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 – 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

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

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 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

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 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

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

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 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

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

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

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 cos2kx

– The probability density periodically varies between 0 and 4|A|2

– 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

= 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

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

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

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)