Topic I: Quantum theory Chapter 7 Introduction to Quantum Theory

Classical physics Before 1900 A particle travels in a trajectory with a precise position and momentum at each instant. Any type of motion can be excited to a state of arbitrary energy. Energy is continuous. Wave and particle are two distinct concepts.

Failures of classical physics Crucial experimental observations: 1. Black-body radiation (Planck in 1900) 2. Heat capacities of solids (Einstein 1905) 3. Photoelectric effect (Einstein in 1905) 4. Diffraction of electrons (Davisson and Germer in 1925)

Black-body radiation The radiation from the pinhole in a close contained is characteristic of the EM waves within the container. Wien’s displacement law  max : Wavelength at the maximum of the distribution Stefan-Boltzmann law M: Total emitted power

Classical theory of EM waves for the black-body radiation Rayleigh – Jeans law  ( T): Density of states The Rayleigh – Jeans law is quite successful at long wavelengths (low frequencies). It fails badly at short wavelengths (high frequencies). In classical theory, the energies of the EM waves are continuous.

Planck’s postulation (in 1900) E : The energy of the EM wave  : The frequency of the EM wave h (Planck’s const) = * J·s Planck’s distribution The energy of the EM wave is limited to discrete values and can not varied arbitrary. Quantization of Energy !

Heat capacities of solids Classical physics predicts the molar heat capacities of solids close to 3R = 25 J K -1 mol -1. This is called Dulong-Petit law. Experimental observation

Einstein’s correction Einstein uses Planck’s hypothesis for quantization of energy. The permitted oscillation energy of an atom is an integral multiple of h, with being the frequency of the oscillation.   = h  /k B : Einstein temperature

Evidences for quantization of energy: Atomic and molecular spectra Emission spectrum of iron atoms Emission spectrum of H atoms Emission spectrum of SO 2 molecules

Discrete energy levels of atoms and molecules The features of these spectra are a series of discrete frequencies. The energy of an atom or a molecule is confined to discrete values, called the allowed energy states or levels. The atom or molecule only jumps between the discrete energy levels. As an atom jumps from one energy level E i to another level E j, the frequency of the radiation is related to the energy difference  E of the two levels. Bohr frequency condition

Quantization of Energy Planck’s distribution agrees well with the experiment results and accounts for Wien’s law and Stefan-Boltzmann law. Energy of a EM wave is quantized and should be discrete, but not continuous. What is the physical reason for the success of Planck’s quantization hypothesis for energy?

Two hypotheses for interpretation of light Wave hypothesis: Light is an electromagnetic wave. Quantity: frequency  and wave length Supporting experiments: Diffraction, interference and EM waves Particle hypothesis: Light is a stream of particles, called photons. Quantity: energy E and momentum p Supporting experiments: Reflection, refraction and photoelectric effect

The photoelectric effect Kinetic energy of photoelectrons

Work function of a metal  Work function of a metal Definition: Work function is the energy required to remove an electron from the metal to infinity. Work function is a characteristic of the metal. No electrons are ejected, regardless of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the metal. The kinetic energy of the ejected electrons varies linearly with the frequency of the incident radiation but is independent of its intensity. Even at low intensities of light, electrons are ejected immediately if the frequency is above the threshold value. These three conclusions from experiments can not be understood by the theory of electromagnetic waves.

Einstein’s interpretation Light is a stream of particles, called photons, rather than EM waves. The energy of each photon is h. As h < , no electrons are ejected out. So, the threshold frequency c is  /h, which is decided by the metal. As h > , the kinetic energy of the ejected electron increases linearly with the frequency of the radiation. Interpretation for the photoelectric effect: right after an electron collides with a photon with sufficient energy, the electron is ejected out from the metal. After the collision, the electron is ejected out immediately. The particle character of electromagnetic radiation

Diffraction of electrons Diffraction: Interference between waves caused by an object in their path Diffraction is a typical characteristic of wave Electrons are particles. In 1925, Davisson and Germer first observed the diffraction of electrons by a crystal. Conclusion: electrons have wave-like properties. The Davisson-Germer experiment has been repeated with other particles, including  particles and molecular H. The wave character of particles

Wave - particle duality: de Broglie’s hypothesis Proposed by de Broglie in 1924 “Wave” have particle-like properties 1. Black-body radiation 2. Photoelectric effect “Particles” have wave-like properties Diffraction of electrons A particle traveling with a momentum p should have a wavelength de Broglie’s relation between p and Momentum p: quantity of particle Wavelength : quantity of wave

Classical physics failed to account for the existence of discrete energies of atoms and other experiments in the early 20th century. Such total failures show that the basic concept of classical mechanics need to be corrected fundamentally. A new mechanics had to be developed to take its place. The new mechanism is called quantum mechanics, and opens a new era of physics. Microscopic dynamics: Quantum mechanics

In 1926, Schrodinger’s interpretation for matter wave Rather than traveling along a path, a particle is spread through space like a wave described by a wave function.  x, t  Wave function of the matter wave  x, t  is a complex-variable function of x and t Wave mechanics: Schrodinger’s equation

What’s the equation for the wave function of a particle moving in a potential? In quantum mechanicsIn classical mechanics x, p and H are scalar variables. Time-dependent Schrodinger equation Time-independent Schrodinger equation

Born interpretation of the wave function  x, t  Probability amplitude |  x, t  | 2 : Probability density to find the particle at x and at time t Since the total probability of finding a particle is one, the integration of the probability density over all space should be one. This is the normalization of a wave function. Normalization of a wave function

Properties of wave functions The curvature of a wave function is related to the kinetic energy of a particle: A higher curvature of a wave function implies the particle has a higher kinetic energy at that region. (a) The wave function should be single value everywhere. (b) The wave function can not be infinite over a finite region of space. (c) The wave function and it’s slope should be continuous everywhere. The wave function must satisfy certain boundary conditions. Only certain values of energy E give the acceptable solutions of Schrodinger’s equation subject to the boundary conditions. This leads to the quantization in energy of the system.

A particle freely moving in an infinite square-well potential Schrodinger equation General solutions Subject to boundary conditions Acceptable normalized wave functions n: quantum number

The uncertainty principle In 1927, Heisenberg proposed that It is impossible to specify simultaneously, with arbitrary precision, both momentum and position of a particle. I. If the momentum of the particle is known exactly For a particle with a definite linear momentum p in free space, the wave function  x  Asin(kx), with k=2  p/h, spreads out through the whole space so that the position of the particle is completely undetermined. II. If the position of the particle is known exactly For a particle with a precise location, the wave function  (x) is sharply localized and is only constructed from the superposition of many functions of Asin(kx) with different k. So, the momentum of the particle is completely undetermined.

A particle with well-defined positionA particle with ill-defined position Particle with position quite uncertain but momentum quite certain Particle with position quite certain but momentum quite uncertain

Exercises 7A.2, 7A.3, 7A.9 7B.4, 7B.7 7C.3, 7C.7, 7C.14