# The Failures of Classical Physics

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```					               The Failures of Classical Physics

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

Chapter 11                           1

• 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

Chapter 11                                   2
Black-body Radiation – Planck’s 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:
E=nh (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 Planck’s hypothesis
– High frequency oscillators can not be excited – the energy is too large
for the walls to supply

Chapter 11                                   3
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
• Einstein’s formula was later modified by Debye
– Debye’s formula closely describes actual heat capacities

Chapter 11                         4
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

Chapter 11                            5
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

Chapter 11                      6
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

Chapter 11                                    7
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

Chapter 11                          8
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

Chapter 11                       9
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

Chapter 11                10
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

Chapter 11                           11
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

Chapter 11                           12
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)

Chapter 11                            13
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

Chapter 11                    14
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

Chapter 11                      15
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

Chapter 11                             16
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

Chapter 11                           17
The Probability Density

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

Chapter 11                                 18
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

Chapter 11                    19
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

Chapter 11                        20
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, |ck|2, of the
corresponding coefficient in the linear combination

Chapter 11                          21
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

Chapter 11                       22
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

Chapter 11                          23
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

Chapter 11                                24
The Uncertainty Principle -A Quantitative Version

• pq  ½h
– p = uncertainty in linear momentum
– q = uncertainty in position
– h = h/2
• `Heisenberg’s 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)

Chapter 11                               25

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