CHAPTER 2: POSTULATES OF QUANTUM MECHANICS

Basics of Quantum Mechanics - Why Quantum Physics? - Classical mechanics (Newton's mechanics) and Maxwell's equations (electromagnetics theory) can explain MACROSCOPIC phenomena such as motion of billiard balls or rockets. Quantum mechanics is used to explain microscopic phenomena such as photon-atom scattering and flow of the electrons in a semiconductor. QUANTUM MECHANICS is a collection of postulates based on a huge number of experimental observations. The differences between the classical and quantum mechanics can be understood by examining both The classical point of view The quantum point of view

Basics of Quantum Mechanics - Classical Point of View - In Newtonian mechanics, the laws are written in terms of PARTICLE TRAJECTORIES. A PARTICLE is an indivisible mass point object that has a variety of properties that can be measured, which we call observables. The observables specify the state of the particle (position and momentum). A SYSTEM is a collection of particles, which interact among themselves via internal forces, and can also interact with the outside world via external forces. The STATE OF A SYSTEM is a collection of the states of the particles that comprise the system. All properties of a particle can be known to infinite precision. Conclusions: TRAJECTORY state descriptor of Newtonian physics, EVOLUTION OF THE STATE Use Newton's second law PRINCIPLE OF CAUSALITY Two identical systems with the same initial conditions, subject to the same measurement will yield the same result.

Basics of Quantum Mechanics - Quantum Point of View - Quantum particles can act as both particles and waves WAVE-PARTICLE DUALITY. Quantum state is a conglomeration of several possible outcomes of measurement of physical properties Quantum mechanics uses the language of PROBABILITY theory (random chance). An observer cannot observe a microscopic system without altering some of its properties. Neither one can predict how the state of the system will change. QUANTIZATION of energy is yet another property of "microscopic" particles.

Basics of Quantum Mechanics - Heisenberg Uncertainty Principle - One cannot unambiguously specify the values of particle's position and its momentum for a microscopic particle, i.e (H.U.P): x ( t 1 0 ) p ( 0) h x t 2 2 Position and momentum are, therefore, considered as incompatible variables. The Heisenberg Uncertainty Principle (H.U.P) strikes at the very heart of the classical physics => the particle trajectory.

Basics of Quantum Mechanics - The Correspondence Principle - When Quantum physics is applied to macroscopic systems, it must reduce to the classical physics. Therefore, the nonclassical phenomena, such as uncertainty and duality, must become undetectable. Niels Bohr codified this requirement into his Correspondence principle:

Basics of Quantum Mechanics - Particle-Wave Duality - The behavior of a "microscopic" particle is very different from that of a classical particle: in some experiments it resembles the behavior of a classical wave (not localized in space), in other experiments it behaves as a classical particle (localized in space). Corpuscular theories of light treat light as though it were composed of particles, but can not explain DIFRACTION and INTERFERENCE. Maxwell's theory of electromagnetic radiation can explain these two phenomena, which was the reason why the corpuscular theory of light was abandoned.

Basics of Quantum Mechanics - Particle-Wave Duality - Waves as particles: Max Plank work on black-body radiation, in which he assumed that the molecules of the cavity walls, described using a simple oscillator model, can only exchange energy in quantized units. 1905 Einstein proposed that the energy in an electromagnetic field is not spread out over a spherical wavefront, but instead is localized in individual clumbs - quanta. Each quantum of frequency n travels through space with speed of light, carrying a discrete amount of energy and momentum =photon => used to explain the photoelectric effect, later to be confirmed by the x-ray experiments of Compton. Particles as waves Double-slit experiment, in which instead of using a light source, one uses the electron gun. The electrons are diffracted by the slit and then interfere in the region between the diaphragm and the detector. Aharonov-Bohm effect

Planck s constant in action The classical picture began to crumble in 1900 when Max Planck published a theory of black-body radiation (theory of thermal radiation in equilibrium with a perfectly absorbing body). Planck provided an explanation of the observed properties of black-body radiation by assuming that atoms emit and absorb DISCRETE quanta of the radiation with energy E = hv, where v freq. of the radiation and h is the fundamental constant of nature with value: h = 6.626 x 10-34 J.s => Planck s constant

Recall: Photons PHOTONS - light particle. - particle-like quanta of EM radiation. - travel at speed of light c with momentum p and energy E, given by: p=h/ and E = hc/ For visible photon with wavelength = 663 nm; => p = 10-27 Js and E = 3 x 10-19 J # 1ev = 1.602 x 10-19 J ; x-ray photons ~ 10 kev. Evidence become compelling when A.H. Compton s experiment, later called Compton scattering.

Recall: Atoms It s well known that atoms can exist in states with DISCRETE of QUANTIZED energy. For example, the energy levels for H atom, consisting of an electron and a proton, where a good approx. the bound states have quantized energies given by E n = -13.6/n 2 ev, n the principle quantum number, n = 1,2,3, When the excitation energy > 13.6 ev, the atom is ionized (e is no longer bound to the proton) and its energy can, in principle, take on any value in the continuum between E = 0 and E =. The ground state, E g or 0> of H has n = 1, and the 1 st excitation state has n = 2 and so on.

Recall: Atoms Transition between hydrogen-atoms states with n i and n f leads to a spectral line (in EM spectra) with a wavelength given by hc/ = E ni E nf Atoms => Bohr atomic model => Bohr atomic radius, i.e., H atom, a 0 = 0.529 x 10-10 m. => Rydberg energy, i.e., H atom, E R = 13.6 ev. => Pauli Exclusion Principle => Heisenberg Uncertainty Principle (measurement) => Postulates of Quantum Mechanics (measurement)

POSTULATES OF QUANTUM MECHANICS Postulate: A fundamental element; a basic principle.

Postulates of Quantum Mechanics The foundation of Quantum Mechanics (QM) are the postulates, and the applications of QM are built on this foundation. Briefly, the postulates are: Postulate #1 (the wave function); State of a System Postulate #2 (Operators); Observables & Operators Postulate #3 (Measurements); Measurements & Eigenvalues of Operators Postulate #4 (Expectation values); Probabilistic outcome of measurements Postulate #5 (Time dependence) Time evolution of a system QM: Concepts & Applications, Nouredine Zettili.

The Postulates of Quantum Mechanics POSTULATES Definition #1 The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through its spatial location and time dependence, i.e. the wavefunction is in the most general sense dependent on time and space; = (x, t). #2 For every measurable property of the system in classical mechanics such as position, momentum, and energy, there exists a corresponding operator in quantum mechanics. An experiment in the lab to measure a value for such an observable is simulated in theory by operating on the wavefunction of the system with the corresponding operator. #3 For a single measurement of an observable corresponding to a quantum mechanical operator, only values that are eigenvalues of the operator will be measured. #4 The average, or expectation, value of an observable corresponding to a quantum mechanical operator (A) is given by: #5 The time-dependent Schrodinger equation governs the time evolution of a quantum mechanical system: x t Hˆ (, ) ( x, t) i t a *( x, t) Aˆ ( x, t) dx *( x, t) ( x, t) dx

Basics of Quantum Mechanics - What is Quantum Mechanics? - Quantum Mechanics is nothing more but linear algebra and Hilbert spaces. What makes quantum mechanics quantum mechanics is the physical interpretation of the results that are obtained.

Basics of Quantum Mechanics - First Postulate of Quantum Mechanics - The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through its spatial location and time dependence, i.e. the wavefunction is in the most general sense dependent on time and space; = (x, t) The state of a quantum mechanical system is completely specified by the wavefunction (x, t). First postulate of Quantum mechanics: Every physically-realizable state of the system is described in quantum mechanics by a state function that contains all accessible physical information about the system in that state. Physically realizable states states that can be studied in laboratory Accesible information the information we can extract from the wavefunction State function function of position, momentum, energy that is spatially localized.

Basics of Quantum Mechanics - First Postulate of Quantum Mechanics - If 1 and 2 represent two physically-realizable states of the system, then the linear combination c 11 c22 where c 1 and c 2 are arbitrary complex constants, represents a third physically realizable state of the system. Note: Wavefunction (x,t) position and time probability amplitude Quantum mechanics describes the outcome of an ensemble of measurements, where an ensemble of measurements consists of a very large number of identical experiments performed on identical non-interacting systems, all of which have been identically prepared so as to be in the same state.

First Postulate of Quantum Mechanics

First Postulate of Quantum Mechanics PdV ( x, y, z) 2 dv * ( x, y, z) ( x, y, z) dv 1 Limitations on the wavefunction: Only normalizable functions can represent a quantum state and these are called physically admissible functions. State function must be continuous and single valued function. State function must be a smoothly-varying function (continuous derivative).

First Postulate of Quantum Mechanics

Basics of Quantum Mechanics - Second Postulate of Quantum Mechanics - A The line segments AB and CD are orthogonal to each other. C B D

Second Postulate of Quantum Mechanics List of classical observables and QM operator

Basics of Quantum Mechanics - Third Postulate of Quantum Mechanics - Third Postulate:

Forth Postulate of Quantum Mechanics

Fifth Postulate of Quantum Mechanics

Postulates of Quantum Mechanics

Postulates of Quantum Mechanics EXAMPLE: Which among the following functions represent physically acceptable wave functions: 2 2 ) ( ; 5 ) ( ; 4 ) ( ; 3sin ) ( x x e x x h x x g x x f

Postulates of Quantum Mechanics

Postulates of Quantum Mechanics Orthonormal: Two vectors are orthonormal if: 1. Their dot product is zero. 2. The two vectors are unit vectors. j 0 90 i i j i j 0 1

Postulates of Quantum Mechanics EXAMPLE:

Postulates of Quantum Mechanics

Observable & Operators An observable is a dynamical variable that can be measured; the dynamical variables encountered most in classical mechanics are the position, linear momentum, angular momentum, and energy. How do we mathematically represent these and other variables in Quantum Mechanics?

Observable & Operators

Observable & Operators Formulate the operator representing the classical orbital angular momentum.

Measurement in Quantum Mechanics Quantum theory is about the results of measurement; it says nothing about what might happen in the physical world outside the context of measurement. So, the emphasis is on measurement.

Measurement in Quantum Mechanics How measurements disturb systems?

Measurement in Quantum Mechanics

Measurement in Quantum Mechanics Expectation Values

Measurement in Quantum Mechanics

Measurement in Quantum Mechanics EXAMPLE:

Measurement in Quantum Mechanics