Introduction to quantum mechanics

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1 Introduction to quantum mechanics Lecture 3 MTX9100 Nanomaterjalid OUTLINE -What is electron particle or wave? - How large is a potential well? -What happens at nanoscale?

2 What is inside? Matter Molecule Atom Nucleus Baryon (Hadron) Quark u 10-2 m 10-9 m m m m <10-19 m Condensed matter/nano-science/chemistry Atomic Physics Nuclear Physics protons, neutrons, mesons, etc. π,ω,λ... Electron (Lepton) <10-18 m top, bottom, charm, strange, up, down High Energy Physics

3 Quantum mechanics milestones the Bohr atomic model Each energy level or shell is represented by the principal quantum number n the ground energy for the hydrogen atom is 13.6 ev Electron energy states for a hydrogen atom

4 Quantum mechanics basics The simplified Bohr atomic model, in which electrons are assumed to revolve around the atomic nucleus in discrete orbitals, and the position of any particular electron is more or less well defined in terms of its orbital. The energies of electrons are quantized; electrons are permitted to have only specific values of energy. If an atom is in one of the excited states E1, E2, and so on, it does not remain in that state forever. Sooner or later it drops to a lower state and radiates energy in the form of light. The frequency of light υ that is liberated in a transition, for example, from energy E3 to energy E1, h is Planck s constant

5 Quantum mechanics milestones Between 1900 and 1925 Quantum Physics was developed by a number of physicists, including Planck, Einstein, Bohr and de Broglie. Werner Heisenberg and Erwin Schrödinger, founders of Quantum Mechanics -Wikipedia From 1925 onwards a more mathematical approach was developed by Schrödinger (wave mechanics), Heisenberg (matrix mechanics) and Dirac (who developed a more general formulation). If you are not confused by Quantum Physics then you haven't really understood it. N.Bohr

6 Quantum mechanics basics (2) Quantum mechanics is the study of mechanical systems whose dimensions are close to the atomic scale. Quantum mechanics is a fundamental branch of physics with wide applications. Quantum theory generalizes classical mechanics to provide accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits. The effects of quantum mechanics become evident at the atomic and subatomic level, and they are typically not observable on macroscopic scales. - Wikipedia

7 What is quantum mechanics good for? Atoms are governed by the laws of quantum mechanics, and quantum mechanics is essential for an understanding of atomic physics. The interactions between atoms are governed by quantum mechanics, and so an understanding of quantum mechanics is a prerequisite for understanding the science This information enables us to calculate the average value of the measurement of a physical variable. Quantum mechanics does not explain how a quantum particle behaves. Instead, it gives a recipe for determining the probability of the measurement of the value of a physical variable (e.g. energy, position or momentum).

8 Wave-particle duality In physics and chemistry, wave particle duality is the concept that all matter and energy exhibits both wave-like and particle-like properties. Particles are waves, waves are particles. Probability distribution of the Bohr atom Wikipedia In quantum mechanics, the motion of particles is described with probabilities. 8

9 Exclusion principle The exclusion principle says that two electrons cannot get into exactly the same energy state. In other words, it is not possible for two electrons to have the same momentum, be at the same location, and spin in the same direction.

10 How many quantum numbers? Using wave mechanics, every electron in an atom is characterized by four parameters called quantum numbers. 10 The Bohr model was a one-dimensional model that used one quantum number to describe the distribution of electrons in the atom. The only information that was important was the size of the orbit, which was described by the n quantum number. Schrödinger's model allowed the electron to occupy three-dimensional space. It therefore required three coordinates, or three quantum numbers, to describe the orbitals in which electrons can be found. The three coordinates that come from Schrödinger's wave equations are the principal (n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the size, shape, and orientation in space of the orbitals on an atom.

11 Quantum numbers The principal quantum number (n) describes the size (distance of an electron from the nucleus) of the orbital. Orbitals for which n = 2 are larger than those for which n = 1. The principal quantum number therefore indirectly describes the energy of an orbital. The angular quantum number (l) describes the shape of the orbital. Orbitals have shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l = 2). They can even take on more complex shapes as the value of the angular quantum number becomes larger. A third quantum number, known as the magnetic quantum number (m), describes the orientation in space of a particular orbital and shows the energy states in sub-shells. 11

12 Shells and sub-shells of orbitals Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same value of the angular quantum number. Bohr and (b) wavemechanical atom models in terms of electron distribution. 12

13 Sub-shells s, p, d and f signify the subshells which the electrons occupy. Different types of subshells have different numbers of energy states Within each energy state there are two possible spin orientations Schematic representation of the relative energies of the electrons for the various shells and subshells 13

14 14 Electron states

15 15 Electron configurations

16 Chemical Bonding Between Atoms Electron states is controlling factor for atomic bonding Types of primary (strong) bonds: ionic, covalent, metallic Types of secondary (weak) bonds: van der Waals, hydrogen Properties that are controlled by interatomic potentials: melting point, bond stiffness, thermal expansion coefficient 16

17 Bonding forces Many properties of materials are determined by the interatomic forces that bind the atoms together. Equilibrium spacing These forces are of two types, attractive and repulsive, and the magnitude of each is a function of the separation or interatomic distance. 17

18 Energy of bonding Bonding energy 18 Once in this position, the two atoms will counteract any attempt to separate them by an attractive force, or to push them together by a repulsive action.

19 This typical curve has a minimum at equilibrium distance R 0 R > R 0 ; the potential increases gradually, approaching 0 as R the force is attractive R < R 0 ; the potential increases very rapidly, approaching at small separation. the force is repulsive V(R) r 0 R 0 Repulsive R Attractive R Force between the atoms is the negative of the slope of this curve. At equlibrium, repulsive force becomes equals to the attractive part.

20 Heinserberg Uncertainty Principle This idea claims that we are not allowed to know simultaneously the definite location and the definite speed of a particle. The laws of motion for a quantum particle have to be framed in such a way that lets us make predictions only for quantities that are the average of many individual measurements. We can only say that there is a probability that a particle will have a position near some coordinate x. the uncertainty in position, x, and the uncertainty in momentum, p, The probability that a particle is at a certain point in space and time is given by the square of a complex number called the probability amplitude, ψ - wavefunction

21 Wavefunction (a) Si is in Group IV in the Periodic Table. An isolated Si atom has two electrons in the 3s and two electrons in the 3p orbitals. (b) When Si is about to bond, the one 3s orbital and the three 3p orbitals become perturbed and mixed to form four hybridized orbitals, ψ hyb, called sp 3 orbitals, which are directed toward the corners of a tetrahedron. The ψ hyb orbital has a large major lobe and a small back lobe. Each ψ hyb orbital takes one of the four valence electrons.

22 Probability In quantum mechanics the probability of finding a particle at a certain position at time t, P(x,y,z,t), is the square of the wave function. Probability densities for the electron of a hydrogen atom in different quantum states.-wikipedia

23 Wavefunctions and Probabilities

24 Summary The quantum state of a particle is characterized by a wave function, which contains all the information about the system an observer can possibly obtain. The wave function is interpreted as a probability amplitude of the particles presence. Ψ(r,t) 2 is the probability density. For a single particle the total probability of finding it anywhere in space at time t is equal to 1. A proper wave function must be square-integrable.

25 The Schrödinger equation - Intro The Schrodinger equation plays the role of Newton's laws and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wavefunction It is a wave equation in terms of the wavefunction which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but given a large number of events, the Schrodinger equation will predict the distribution of results.

26 The Schrödinger equation - basics

27 Schrodinger equation The Schrodinger equation is a differential equation that describes the time evolution of Kinetic energy of a free particle For a particle moving in a potential V(x,t)

28 Schrodinger equation Js = = π h h Planck s constant wave function ( ) ( ) ( ) t t i t t V m,,, r r r ψ ψ = + h h

29 Time-independent Schrödinger equation values of E constitute the allowed energies H is the Hamiltonian operator The word operator means that it is a mathematical set of operations to be carried out on a function placed to its right in this case

30 Solution of the time-independent Schrodinger equation This is the quantum number of the state.

31 Application of Schrodinger equation

32 Summary

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