 Heisenberg’s Matrix Mechanics Schrödinger’s Wave Mechanics The Schrödinger Wave Equation - Each particle in a physical system is describable by a wave function Ψ that is a function of position and time, that is Ψ(x,t). - In its time-dependent form for a particle of total energy E moving in a potential field V in one dimension: (2-23) in three dimensions (2-24)

Basic Postulates of Quantum Mechanics ||2 or p(x) is called the probability density function. It is measurable and is just the probability per unit length. P(x) must be normalized: The wave function (x,t) and its space derivative must be single-valued, finite, and continuous. Each physical observable Q is related to a quantum mechanics operator Qop, from which the expectation value Q can be calculated:

separated to obtain an equation containing (t), Let (x,t) be represented by (x)(t), the Schrödinger equation (2-23) can be separated to obtain an equation containing (t), (2-26) And the time-independent Schrodinger equation: (2-27) Total derivatives! The Potential Well Problem L x V(x) ∞ V(x) = 0, 0<x<L V(x) = ∞, x = 0, L (2-28)

Inside the well we set V(x)=0 in Eq. (2-27) (2-29) Possible solution of Eq. (2-29) Since outside the well =0 and the interior wave must match the exterior wave at the walls In order that (x) be continuous everywhere, Then, k must be some integral multiple of π/L The total energy En for each value of integer n can be solved. (2-33)

The nth wave function The constant A is found by normalizing n (2-35) Energy is quantized; the possible energy of En are called energy levels The integer n is called a quantum number The particular n and corresponding En describe the quantum state of the particle n=1 defines the ground state; its energy E1 is not zero Quantum states defined by n>1 are called excited states (t)=e-jt by solving (2-26)stationary state

Device application Chap 10.1 Scanning Tunneling Microscope (STM) Figure 2—6 Quantum mechanical tunneling: (a) potential barrier of height V0 and thickness W; (b) probability density for an electron with energy E < V 0, indicating a non-zero value of the wave function beyond the barrier. - Even for a particle with E<V0, quantum mechanics predicts that there is a finite probability the particle will be found beyond the barrier. The mechanism by which the particle “penetrates” the barrier is called quantum mechanical tunneling. Zener effect Chap 5.4. Device application Chap 10.1 Scanning Tunneling Microscope (STM)

2.5 Atomic Structure and The Periodic Table The Hydrogen Atom (2-45) The spherical coordinate system Quantum Numbers n principle quantum number n=1, 2, 3, … l orbital angular momentum quantum number l=0, 1, 2, 3,…, n-1 m magnetic quantum number m= -l, -l+1, …0, 1, …, l-1, l s spin quantum number Every allowed energy state of the electron in a H atom is uniquely described by the above four quantum numbers

The Periodic Table—Quantum Mechanics Explanation —The discovery of an empirical periodic table of elements, 1869, Demitri Mendeleev —The elucidation of the underlying physical basis of the periodic table, late 1920s Pauli Exclusion Principle: No two electrons in an atom may have the same set of quantum numbers (n, l, m, s) The electrons in an atom tend to occupy the lowest energy levels available to them Only one electron can be in a state with a given (complete) set of quantum numbers (Pauli exclusion principle) —Some conventions: Principle quantum number n=1, 2, 3, 4, … shells The naming of l values l=0, 1, 2, 3, 4, … s, p, d, f, g, … subshells Electron state 3p6 Total electronic configuration of Si 1s22s22p63s23p2