Brook Abegaz, Tennessee Technological University, Fall 2013 1 Tennessee Technological University Saturday, October 05, 2013
C hapt er 2 – I nt r oduct i on t o Q uant um M echani cs Quantum Mechanics: Used to understand the current – voltage characteristics of materials. Explains the electron behavior when the material is subject to various potential functions. The behavior and characteristics of sub-atomic particles can be de described by the formulation of quantum mechanics called Wave Mechanics . 2 Tennessee Technological University Saturday, October 05, 2013
Principles of Quantum Mechanics Three basic principles for Quantum Mechanics. 1. The Principle of Energy Quanta, 2. The Principle of Wave-Particle Duality, 3. The Principle of Uncertainty Louis de Broglie Werner Heisenberg Albert Einstein 3 Tennessee Technological University Saturday, October 05, 2013
4 Tennessee Technological University Saturday, October 05, 2013
The Principle of Energy Quanta: 1. Photo-Electric Effect experiment: if monochromatic light is incident on a clean surface a. of a material, electrons or photoelectrons are emitted from the surface. At constant incident intensity, the maximum kinetic b. energy of the photoelectron varies linearly with frequency with a limiting frequency υ = υ o below which no photoelectronsare produced. In 1900, Plank postulated that such thermal radiation c. is emitted from a heated surface in discrete packets of energy called Quanta whose energy is: where h = 6.625x10 -34 J* sec, E= h* υ , υ = frequency of incident light 5 Tennessee Technological University Saturday, October 05, 2013
The Principle of Energy Quanta: 1. In 1905, Einstein interpreted the photoelectric results by suggesting that the energy in a light wave is also contained in discrete packets or bundles called photon whose energy is E= h* υ . Work Function = minimum energy required to remove an electron. If Incoming Energy > Work Function, excess photon energy goes to the kinetic energy of the photoelectron: T = ½ mv 2 = h* υ – Φ = h* υ - h* υ o ( υ ≥ υ o ) WhereT = maximum Kinetic Energy of a photoelectron Φ = h* υ o = work function 6 Tennessee Technological University Saturday, October 05, 2013
2. Wave-Particle Duality (de Broglie) Photo-electric effect = light waves behave as particles. Compton effect = electromagnetic waves act as particles. De Broglie’s experiment = particles also act like waves with a wavelength λ = h/p where h = Plank’s constant, p = momentum of a particle. Wave-Particle Duality principle of Quantum Mechanics applies to small particles such as electrons and for very large particles, the equations reduce to classical mechanics. 7 Tennessee Technological University Saturday, October 05, 2013
3. The Uncertainty Principle (Heisenberg) It is not possible to describe the absolute accuracy of the behavior of small particles, including [position and momentum] and [energy and time]. It is impossible to simultaneously describe the position and momentum of a particle accurately. It is impossible to accurately describe energy and time for a particle simultaneously. ∆p∆x ≥ ħ where ∆p = uncertainty in momentum, ∆x = uncertainty in position. ∆E∆t ≥ ħ where ∆E = uncertainty in energy, ∆t = uncertainty in time. 8 Tennessee Technological University Saturday, October 05, 2013
Exercise The uncertainty in position of an electron is ∆x = 1. 8Å where 1Å =0.1nm. Determine the minimum uncertainty in a) momentum. If the nominal value of momentum is p = 1.2x10 - b) 23 kgms -1 , determine the corresponding uncertainty in Kinetic energy where the uncertainty in Kinetic energy is ∆E = ( dE/dp )∆p = p ∆p/m. Solution a) Minimum uncertainty in momentum: ∆p∆x ≥ ħ, ∆p ≥ 6.625x10 -34 Js / (2 π *8*0.1*10 -9 m) ∆p ≥ 1.319*10 -25 Kgm/s b) Corresponding uncertainty in Kinetic Energy for ∆p ≥ 1.2*10 - 23 Kgms -1 , ∆E = p ∆p/m e = 1.2x10 -23 Kgms -1 * 1.319x10 -25 Kgms -1 /m e ∆E = 1.737x10 -18 Kgm 2 s -2 ∆E = (1.737x10 -18 Kgm 2 s -2 )/(1.6x10 -19 J) = 10.86eV. 9 Tennessee Technological University Saturday, October 05, 2013
Exercise 2. A proton’s energy is measured with an uncertainty of 0.8eV. a) Determine the minimum uncertainty in time over which this energy is measured. b) Compute the same problem for an electron . Solution a) Minimum uncertainty in time: ∆E∆t ≥ ħ, ∆t ≥ 6.625x10 -34 Js / (2 π *0.8*1.6*10 -19 C) ∆t ≥ 8.25*10 -16 s b) Corresponding uncertainty in time for an electron, ∆t ≥ 8.25*10 -16 s. (It is the same!) 10 Tennessee Technological University Saturday, October 05, 2013
Shrödinger’sWave Equation Incorporates the principle of Quanta introduced by Plank and the Wave-Particle Duality principle Introduced by De Broglie to formulate Wave Mechanics. Wave Mechanics = a formulation of Quantum Mechanics that describes the behavior and characterization of electron movement and subatomic particles in materials and devices. Wave Equation = a one-dimensional, non-relativistic Shrödinger’swave equation: 2 2 (1) ( , ) ( , ) x t x t ( ) ( , ) V x x t j 2 2 m x t where (x,t) is the wave function, V(x) is the potential function assumed to be independent of time, m is the mass of the particle and j is √( -1). 11 Tennessee Technological University Saturday, October 05, 2013
Shrödinger’sWave Equation From Wave-Particle Duality Principles ( , ) ( ) ( ) x t x t Therefore, we would like to split equation 1to A left side that isindependent of time and a) A right side that isindependent of position. b) 2 2 ( ) ( ) x t ( ) ( ) ( ) ( ) ( ) t V x x t j x 2 2 m x t Dividing by the total wave function gives: 2 2 1 ( ) 1 ( ) x t ( ) V x j (2) 2 2 ( ) ( ) m x x t t The left side of the equation isa function of position only and the right side of the equation is a function of time t, each side must be equal to a constant η : 1 ( ) t . . j ( ) t t 12 Tennessee Technological University Saturday, October 05, 2013
Shrödinger’sWave Equation The solution of such equation is: j t E hv h ( ) t e 2 E j t E ( ) t e Substituting in (2) 2 2 1 ( ) x . . ( ) V x E 2 2 ( ) m x x 2 ( ) 2 (3) x m ( ( )) ( ) 0 E V x x 2 2 x where E is the total energy of the particle, V(x) is the potential experienced by the particle and m is the mass of the particle. 13 Tennessee Technological University Saturday, October 05, 2013
Physical Meaning of theWave Equation The total wave equation is the product of position-dependent, time- independent function and a time-dependent, position-independent function. E ( ) j t ( , ) ( ) ( ) ( ) x t x t x e Max Born in 1926 postulated that the probability of finding a particle between x and x+dx at a given time t is: 2 * | ( , ) | ( , ). ( , ) x t x t x t E ( ) j t * * ( , ) ( ). x t x e And 2 * 2 | ( , ) | ( , ). ( , ) | ( ) | x t x t x t x Therefore, the probability density function is Independent of Time. Main Difference between Classical Physics and Quantum Mechanics: Position of a particle can be determined precisely in Classical Physics; But in Quantum Mechanics, it can be done so only with a probability . 14 Tennessee Technological University Saturday, October 05, 2013
Boundary Conditions Using the probability density function for a single particle. 2 | ( ) | 1 x dx Two additional conditions: Condition 1: Ψ (x) must be finite, single valued and continuous. Condition 2: d Ψ (x) /dx must be finite, single valued and continuous. These conditions are important to have a finite energy E and a finite potential V(x). Potential Functions and Corresponding Wave Function solutions Fig. a) When the potential function is Finite everywhere, Fig. b) When the potential function is Infinite in some regions. ∞ ∞ 15 Tennessee Technological University Saturday, October 05, 2013
Reading Assignment Text Book: Semiconductor Physics and Devices, Basic Principles, Donald A. Neamen Finish Reading Chapter 2: “Introduction to Quantum Mechanics” Discussion on that topic is on Friday, 9/6/13. A Homework per Two Chapters covered (expect a homework after next class, Its due date would be in two weeks.) 16 Tennessee Technological University Saturday, October 05, 2013
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