Materials Physics Lecture 01 PDF

Materials Physics Lecture 01 Nuclei and Electrons Plan for today: Principle assumption of Materials Science and Engineering: All properties of The building block...

Materials Physics Lecture 01 Nuclei and Electrons Plan for today: Principle assumption of Materials Science and Engineering: All properties of The building blocks of matter: materials derive from the structure (spatial distribution) of the nuclei and electrons of the material. Fundamentally, both nuclei and electrons are quantum objects. However, in practice, we describe electrons as quantum objects and nuclei as classical, point- like, in most cases even stationary, charged particles. Reason: De-Broglie wavelength of nuclei is much smaller than the inter-atomic distance.. The first lectures in this course will introduce you to a simplified version of quantum mechanics, just enough to understand the behavior of electrons in the most important classes of materials. We will mention simplifications, in case you later need to expand your knowledge. 12 First postulate of quantum mechanics Measurements in Quantum Mechanics The state of a quantum-mechanical system is fully represented by a function, the Measurements, or observations, are described in quantum mechanics by so-called so-called wavefunction. operators. These are either functions by which the wavefunctions are multiplied, or derivative operations applied to the wavefunction. The average result of a measurement (average over many repetitions of the same experiment on identical systems) is then the so-called expectation value , which The wavefunction fully defines the outcome of all measurements in terms of is calculated as probabilities (statistical likelihood of possible results). Simplification: Here, the star indicates complex conjugation. is in principle time- where the each electron can be described individually by a one-particle wave dependent. function The integrand, , can be understood as a (probability) density of observable. Here, is the spin, which is either up ( ) or down ( ). 13 14  Observable 1: Position Observable 2: Potential Energy The position probability density is simply defined by the absolute The potential energy density of an electron is defined by the potential square of wavefunction itself (technically, ). That is: energy associated with a certain position, , times the probability of finding the electron at that position. Technically, for the potential energy. is the probability (density) of finding the electron at position at time. The total potential energy of the electron is the integral of this density over the entire space: Since the electron must exist somewhere, the integrated probability density must be 100 % = 1. That is: for any wavefunction. This is known as the normalization condition. 15 16 Observable 3: Kinetic Energy Observable 3: Kinetic Energy (ctnd.) The kinetic energy operator is defined by the second order gradient:. This introduces a minus sign. As a result: Now, it matters that the operator is only applied to one of the Hence, the absolute square of the first derivative of the wavefunction, is the kinetic energy density. 17 18 Illustration of position probability and energy density Second postulate of quantum mechanics The state evolves according to with the energy operator This equation is called Schrödinger Equation and is called Hamiltonian. is the gradient operator and is the potential. Simplification: In this lecture, we will focus on time-independent potentials. 19 20 Stationary states (1) Stationary states (2) In this lecture, we are interested in materials at equilibrium. should not It follows (division by , which is non-zero as the particle always exists): depend on time. In this case we can find solutions of the form Since neither nor depends on , the right hand side of the equation is time- With this, the Schrödinger Equation reads: independent. Therefore, the term in must be time-independent. It also does not depend on. The second line follows because is just a pre-factor for the time-independent operator. Similarly, is a pre-factor for the time operator. is the energy of the electron. The remaining time-independent Schrödinger Equation determines the spatial wavefunction: 21 22