No.
• This has led to the development of Quantum
  Mechanics we will contrast classical and
  quantum mechanics.

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• 1.1 Features of Classical Mechanics (CM)
• 1) CM predicts a precise trajectory for a
  particle.
• The exact position (r)and velocity (v) (and hence
  the momentum p mv) of a particle (mass m) can
  be known simultaneously at each point in time.
• Note position (r),velocity (v) and momentum (p)
  are vectors, having magnitude and direction ? v
  (vx,vy,vz).

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• Any type of motion (translation, vibration,
  rotation) can have any value of energy associated
  with it
• i.e. there is a continuum of energy states.
• Particles and waves are distinguishable
  phenomena, with different, characteristic
  properties and behaviour.
• Property Behaviour
• mass momentum
• Particles position ? collisions
• velocity
• Waves wavelength ? diffraction
• frequency interference

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• 1.2 Revision of Some Relevant Equations in CM
• Total energy of particle
• E Kinetic Energy (KE) Potential Energy (PE)
• E ½mv2 V
• ? E p2/2m V (p mv)
• Note strictly E, T, V (and r, v, p) are all
  defined at a particular time (t) E(t) etc..

T - depends on v
V - depends on r
V depends on the system e.g. positional,
electrostatic PE
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• Consider a 1-dimensional system (straight line
  translational motion of a particle under the
  influence of a potential acting parallel to the
  direction of motion)
• Define position r x
• velocity v dx/dt
• momentum p mv m(dx/dt)
• PE V
• force F ?(dV/dx)
• Newtons 2nd Law of Motion
• F ma m(dv/dt) m(d2x/dt2)
• Therefore, if we know the forces acting on a
  particle we can solve a differential equation to
  determine its trajectory x(t),p(t).

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1.3 Example The 1-Dimensional Harmonic
Oscillator
NB assuming no friction or other forces act on
the particle (except F).

• The particle experiences a restoring force (F)
  proportional to its displacement (x) from its
  equilibrium position (x0).
• Hookes Law F ?kx
• k is the stiffness of the spring (or stretching
  force constant of the bond if considering
  molecular vibrations)
• Substituting F into Newtons 2nd Law we get
• m(d2x/dt2) ?kx a (second order)
  differential equation

k
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• Solution
• position x(t) Asin(?t)
• of particle
• frequency ? ?/2?
• (of oscillation)
• Note frequency depends only on characteristics
  of the system (m,k) not the amplitude (A)!

x
time period ? 1/ ?
?
A
t
?A
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• Assuming that the potential energy V 0 at x
  0, it can be shown that the total energy of the
  harmonic oscillator is given by
• E ½kA2
• As the amplitude (A) can take any value, this
  means that the energy (E) can also take any value
  i.e. energy is continuous.
• At any time (t), the position x(t) and velocity
  v(t) can be determined exactly i.e. the
  particle trajectory can be specified precisely.
• We shall see that these ideas of classical
  mechanics fail when we go to the atomic regime
  (where E and m are very small) then we need to
  consider Quantum Mechanics.
• CM also fails when velocity is very large (as v ?
  c), due to relativistic effects.

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1.4 Experimental Evidence for the Breakdown of
Classical Mechanics

• By the early 20th century, there were a number of
  experimental results and phenomena that could not
  be explained by classical mechanics.
• a) Black Body Radiation (Planck 1900)

l
/nm
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Plancks Quantum Theory

• Planck (1900) proposed that the light energy
  emitted by the black body is quantized in units
  of h? (? frequency of light).
• ?E nh? (n 1, 2, 3, )
• High frequency light only emitted if thermal
  energy kT ? h?.
• h? a quantum of energy.
• Plancks constant h 6.626?10?34 Js
• If h ? 0 we regain classical mechanics.
• Conclusions
• Energy is quantized (not continuous).
• Energy can only change by well defined amounts.

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b) Heat Capacities (Einstein, Debye 1905-06)

• Heat capacity relates rise in energy of a
  material with its rise in temperature
• CV (dU/dT)V
• Classical physics ? CV,m 3R (for all T).
• Experiment ? CV,m lt 3R (CV? as T?).
• At low T, heat capacity of solids determined by
• vibrations of solid.
• Einstein and Debye adopted Plancks hypothesis.
• Conclusion vibrational energy in solids is
  quantized
• vibrational frequencies of solids can
• only have certain values (?)
• vibrational energy can only change
• by integer multiples of h?.

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c) Photoelectric Effect (Einstein 1905)

• Ideas of Planck applied to electromagnetic
  radiation.
• No electrons are ejected (regardless of light
  intensity) unless n exceeds a threshold value
  characteristic of the metal.
• Ek independent of light intensity but linearly
  dependent on n.
• Even if light intensity is low, electrons are
  ejected if n is above the threshold. (Number of
  electrons ejected increases with light
  intensity).
• Conclusion Light consists of discrete packets
  (quanta) of energy photons (Lewis, 1922).

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d) Atomic and Molecular Spectroscopy

• It was found that atoms and molecules absorb and
  emit light only at specific discrete
  frequencies ?? spectral lines (not
  continuously!).
• e.g. Hydrogen atom emission spectrum (Balmer
  1885)
• Empirical fit to spectral lines (Rydberg-Ritz)
  n1, n2 (gt n1) integers.
• Rydberg constant RH 109,737.3 cm-1 (but can
  also be expressed in energy or frequency units).

n1 1 ? Lyman n1 2 ? Balmer n1 3 ?
Paschen n1 4 ? Brackett n1 5 ? Pfund
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Revision Electromagnetic Radiation
A Amplitude l wavelength n -
frequency c n x l or n c / l
wavenumber n n / c 1 / l c (velocity of
light in vacuum) 2.9979 x 108 m s-1.
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1.5 The Bohr Model of the Atom

• The H-atom emission spectrum was rationalized by
  Bohr (1913)
• Energies of H atom are restricted to certain
  discrete values
• (i.e. electron is restricted to well-defined
  circular orbits, labelled by quantum number n).
• Energy (light) absorbed in discrete amounts
  (quanta photons), corresponding to differences
  between these restricted values
• ?E E2 ? E1 h?

• Conclusion Spectroscopy provides direct
  evidence for quantization of energies
  (electronic, vibrational, rotational etc.) of
  atoms and molecules.

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• Limitations of Bohr Model Rydberg-Ritz Equation
  
• The model only works for hydrogen (and other one
  electron ions) ignores e-e repulsion.
• Does not explain fine structure of spectral
  lines.
• Note The Bohr model (assuming circular electron
  orbits) is fundamentally incorrect.

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2. Wave-Particle Duality

• Remember Classically, particles and waves are
  distinct
• Particles characterised by position, mass,
  velocity.
• Waves characterised by wavelength, frequency.
• By the 1920s, however, it was becoming apparent
  that sometimes matter (classically particles) can
  behave like waves and radiation (classically
  waves) can behave like particles.

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• 2.1 Waves Behaving as Particles
• The Photoelectric Effect
• Electromagnetic radiation of frequency ? can be
  thought of as being made up of particles
  (photons), each with energy E h ?.
• This is the basis of Photoelectron Spectroscopy
  (PES).
• Spectroscopy
• Discrete spectral lines of atoms and molecules
  correspond to the absorption or emission of a
  photon of energy h ?, causing the atom/molecule
  to change between energy levels ?E h ?.
• Many different types of spectroscopy are
  possible.

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• The Compton Effect (1923)
• Experiment A monochromatic beam of X-rays (?i)
  incident on a graphite block.
• Observation Some of the X-rays passing through
  the block are found to have longer wavelengths
  (?s).

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• Explanation The scattered X-rays undergo elastic
  collisions with electrons in the graphite.
• Momentum (and energy) transferred from X-rays to
  electrons.
• Conclusion Light (electromagnetic radiation)
  possesses momentum.
• Momentum of photon p h/?
• Energy of photon E h? hc/ ?
• Applying the laws of conservation
• of energy and momentum we get

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• 2.2 Particles Behaving as Waves
• Electron Diffraction (Davisson and Germer, 1925)

Davisson and Germer showed that a beam of
electrons could be diffracted from the surface of
a nickel crystal. Diffraction is a wave property
arises due to interference between scattered
waves. This forms the basis of electron
diffraction an analytical technique for
determining the structures of molecules, solids
and surfaces (e.g. LEED).
NB Other particles (e.g. neutrons, protons,
He atoms) can also be diffracted by crystals.
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2.3 The De Broglie Relationship (1924)

• In 1924 (i.e. one year before Davisson and
  Germers experiment), De Broglie predicted that
  all matter has wave-like properties.
• A particle, of mass m, travelling at velocity v,
  has linear momentum p mv.
• By analogy with photons, the associated
  wavelength of the particle (?) is given by

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3. Wavefunctions

• A particle trajectory is a classical concept.
• In Quantum Mechanics, a particle (e.g. an
  electron) does not follow a definite trajectory
  r(t),p(t), but rather it is best described as
  being distributed through space like a wave.
• 3.1 Definitions
• Wavefunction (?) a wave representing the
  spatial distribution of a particle.
• e.g. electrons in an atom are described by a
  wavefunction centred on the nucleus.
• ? is a function of the coordinates defining the
  position of the classical particle
• 1-D ?(x)
• 3-D ?(x,y,z) ?(r) ?(r,?,?) (e.g. atoms)
• ? may be time dependent e.g. ?(x,y,z,t)

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• The Importance of ?
• ? completely defines the system (e.g. electron in
  an atom or molecule).
• If ? is known, we can determine any observable
  property (e.g. energy, vibrational frequencies,
  ) of the system.
• QM provides the tools to determine ?
  computationally, to interpret ? and to use ? to
  determine properties of the system.

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• 3.2 Interpretation of the Wavefunction
• In QM, a particle is distributed in space like
  a wave.
• We cannot define a position for the particle.
• Instead we define a probability of finding the
  particle at any point in space.
• The Born Interpretation (1926)
• The square of the wavefunction at any point in
  space is proportional to the probability of
  finding the particle
• at that point.
• Note the wavefunction (?) itself has no physical
  meaning.

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• 1-D System
• If the wavefunction at point x is ?(x), the
  probability of finding the particle in the
  infinitesimally small region (dx) between x and
  xdx is
• P(x) ? ??(x)?2 dx
• ??(x)? the magnitude of ? at point x.
• Why write ???2 instead of ?2 ?
• Because ? may be imaginary or complex ? ?2 would
  be negative or complex.
• BUT probability must be real and positive (0 ? P
  ? 1).
• For the general case, where ? is complex (? a
  ib) then
• ???2 ?? where ? is the complex conjugate
  of ?.
• (? a ib) (NB )

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• 3-D System
• If the wavefunction at r (x,y,z) is ?(r), the
  probability of finding the particle in the
  infinitesimal volume element d? ( dxdydz) is
• P(r) ? ??(r)?2 d?
• If ?(r) is the wavefunction describing
• the spatial distribution of an electron
• in an atom or molecule, then
• ??(r)?2 ?(r) the electron density at point r

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• 3.3 Normalization of the Wavefunction
• As mentioned above, probability P(r) ? ??(r)?2
  d?
• What is the proportionality constant?
• If ? is such that the sum of ??(r)?2 at all
  points in space 1, then
• P(x) ??(x)?2 dx 1-D
• P(r) ??(r)?2 d? 3-D
• As summing over an infinite number of
  infinitesimal steps integration, this means
• i.e. the probability that the particle is
  somewhere in space 1.
• In this case, ? is said to be a normalized
  wavefunction.

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• How to Normalize the Wavefunction
• If ? is not normalized, then
• A corresponding normalized wavefunction (?Norm)
  can be defined
• such that
• The factor (1/?A) is known as the normalization
  constant (sometimes represented by N).

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• 3.4 Quantization of the Wavefunction
• The Born interpretation of ? places restrictions
  on the form of the wavefunction
• (a) ? must be continuous (no breaks)
• (b) The gradient of ? (d?/dx) must be continuous
  (no kinks)
• (c) ? must have a single value at any point in
  space
• (d) ? must be finite everywhere
• (e) ? cannot be zero everywhere.

• Other restrictions (boundary conditions) depend
  on the exact system.
• These restrictions on ? mean that only certain
  wavefunctions and ? only
• certain energies of the system are allowed.
• ? Quantization of ? ? Quantization of E

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• 3.5 Heisenbergs Uncertainty Principle
• It is impossible to specify simultaneously, with
  precision, both the momentum
• and the position of a particle
• (if it is described by Quantum Mechanics)
• Heisenberg (1927)
• Dpx.Dx ? h / 4p (or ?/2, where ? h/2p).
• Dx uncertainty in position
• Dpx uncertainty in momentum (in the
  x-direction)
• If we know the position (x) exactly, we know
  nothing about momentum (px).
• If we know the momentum (px) exactly, we know
  nothing about position (x).
• i.e. there is no concept of a particle trajectory
  x(t),px(t) in QM (which applies to small
  particles).

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• How to Understand the Uncertainty Principle
• To localize a wavefunction (?) in space (i.e. to
  specify the particles position accurately, small
  Dx) many waves of different wavelengths (?) must
  be superimposed ? large Dpx (p h/?).
• The Uncertainty Principle imposes a fundamental
  (not experimental) limitation on how precisely we
  can know (or determine) various observables.

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• Note to determine a particles position
  accurately requires use of short radiation (high
  momentum) radiation. Photons colliding with the
  particle causes a change of momentum (Compton
  effect) ? uncertainty in p.
• ? The observer perturbs the system.
• Zero-Point Energy (vibrational energy Evib ? 0,
  even at T 0 K) is also a consequence of the
  Uncertainty Principle
• If vibration ceases at T 0, then position and
  momentum both 0 (violating the UP).
• Note There is no restriction on the precision in
  simultaneously knowing/measuring the position
  along a given direction (x) and the momentum
  along another, perpendicular direction (z)
• Dpz.Dx 0

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• Similar uncertainty relationships apply to other
  pairs of observables.
• e.g. the energy (E) and lifetime (?) of a state
• DE.D? ? ?
• (a) This leads to lifetime broadening of
  spectral lines
• Short-lived excited states (? well defined, small
  D?) possess large uncertainty in the energy
  (large DE) of the state.
• Broad peaks in the spectrum.
• Shorter laser pulses (e.g. femtosecond,
  attosecond lasers) have broader energy (therefore
  wavelength) band widths.
• (1 fs 10?15 s, 1 as 10?18 s)

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4. Wave Mechanics

• Recall the wavefunction (?) contains all the
  information we need to know about any particular
  system.
• How do we determine ? and use it to deduce
  properties of the system?
• 4.1 Operators and Observables
• If ? is the wavefunction representing a system,
  we can write
• where Q observable property of system (e.g.
  energy, momentum, dipole moment )
• operator corresponding to observable Q.

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• This is an eigenvalue equation and can be
  rewritten as
• (Note ? cant be cancelled).
• Examples d/dx (eax) a eax
• d2/dx2 (sin ax) ?a2 sin ax

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• To find ? and calculate the properties
  (observables) of a system
• 1. Construct relevant operator
• 2. Set up equation
• 3. Solve equation ? allowed values of ? and Q.
• Quantum Mechanical Position and Momentum
  Operators
• 1. Operator for position in the x-direction is
  just multiplication by x
• 2. Operator for linear momentum in the
  x-direction
• ?
• (solve first order differential equation ? ? ,
  px).

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• Constructing Kinetic and Potential Energy QM
  Operators
• 1. Write down classical expression in terms of
  position and momentum.
• 2. Introduce QM operators for position and
  momentum.
• Examples
• 1. Kinetic Energy Operator in 1-D
• CM ? QM
• 2. KE Operator in 3-D
• CM QM
• 3. Potential Energy Operator (a function
  of position)
• ? PE operator corresponds to multiplication by
  V(x), V(x,y,z) etc.

del-squared
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• 4.2 The Schrödinger Equation (1926)
• The central equation in Quantum Mechanics.
• Observable total energy of system.
• Schrödinger Equation
  Hamiltonian Operator
• E Total Energy
• where and E T V.
• SE can be set up for any physical system.
• The form of depends on the system.
• Solve SE ? ? and corresponding E.

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• Examples
• 1. Particle Moving in 1-D ?(x)
• The form of V(x) depends on the physical
  situation
• Free particle V(x) 0 for all x.
• Harmonic oscillator V(x) ½kx2
• 2. Particle Moving in 3-D ?(x,y,z)
• SE ?
• or

Note The SE is a second order differential
equation
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• 4.3 Particle in a I-D Box
• System
• Particle of mass m in 1-D box of length L.
• Position x 0?L.
• Particle cannot escape from box as PE V(x) ? for
  x 0, L (walls).
• PE inside box V(x) 0 for 0lt x lt L.
• 1-D Schrödinger Eqn.
• (V 0 inside box).

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• This is a second order differential equation
  with general solutions of the form
• ? A sin kx B cos kx
• SE ?
• ? (i.e. E depends on k).

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• Restrictions on ?
• In principle Schrödinger Eqn. has an infinite
  number of solutions.
• So far we have general solutions
• any value of A, B, k ? any value of ?,E.
• BUT due to the Born interpretation of ?, only
  certain values of ? are physically acceptable
• outside box (xlt0, xgtL) V ? ? impossible for
  particle to be outside the box
• ? ???2 0 ? ? 0 outside box.
• ? must be a continuous function
• ? Boundary Conditions ? 0 at x 0
• ? 0 at x L.

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• Effect of Boundary Conditions
• x 0 ? A sin kx B cos kx B
• ? 0 ? B 0
• ? ? A sin kx for all x
• x L ? A sin kL 0
• sin kL 0 ? kL n? n 1, 2, 3,
• (n ? 0, or ? 0 for all x)

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• Allowed Wavefunctions and Energies
• k is restricted to a discrete set of values k
  n?/L
• Allowed wavefunctions ?n A sin(n?x/L)
• Normalization A ?(2/L) ?
• Allowed energies
• ?

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• Quantum Numbers
• There is a discrete energy state (En),
  corresponding to a discrete wavefunction (?n),
  for each integer value of n.
• Quantization occurs due to boundary conditions
  and requirement for ? to be physically reasonable
  (Born interpretation).
• n is a Quantum Number labels each allowed state
  (?n) of the system and determines its energy
  (En).
• Knowing n, we can calculate ?n and En.

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• Properties of the Wavefunction
• Wavefunctions are standing waves
• The first 5 normalized wavefunctions for the
  particle in the 1-D box
• Successive functions possess one more half-wave
  (? they have a shorter wavelength).
• Nodes in the wavefunction points at which ?n
  0 (excluding the ends which are constrained to be
  zero).
• Number of nodes (n-1) ?1 ? 0 ?2 ? 1 ?3 ? 2
  

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• Curvature of the Wavefunction
• If y f(x) dy/dx gradient of y (with respect
  to x).
• d2y/dx2 curvature of y.
• In QM Kinetic Energy ? curvature of ?
• Higher curvature ? (shorter ?) ? higher KE
• For the particle in the 1-D box (V0)

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• Energies
• En ? n2/L2 ? En? as n? (more nodes in ?n)
• En? as L? (shorter box)
• n? (or L?) ? curvature of ?n?
• ? KE? ? En?

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• En ? n2 ? energy levels get further apart as n?
• Zero-Point Energy (ZPE) lowest energy of
  particle in box
• CM Emin 0
• QM E 0 corresponds to ? 0 everywhere
  (forbidden).

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• If V(x) V ? 0, everywhere in box, all energies
  are shifted by V.

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• Density Distribution of the Particle in the 1-D
  Box
• The probability of finding the particle
• between x and xdx (in the state
• represented by ?n) is
• Pn(x) ??n(x)?2 dx (?n(x))2 dx (?n is real)
• ?
• Note probability is not uniform
• ?n2 0 at walls (x 0, L) for all ?n.
• ?n2 0 at nodes (where ?n 0).

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• 4.4 Further Examples
• (a) Particle in a 2-D Square or 3-D Cubic Box
• Similar to 1-D case, but ? ? ?(x,y) or ?(x,y,z).
• Solutions are now defined by 2 or 3 quantum
  numbers
• e.g. ?n,m, En,m ?n,m,l, En,m,l.
• Wavefunctions can be represented as contour plots
  in 2-D
• (b) Harmonic Oscillator
• Similar to particle in 1-D box, but PE V(x)
  ½kx2
• (c) Electron in an Atom or Molecule
• 3-D KE operator
• PE due to electrostatic interactions between
  electron and all other electrons and nuclei.