1. 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. 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. 3
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
4. 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
• Einstein’s formula was later modified by Debye
– Debye’s formula closely describes actual heat capacities
5. 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. 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. 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 ho =
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. 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. Chapter 11 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. Chapter 11 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. 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 2
dx
2
is the probability density
– is called the probability amplitude
12. Chapter 11 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 2
d
– d = dx dy dz
– d is an infinitesimal volume element
• P 2
d
– P is the probability
13. Chapter 11 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. 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. 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. 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. 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 = k2
h2
/2m
• h = h/2
18. Chapter 11 18
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
cos2
kx
– The probability density periodically varies between 0 and 4|A|2
– Locations where ||2
= 0 corresponds to nodes – nodal points
19. 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. 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. 21
Quantum Mechanical Rules
• The following rules apply for a wavefunction, , that can be
written as a linear combination of eigenfunctions of an operator
= c11 + c22 + …….. = ckk
– 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
22. Chapter 11 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. 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. 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. 25
The Uncertainty Principle -A Quantitative Version
pq 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 12 21
– The two operators do not commute (the effect of the two operators
depends on their order)
