The Failures of Classical Physics Observations of the

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 1

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 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 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= 3 R) • Heat capacities of all metals are lower than 3 R 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 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 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 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: • ½mv 2 = h - • The photoelectric effect shows that the incident radiation is composed of photons that have energy proportional to the frequency of the radiation 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 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): • (-h 2/2 m) d 2 /dx 2 + V(x) = E – V(x) is the potential energy of the particle at the point x – h = h/2 – E is the energy of the particle Chapter 11 9

The Schrödinger Equation • The Schrödinger equation for a particle moving in three dimensions can be written: • (-h 2/2 m) 2 + V = E – 2 = 2/ x 2 + 2/ y 2 + 2/ z 2 • The Schrödinger equation is often written: • H = E – H is the hamiltonian operator – H = -h 2/2 m 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 2 dx – 2 is the probability density – is called the probability amplitude 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 2 d – 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 14

Spherical Polar Coordinates • For systems with spherical symmetry, we often use spherical polar coordinates ( r, , and ) – x = r sin cos – y = r sin – z = r cos • The volume element , d = r 2 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 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 • • 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: • (-h 2/2 m) d 2 /dx 2 = 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 = k 2 h 2/2 m • h = h/2 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 cos 2 kx – 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 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 = 2 A 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 – = 2 A 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 20

Quantum Mechanical Rules • The following rules apply for a wavefunction, , that can be written as a linear combination of eigenfunctions of an operator • = c 1 1 + c 2 2 + ……. . = ck k – c 1 , c 2 , …. 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 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 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 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) 25