The Failures of Classical Physics

1
The Failures of Classical Physics

• Observations of the following phenomena indicate
  that systems can take up energy only in discrete
  amounts (quantization of energy)
• Black-body radiation
• Heat capacities of solids
• Atomic spectra

2
Black-body Radiation

• Hot objects emit electromagnetic radiation
• An ideal emitter is called a black-body
• The energy distribution plotted versus the
  wavelength exhibits a maximum.
• The peak of the energy of emission shifts to
  shorter wavelengths as the temperature is
  increased
• The maximum in energy for the black-body spectrum
  is not explained by classical physics
• The energy density is predicted to be
  proportional to ?-4 according to the
  Rayleigh-Jeans law
• The energy density should increase without bound
  as ??0

3
Black-body Radiation Plancks Explanation of
the Energy Distribution

• Planck proposed that the energy of each
  electromagnetic oscillator is limited to discrete
  values and cannot be varied arbitrarily
• According to Planck, the quantization of cavity
  modes is given by Enh? (n 0,1,2,)
• h is the Planck constant
• ? is the frequency of the oscillator
• Based on this assumption, Planck derived an
  equation, the Planck distribution, which fits the
  experimental curve at all wavelengths
• Oscillators are excited only if they can acquire
  an energy of at least h? according to Plancks
  hypothesis
• High frequency oscillators can not be excited
  the energy is too large for the walls to supply

4
Heat Capacities of Solids

• Based on experimental data, Dulong and Petit
  proposed that molar heat capacities of
  mono-atomic solids are 25 J/K mol
• This value agrees with the molar constant-volume
  heat capacity value predicted from classical
  physics ( cv,m 3R)
• Heat capacities of all metals are lower than 3R
  at low temperatures
• The values approach 0 as T? 0
• By using the same quantization assumption as
  Planck, Einstein derived an equation that follows
  the trends seen in the experiments
• Einsteins formula was later modified by Debye
• Debyes formula closely describes actual heat
  capacities

5
Atomic Spectra

• Atomic spectra consists of series of narrow lines
• This observation can be understood if the energy
  of the atoms is confined to discrete values
• Energy can be emitted or absorbed only in
  discrete amounts
• A line of a certain frequency (and wavelength)
  appears for each transition

6
Wave-Particle Duality

• Particle-like behavior of waves is shown by
• Quantization of energy (energy packets called
  photons)
• The photoelectric effect
• Wave-like behavior of waves is shown by electron
  diffraction

7
The Photoelectric Effect

• Electrons are ejected from a metal surface by
  absorption of a photon
• Electron ejection depends on frequency not on
  intensity
• The threshold frequency corresponds to h?o ?
• ? is the work function (essentially equal to the
  ionization potential of the metal)
• The kinetic energy of the ejected particle is
  given by
• ½mv2 h? - ?
• The photoelectric effect shows that the incident
  radiation is composed of photons that have energy
  proportional to the frequency of the radiation

8
Diffraction of electrons

• Electrons can be diffracted by a crystal
• A nickel crystal was used in the Davisson-Germer
  experiment
• The diffraction experiment shows that electrons
  have wave-like properties as well as particle
  properties
• We can assign a wavelength, ?, to the electron
• ? h/p (the de Broglie relation)
• A particle with a high linear momentum has a
  short wavelength
• Macroscopic bodies have such high momenta (even
  et low speed) that their wavelengths are
  undetectably small

9
The Schrödinger Equation

• Schrödinger proposed an equation for finding the
  wavefunction of any system
• The time-independent Schrödinger equation for a
  particle of mass m moving in one dimension (along
  the x-axis)
• (-h2/2m) d2?/dx2 V(x)? E?
• V(x) is the potential energy of the particle at
  the point x
• h h/2?
• E is the the energy of the particle

10
The Schrödinger Equation

• The Schrödinger equation for a particle moving in
  three dimensions can be written
• (-h2/2m) ?2? V? E?
• ?2 ?2/?x2 ?2/?y2 ?2/?z2
• The Schrödinger equation is often written
• H? E?
• H is the hamiltonian operator
• H -h2/2m ?2 V

11
The Born Interpretation of the Wavefunction

• Max Born suggested that the square of the
  wavefunction, ?2, at a given point is
  proportional to the probability of finding the
  particle at that point
• ?? is used rather than ?2 if ? is complex
• ? ? conjugate
• In one dimension, if the wavefunction of a
  particle is ? at some point x, the probability of
  finding the particle between x and (x
  dx) is proportional to ?2dx
• ?2 is the probability density
• ? is called the probability amplitude

12
The Born Interpretation, Continued

• For a particle free to move in three dimensions,
  if the wavefunction of the particle has the value
  ? at some point r, the probability of finding the
  particle in a volume element, d?, is proportional
  to ?2d?
• d? dx dy dz
• d? is an infinitesimal volume element
• P ? ?2 d?
• P is the probability

13
Normalization of Wavefunction

• If ? is a solution to the Schrödinger equation,
  so is N?
• N is a constant
• ? appears in each term in the equation
• We can find a normalization constant, so that the
  probability of finding the particle becomes an
  equality
• P ? (N?)(N?)dx
• For a particle moving in one dimension
• ? (N?)(N?)dx 1
• Integrated from x -? to x?
• The probability of finding the particle somewhere
  1
• By evaluating the integral, we can find the value
  of N (we can normalize the wavefunction)

14
Normalized Wavefunctions

• A wavefunction for a particle moving in one
  dimension is normalized if
• ? ?? dx 1
• Integrated over entire x-axis
• A wavefunction for a particle moving in three
  dimensions is normalized if
• ? ?? d? 1
• Integrated over all space

15
Spherical Polar Coordinates

• For systems with spherical symmetry, we often use
  spherical polar coordinates ( r, ?, and ? )
• x r sin? cos?
• y r sin? sin?
• z r cos?
• The volume element , d? r2 sin? dr d? d?
• To cover all space
• The radius r ranges from 0 to ?
• The colatitude, ?, ranges from 0 to ?
• The azimuth, ?, ranges from 0 to 2?

16
Quantization

• The Born interpretation puts restrictions on the
  acceptability of the wavefunction
• 1. ? must be finite
• ? ? ?
• 2. ? must be single-valued at each point
• 3. ? must be continuous
• 4. Its first derivative (its slope) must be
  continuous
• These requirements lead to severe restrictions on
  acceptable solutions to the Schrödinger equation
• A particle may possess only certain energies, for
  otherwise its wavefunction would be physically
  impossible
• The energy of the particle is quantized

17
Solutions to the Schrödinger equation

• The Schrödinger equation for a particle of mass m
  free to move along the x-axis with zero potential
  energy is
• (-h2/2m) d2?/dx2 E?
• V(x) 0
• h h/2?
• Solutions of the equation have the form
• ? A eikx B e-ikx
• A and B are constants
• E k2h2/2m
• h h/2?

18
The Probability Density

• ? A eikx B e-ikx
• 1. Assume B0
• ? A eikx
• ?2 ?? A2
• The probability density is constant (independent
  of x)
• Equal probability of finding the particle at each
  point along x-axis
• 2. Assume A0
• ?2 B2
• 3. Assume A B
• ?2 4A2 cos2kx
• The probability density periodically varies
  between 0 and 4A2
• Locations where ?2 0 corresponds to nodes
  nodal points

19
Eigenvalues and Eigenfunctions

• The Schrödinger equation is an eigenvalue
  equation
• An eigenvalue equation has the form
• (Operator)(function) (Constant factor) ? (same
  function)
• ?? ??
• ? is the eigenvalue of the operator ?
• the function ? is called an eigenfunction
• ? is different for each eigenvalue
• In the Schrödinger equation, the wavefunctions
  are the eigenfunctions of the hamiltonian
  operator, and the corresponding eigenvalues are
  the allowed energies

20
Superpositions and Expectation Values

• When the wave function of a particle is not an
  eigenfunction of an operator, the property to
  which the operator corresponds does not have a
  definite value
• For example, the wavefunction ? 2A coskx is not
  an eigenfunction of the linear momentum operator
• This wavefunction can be written as a linear
  combination of two wavefunctions with definite
  eigenvalues, kh and -kh
• ? 2A coskx A eikx A e-ikx
• h h/2?
• The particle will always have a linear momentum
  of magnitude kh (kh or kh)
• The same interpretation applies for any
  wavefunction written as a linear combination or
  superposition of wavefunctions

21
Quantum Mechanical Rules

• The following rules apply for a wavefunction, ?,
  that can be written as a linear combination of
  eigenfunctions of an operator
• ? c1?1 c2?2 .. ? ck?k
• c1 , c2 , . are numerical coefficients
• ?1 , ?2 , . are eigenfunctions with different
  eigenvalues
• 1. When the momentum (or other observable) is
  measured in a single observation, one of the
  eigenvalues corresponding to the ?k that
  contribute to the superposition will be found
• 2. The probability of measuring a particular
  eigenvalue in a series of observations is
  proportional to the square modulus, ck2, of the
  corresponding coefficient in the linear
  combination

22
Quantum Mechanical Rules

• 3. The average value of a large number of
  observations is given by the expectation value,
  ???, of the operator ? corresponding to the
  observable of interest
• The expectation value of an operator ? is defined
  as
• ??? ? ??? d?
• the formula is valid for normalized wavefunctions

23
Orthogonal Wavefunctions

• Wave functions ?i and ?j are orthogonal if
• ? ?i?j d? 0
• Eigenfunctions corresponding to different
  eigenvalues of the same operator are orthogonal

24
The Uncertainty Principle

• It is impossible to specify simultaneously with
  arbitrary precision both the momentum and
  position of a particle (The Heisenberg
  Uncertainty Principle)
• If the momentum is specified precisely, then it
  is impossible to predict the location of the
  particle
• By superimposing a large number of wavefunctions
  it is possible to accurately know the position of
  the particle (the resulting wave function has a
  sharp, narrow spike)
• Each wavefunction has its own linear momentum.
• Information about the linear momentum is lost

25
The Uncertainty Principle -A Quantitative Version

• ?p?q ? ½h
• ?p uncertainty in linear momentum
• ?q uncertainty in position
• h h/2?
• Heisenbergs Uncertainty Principle applies to
  any pair of complementary observables
• Two observables are complementary if ?1?2 ? ?2?1
• The two operators do not commute (the effect of
  the two operators depends on their order)