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Chap 2. Introduction to Quantum Mechanics Principles of Quantum Mechanics Schrödinger’s Wave Equation Application of Schrödinger’s Wave Equation Homework Solid-State Electronics Chap. 2 1 Instructor: Pei-Wen Li Dept. of E. E. NCU Introduction In solids, there are about 1023 electrons and ions packed in a volume of 1 cm3. The consequences of this highly packing density : – Interparticle distance is very small: ~2x10-8 cm. the instantaneous position and velocity of the particle are no longer deterministic. Thus, the electrons motion in solids must be analyzed by a probability theory. Quantum mechanics Newtonian mechanics Schrodinger’s equation: to describe the position probability of a particle. Solid-State Electronics Chap. 2 2 Instructor: Pei-Wen Li Dept. of E. E. NCU Introduction – The force acting on the j-th particle comes from all the other 1023-1 particles. – The rate of collision between particles is very high, 1013 collisions/sec average electron motion instead of the motion of each electron at a given instance of time are interested. (Statistical Mechanics) equilibrium statistical mechanics: Fermi-Dirac quantum-distribution Boltzmann classical distribution Solid-State Electronics Chap. 2 3 Instructor: Pei-Wen Li Dept. of E. E. NCU Principles of Quantum Mechanics Principle of energy quanta Wave-Particle duality principle Uncertainty principle Solid-State Electronics Chap. 2 4 Instructor: Pei-Wen Li Dept. of E. E. NCU Energy Quanta Consider a light incident on a surface of a material as shown below: Classical theory: as long as the intensity of light is strong enough photoelectrons will be emitted from the material. Photoelectric Effect: experimental results shows “NOT”. Observation: – as the frequency of incident light < o: no electron emitted. – as > o:at const. frequency, intensity, emission rate, K.E. unchanged. at const. intensity, the max. K. E. the frequency of incident light. Solid-State Electronics Chap. 2 5 Instructor: Pei-Wen Li Dept. of E. E. NCU Quanta and Photon Planck postulated that thermal radiation is emitted from a heated surface in discrete energy called quanta. The energy of these quanta is given by E = h, h = 6.625 x 10-34 J-sec (Planck’s constant) According to the photoelectric results, Einstein suggested that the energy in a light wave is also contained in discrete packets called photon whose energy is also given by E = h. The maximum K.E. of the photoelectron is Tmax = ½mv2 = h - ho The momentum of a photon, p = h/ Solid-State Electronics Chap. 2 6 Instructor: Pei-Wen Li Dept. of E. E. NCU Wave-Particle Duality de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. de Broglie suggested that the wavelength of a particle is expressed as = h /p, where p is the momentum of a particle Davisson-Germer experimentally proved de Broglie postulation of “Wave Nature of Electrons”. Solid-State Electronics Chap. 2 7 Instructor: Pei-Wen Li Dept. of E. E. NCU Davisson-Germer Experiment Consider the experimental setup below: Observation: – the existence of a peak in the density of scattered electrons can be explained as a constructive interference of waves scattered by the periodic atoms. – the angular distribution of the deflected electrons is very similar to an interference pattern produced by light diffracted from a grating. Solid-State Electronics Chap. 2 8 Instructor: Pei-Wen Li Dept. of E. E. NCU Conclusion In some cases, EM wave behaves like particles (photons) and sometimes particles behave as if they are waves. Wave-particle duality principle applies primarily to SMALL particles, e.g., electrons, protons, neutrons. For large particles, classical mechanics still apply. Solid-State Electronics Chap. 2 9 Instructor: Pei-Wen Li Dept. of E. E. NCU Uncertainty Principle Heisenberg states that we cannot describe with absolute accuracy the behavior of the subatomic particles. 1. It is impossible to simultaneously describe with the absolute accuracy the position and momentum of a particle. p x ħ. (ħ = h/2 = 1.054x10-34 J-sec) 2. It is impossible to simultaneously describe with the absolute accuracy the energy of a particle and the instant of time the particle has this energy. E t ħ The uncertainty principle implies that these simultaneous measurements are in error to a certain extent. However, ħ is very small, the uncertainty principle is only significant for small particles. Solid-State Electronics Chap. 2 10 Instructor: Pei-Wen Li Dept. of E. E. NCU Schrodinger’s Wave Equation Based on the principle of quanta and the wave-particle duality principle, Schrodinger’s equation describes the motion of electrons in a crystal. 1-D Schrodinger’s equation, 2 2 ( x, t ) ( x, t ) V ( x ) ( x , t ) j 2m x 2 t Where (x,t) is the wave function, which is used to describe the behavior of the system, and mathematically can be a complex quantity. V(x) is the potential function. Assume the wave function (x,t) = (x)(t), then the Schrodinger eq. Becomes 2 2 ( x) (t ) 2m Solid-State Electronics Chap. 2 (t ) x 2 V ( x) ( x) (t ) j ( x) 11 t Instructor: Pei-Wen Li Dept. of E. E. NCU Schrodinger’s Wave Equation 2 1 2 ( x) 1 (t ) V ( x ) j E 2 2m ( x) x (t ) t where E is the total energy, and the solution of the eq. is and the time-indep. Schrodinger equation can be written as (t ) e j ( E / )t 2 ( x) 2m 2 ( E V ( x)) ( x) 0 2 x The physical meaning of wave function: – (x,t) is a complex function, so it can not by itself represent a real physical quantity. – |2(x,t)| is the probability of finding the particle between x and x+dx at a given time, or is a probability density function. – |2(x,t)|= (x,t) *(x,t) =(x)* (x) = |(x)|2 -- indep. of time Solid-State Electronics Chap. 2 12 Instructor: Pei-Wen Li Dept. of E. E. NCU Boundary Conditions 2 ( x) dx 1 since |(x)|2 represents the probability density function, then for a single particle, the probability of finding the particle somewhere is certain. If the total energy E and the potential V(x) are finite everywhere, 2. (x) must be finite, single-valued, and continuous. 3. (x)/x must be finite, single-valued, and continuous. 1. Solid-State Electronics Chap. 2 13 Instructor: Pei-Wen Li Dept. of E. E. NCU Applications of Schrodinger’s Eq. The infinite Potential Well In region I, III, (x) = 0, since E is finite and a particle cannot penetrate the infinite potential barriers. In region II, the particle is contained within a finite region of space and V = 0. 1-D time-indep. Schrodinger’s eq. becomes 2 ( x) 2m E 2 ( x) 0 x 2 the solution is given by ( x) A1 cos Kx A2 sin Kx, where K Solid-State Electronics Chap. 2 14 2m E 2 Instructor: Pei-Wen Li Dept. of E. E. NCU Infinite Potential Well Boundary conditions: 1. (x) must be continuous, so that (x = 0) = (x = a) = 0 A1 = A2sinKa 0 K = n/a, where n is a positive integer. 2. 2 ( x) dx 1 A22 sin 2 Kxdx 1 A2 a 0 2 a So the time-indep. Wave equation is given by ( x) 2 nx sin( ) wheren 1,2,3... a a The solution represents the electron in the infinite potential well is in a standing waveform. The parameter K is related to the total energy E, therefore, 2 n 2 2 E En Solid-State Electronics Chap. 2 wheren is a positiveinteger 2m a2 15 Instructor: Pei-Wen Li Dept. of E. E. NCU Infinite Potential Well That means that the energy of the particle in the infinite potential well is “quantized”. That is, the energy of the particle can only have particular discrete values. Solid-State Electronics Chap. 2 16 Instructor: Pei-Wen Li Dept. of E. E. NCU The Step Potential Function Consider a particle being incident on a step potential barrier: In region I, V = 0, And the general solution of this equation is 2 1 ( x) 2m E 2 1 ( x) 0 2 x 1 ( x) A1e jK x B1e jK x ( x 0) where K1 1 1 2mE 2 In region II, V = Vo, if we assume E < Vo, then 2 2 ( x) x Solid-State Electronics Chap. 2 2 2m 2 (Vo E ) 2 ( x) 0 17 Instructor: Pei-Wen Li Dept. of E. E. NCU The Step Potential Function The general solution is in the form 2 ( x) A2e K2 x B2e K 2 x ( x 0) where K2 Boundary Conditions: 2m(Vo E) 2 2 ( x) A2e K 2 x ( x 0) – 2(x) must remain finite, B2 0 – (x) must be continuous, i.e., 1(x = 0) = 2(x = 0) A1+B1 = A2 – (x)/ x must be continuous, i.e., 1 x x 0 21 x jK1 A1 jK1B1 K2 A2 x 0 A1, B1, and A2 could be solved from the above equations. Solid-State Electronics Chap. 2 18 Instructor: Pei-Wen Li Dept. of E. E. NCU The Potential Barrier Consider the potential barrier function as shown: Assume the total energy of an incident particle E < Vo, as before, we could solve the Schrodinger’s equations in each region, and obtain 1 ( x) A1e jK x B1e jK x 1 1 2 ( x) A2e K 2 x B2e K 2 x where K1 3 ( x) A3e jK x B3e jK x 1 1 2m(Vo E ) 2m E and K 2 2 2 We can solve B1, A2, B2, and A3 in terms of A1 from boundary conditions: – B3 = 0 , once a particle enters in region III, there is no potential changes to cause a reflection, therefore, B3 must be zero. – At x = 0 and x = a, the corresponding wave function and its first derivative must be continuous. Solid-State Electronics Chap. 2 19 Instructor: Pei-Wen Li Dept. of E. E. NCU The Potential Barrier The results implies that there is a finite probability that a particle will penetrate the barrier, that is so called “tunneling”. * The transmission coefficient is defined by T A3 A3* A1 A1 If E<<Vo, E E T 16 1 exp 2 K 2 a Vo Vo This phenomenon is called “tunneling” and it violates classical mechanics. Solid-State Electronics Chap. 2 20 Instructor: Pei-Wen Li Dept. of E. E. NCU One-Electron Atom Consider the one-electron atom potential function due to the coulomb 2 attraction between the proton and electron: V (r ) e 4 o r Then we can generalize the Schrodinger’s eq. to 3-D in spherical coordinates: 2mo 1 2 1 2 1 ( r ) (sin ) ( E V (r )) 0 r 2 r r r 2 sin 2 2 r 2 sin 2 2 Assume the solution to the equation can be written as (r, , ) R(r ) ( ) ( ) Then the solution is of the form, = ejm, where m is an integer. Solid-State Electronics Chap. 2 21 Instructor: Pei-Wen Li Dept. of E. E. NCU One-Electron Atom Similarly, we can generate two additional constants n and l for the variables and r. n, l, and m are known as quantum numbers (integers) n 1,2,3,... l n 1, n 2, n 3,...,0 , each set of quantum numbers corresponds to a m l , l 1,...,0 quantum state which the electron may occupy. The solution of the wave equation is designated by nlm. For the lowest energy state (n=1, l=0, m=0), 100 1 ao 1 3/ 2 e r / ao where ao 0.529 angstrom The electron energy E is quantized, Solid-State Electronics Chap. 2 22 mo e 4 En 4 o 2 2 2n2 Instructor: Pei-Wen Li Dept. of E. E. NCU One Electron Atom The probability density function, or the probability of finding the electron at a particular distance form the nucleus, is proportional to 100*100 and also to the differential volume of the shell around the nucleus. The electron is not localized at a given radius. Solid-State Electronics Chap. 2 23 Instructor: Pei-Wen Li Dept. of E. E. NCU Homework 2.1 2.15 2.23 Solid-State Electronics Chap. 2 24 Instructor: Pei-Wen Li Dept. of E. E. NCU