The Failures of Classical Physics

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

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

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

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

Atomic Spectra • Atomic spectraconsists 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

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

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

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

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

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

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

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

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

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

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

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

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 • Solutionsof the equation have the form: •  = A eikx + B e-ikx • A and B are constants • E = k2h2/2m • h = h/2

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|2cos2kx • The probability density periodically varies between 0 and 4|A|2 • Locations where ||2 = 0 corresponds to nodes – nodal points Chapter 11

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

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 interpretationapplies for any wavefunction written as a linear combination or superposition of wavefunctions

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

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

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

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

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)

The Failures of Classical Physics