Lecture notes on Quantum Physics

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Introduction to quantum and solid state physics for engineers Lecture notes by Amir Erez Ben Gurion University January 15, 2014 Watercolors by Anna Ayelet EppelContents 0 Why study quantum physics? 1 I Introduction to quantum mechanics 5 1 Experimental background 7 1.1 The photoelectric e ect . . . . . . . . . . . . . . . . . . . . 7 1.2 Waves - a brief summary. . . . . . . . . . . . . . . . . . . . 8 1.2.1 The wave equation . . . . . . . . . . . . . . . . . . . 8 1.2.2 Solution by separation of variables . . . . . . . . . . 9 1.2.3 Superposition . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Double-slit experiment: a qualitative description. . . . . . . 12 1.4 Linear Polarization . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Spectral Decomposition . . . . . . . . . . . . . . . . 15 1.5 de Broglie Principle. . . . . . . . . . . . . . . . . . . . . . . 17 2 The Schr odinger equation 19 2.1 Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.2 Delta functions . . . . . . . . . . . . . . . . . . . . . 20 2.1.3 Application to the wave equation . . . . . . . . . . . 20 2.2 The Schr odinger equation . . . . . . . . . . . . . . . . . . . 21 2.2.1 Free particle . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Probability and normalization . . . . . . . . . . . . . . . . . 23 iiCONTENTS iii 3 Uncertainty principle 25 3.1 Wave packet description of a free particle. . . . . . . . . . . 26 3.2 Uncertainty principle for a wave packet. . . . . . . . . . . . 28 3.3 Time independent Schr odinger equation . . . . . . . . . . . 30 4 Particle in a potential well 33 4.1 An in nite potential well in one dimension . . . . . . . . . . 33 4.1.1 Quantization . . . . . . . . . . . . . . . . . . . . . . 35 4.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . 36 4.1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . 37 4.2 Scattering from a potential step . . . . . . . . . . . . . . . . 40 4.2.1 E V , particle comes from the left . . . . . . . . . 41 0 4.2.2 E V , particle comes from the left . . . . . . . . . 42 0 4.2.3 E V , particle comes from the right . . . . . . . . 44 0 4.3 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Energy states, degeneracy, fermion statistics 49 5.1 In nite well in higher dimensions . . . . . . . . . . . . . . . 49 5.2 De nition of degeneracy . . . . . . . . . . . . . . . . . . . . 50 5.3 Periodic boundary conditions . . . . . . . . . . . . . . . . . 50 5.4 Describing the quantum state . . . . . . . . . . . . . . . . . 51 5.5 Pauli exclusion principle . . . . . . . . . . . . . . . . . . . . 51 5.6 Number of states . . . . . . . . . . . . . . . . . . . . . . . . 52 5.7 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.8 Fermi momentum . . . . . . . . . . . . . . . . . . . . . . . . 55 5.9 Fermi Energy, Fermi-Dirac distribution . . . . . . . . . . . . 56 5.10 Probability current . . . . . . . . . . . . . . . . . . . . . . . 59 II The foundations of quantum mechanics 61 6 The basic concepts of quantum mechanics I 63iv CONTENTS 6.1 Some linear algebra . . . . . . . . . . . . . . . . . . . . . . . 63 6.1.1 Transpose vs. hermitian conjugate . . . . . . . . . . 63 6.1.2 Hermitian and unitary operators . . . . . . . . . . . 64 6.1.3 Inner product of functions . . . . . . . . . . . . . . . 64 6.1.4 Delta functions: Dirac vs. Kronecker . . . . . . . . . 64 6.1.5 Dual vector spaces / dual basis . . . . . . . . . . . . 66 6.2 Dirac bra-ket notation . . . . . . . . . . . . . . . . . . . . . 66 6.2.1 Bra and Ket . . . . . . . . . . . . . . . . . . . . . . 67 6.2.2 Scalar product . . . . . . . . . . . . . . . . . . . . . 67 6.2.3 Projection, projection operators . . . . . . . . . . . 68 6.2.4 Completeness relation . . . . . . . . . . . . . . . . . 68 6.2.5 Operators . . . . . . . . . . . . . . . . . . . . . . . . 69 6.3 Physical variables as operators . . . . . . . . . . . . . . . . 70 6.3.1 Representation of bras and kets . . . . . . . . . . . . 70 6.3.2 Representation of operators . . . . . . . . . . . . . . 71 6.3.3 Back to linear polarization . . . . . . . . . . . . . . 73 6.3.4 Momentum and position bases . . . . . . . . . . . . 73 6.3.5 Momentum and position operators . . . . . . . . . . 75 6.3.6 Commutation relations . . . . . . . . . . . . . . . . . 76 6.3.7 Sets of commuting variables . . . . . . . . . . . . . . 76 7 The basic concepts of quantum mechanics II 79 7.1 The Postulates of Quantum Mechanic . . . . . . . . . . . . 79 7.1.1 Postulate I: . . . . . . . . . . . . . . . . . . . . . . . 79 7.1.2 Postulate II: . . . . . . . . . . . . . . . . . . . . . . 80 7.1.3 Postulate III: . . . . . . . . . . . . . . . . . . . . . . 80 7.1.4 Postulate IV: (discrete non-degenerate spectrum) . . 80 7.1.5 Postulate V: . . . . . . . . . . . . . . . . . . . . . . 81 7.1.6 Postulate VI: . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 82CONTENTS v 7.2.1 Expectation value of the Hamiltonian . . . . . . . . 82 7.2.2 Unitary evolution . . . . . . . . . . . . . . . . . . . . 83 7.3 Expectation value . . . . . . . . . . . . . . . . . . . . . . . 84 7.3.1 Ehrenfest's Theorem . . . . . . . . . . . . . . . . . . 85 7.3.2 Standard deviation and uncertainty . . . . . . . . . 87 7.4 Site system, position, translation . . . . . . . . . . . . . . . 88 8 Harmonic Oscillator 91 8.1 Classical oscillator . . . . . . . . . . . . . . . . . . . . . . . 91 8.2 Quantum harmonic oscillator in one dimension . . . . . . . 92 8.2.1 Quantization . . . . . . . . . . . . . . . . . . . . . . 92 8.2.2 Wavefunctions . . . . . . . . . . . . . . . . . . . . . 93 8.3 Raising or lowering operators . . . . . . . . . . . . . . . . . 94 8.3.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . 94 8.3.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 95 8.3.3 N basis . . . . . . . . . . . . . . . . . . . . . . . . . 95 III Introduction to solid state physics 99 9 Lattice 101 9.1 Bravais lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.2 Primitive vectors, BCC, FCC . . . . . . . . . . . . . . . . . 104 9.3 Wigner-Seitz Primitive Cell . . . . . . . . . . . . . . . . . . 105 9.4 The reciprocal lattice. . . . . . . . . . . . . . . . . . . . . . 106 9.4.1 Examples: . . . . . . . . . . . . . . . . . . . . . . . . 108 10 Electrons in a periodic potential 109 10.1 Bloch's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.1.1 Crystal Momentum . . . . . . . . . . . . . . . . . . . 113 10.1.2 First Brillouin Zone . . . . . . . . . . . . . . . . . . 113 10.1.3 The band index n . . . . . . . . . . . . . . . . . . . 113vi CONTENTS 10.1.4 Energy bands . . . . . . . . . . . . . . . . . . . . . . 114 10.1.5 Comparison between free and Bloch electrons . . . . 115 10.2 The Kronig-Penney model . . . . . . . . . . . . . . . . . . . 115 11 Energy gap 121 11.1 Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.2 Metal vs. Insulator . . . . . . . . . . . . . . . . . . . . . . . 122 11.3 Measuring the gap, optical absorption . . . . . . . . . . . . 123 11.4 Insulator vs. Semiconductor . . . . . . . . . . . . . . . . . . 124 12 Conduction 127 12.1 Drude - classically explaining conduction. . . . . . . . . . . 127 12.2 The Hall E ect . . . . . . . . . . . . . . . . . . . . . . . . . 130 13 Summary 133Chapter 0 Why study quantum physics? any engineering students consider the compulsory physics courses in their degree as a form of re ned torture. The M reasons for this a numerous; not least because university- level physics is dicult and requires a way of thinking not usually taught in school. Still, probably all engineers would agree that mechanics and elec- tromagnetic courses are important. But why study quantum physics ? Here are some thoughts: You are now investing in your future career. Which means you expect to be working in the next 40 years... Every year, quantum physics plays a larger role in engineering applications: semiconductors, nan- otechnology, lasers, superconductors, MRI machines, CCD cameras, computers/mobile phones; this is a partial list of technologies which rely on quantum mechanics to work. As an engineer you should have an understanding of how they work. If you don't, you'll be left behind. It's true that engineering software often calculates the quantum e ects for you. It's unlikely that you'll need to solve quantum mechanical problems in your professional life. But the di erence between an ok engineer and a good engineer is that the latter knows what the software 12 Chapter 0 Why study quantum physics? does, not just how to use it. One day we may succeed in building quantum computers. When that day comes, it will bring a technological revolution which hard to imag- ine. eg. Shor's algorithm factors numbers, which will make current RSA cryptography insecure. Quantum cryptography will allow pri- vate communication, ensuring that you and your communication part- ner are not eavesdropped on. When this revolution happens - you'll de nitely need to understand the basics of quantum theory. Why did you go to university in the rst place ? Was it just to get your engineering degree and get a job? Or are you interested in expanding your horizons ? Quantum theory is one of the most interesting, bizarre, elegant and successful scienti c theories ever made. Unlike most sci- enti c theories, it is not the result of one person - it is the result of the th combined e ort of the best genius minds of the 20 century. This is your chance to learn it. Take this chance - life is busy - you may not get another. How is this engineering course di erent from a quantum physics course for physics students ? We will solve only the simplest cases possible. They require under- standing but not a high technical skill. This is partly why in this text I refer to wikipedia often. We will omit some important basic quantum theory topics (such as spin, hydrogen atom, angular momentum) so that we can learn some basic solid state physics. You will need this knowledge later on when you learn about semiconductors. We will not compromise on the fundamentals. When the course is done - you will be able to open any "serious" quantum physics textbook3 and feel at home. You will not receive a simpli ed version of quantum mechanics (I don't know if it's even possible). You will receive a quick (and sometimes dirty) introduction to the real thing.Part I Introduction to quantum mechanics 5Chapter 1 Experimental background 1.1 The photoelectric e ect t the end of the 19th century, light was thought to consist Max Karl Ernst Ludwig Planck (1858- of waves of electromagnetic elds which propagate accord- 1947) was a German theoretical physi- cist who originated quantum theory, A ing to Maxwell's equations, while matter was thought to which won him the Nobel Prize in consist of localized particles. Physics in 1918. In 1894 Planck turned his attention to the problem of black- The photoelectric e ect was rst observed in 1887 by Heinrich Hertz. body radiation, commissioned by electric companies to create maximum light Hertz found that shining ultraviolet light on metals facilitated the creation of from lightbulbs with minimum energy. sparks. Later experiments by others, most notably Robert Millikan, observed The question had been explored exper- imentally, but no theoretical treatment that shining light on a metal makes it emit electrons, called photoelectrons. agreed with experiments. Finally, in But frequencies below a threshold frequency, would not eject photoelectrons 1900, he presented his theory, now known as the Planck postulate, that from the metal surface no matter how bright the source. The intensity of the electromagnetic energy could be emitted light only mattered when the frequency was higher than the threshold. Below only in quantized form. Planck stayed in Germany throughout WWI and WWII. the theshold frequency, no matter how high the intensity, no photoelectrons When the Nazis seized power in 1933, would be emitted. Planck was 74. In 1945, his son, Erwin, In classical physics, light was considered wave-like. But the energy ux was executed by the Nazis because of his participation in the failed attempt to of a wave depends on its amplitude (intensity) according to the Poynting assassinate Hitler.(Wikipedia) 1 2 vectorS = EB/jEj . So how was it possible that the high amplitude  0 78 Chapter 1 Experimental background didn't provide enough energy to eject electrons from the metal ? Einstein was the one to explain the photoelectric e ect in one of his famous 1905 papers. He won the Nobel Prize for this work. Einstein's idea was simple: suppose light is composed of particles (later called photons), the intensity of the light is proportional to the number of photons. Each photon could knock out one electron, with a relationship between the electron's kinetic energy and the photon frequency. Einstein suggested that, Einstein's photoelectric h =  +K (1.1) equation with the frequency of the light,h a constant (later called Planck's constant),K the photoelectron kinetic energy and  a minimal energy (\work function") below which no photoelectrons are emitted. Since  0, and by de nition K 0, the photon frequency had to satisfy  =h. This link between frequency and energy and the idea that light is composed of particles later led to the development of quantum mechanics. Instead of looking at frequency  we can instead think of angular velocity = 2 so that Figure 1.1 Photoelectric e ect (Wikipedia) E = h = (1.2) h 34 Planck's Constant =  1:05 10 Js 2 Note that the units Js are units of angular momentum but are also called as units of action. is a quantum of action. As we shall see later in the course, in many ways, the fact that 6= 0 makes the di erence between classical physics and quantum physics. 1.2 Waves - a brief summary. 1.2.1 The wave equation To understand quantum mechanics we must rst understand the basics of wave mechanics. A wave propagates energy in time and space. There are1.2 Waves - a brief summary. 9 two types of waves: transverse and longitudinal. In transverse waves the movement is perpendicular to the direction of the motion of the wave (like in the ocean) whereas in longitudinal waves the movement is parallel (eg. sound waves). The (simplest) one-dimensional linear wave equation is 2 2 (x;t) 1 (x;t) = (1.3) The wave equation 2 2 2 x c t This linear wave equation can be derived from eg: Taking a string, dividing it into di erential segments of weights con- nected by springs and writing Newton's laws. In this case, c is the speed of sound in the medium. Playing with Maxwell's electromagnetic equations. In this casec is the speed of light in the medium. 1.2.2 Solution by separation of variables We can solve the wave equation by using a technique called \separation of variables" which means that we assume that the function (x;t) =X(x)T (t) is a multiplication of two functions, one that depends solely on time and the other solely on space. Separation of variables (x;t) = X(x)T (t) (1.4) 2 2 (X(x)T (t)) 1 X(x)T (t) = 2 2 2 x c t 2 2 (X(x)) 1 T (t) T (t) = X(x) 2 2 2 x c t 2 2 1 (X(x)) 1 1 T (t) 2 = =k 2 2 2 X(x) x c T (t) t10 Chapter 1 Experimental background Let's look at the last line: on the left we have a expression only of x, in the middle we have a expression only of t. But x;t are independent variables, the only way this equation can work is if both expressions equal 2 a constant, which I callk . Now solving each equation separately is easy, by integrating: Interating the separated variables 2 1 (X(x)) 2 = k (1.5) 2 X(x) x 2 (X(x)) 2 = k X(x) 2 x ikx ikx X(x) = A e +A e + 2 1 1 T (t) 2 = k 2 2 c T (t) t 2 T (t) 2 2 = c k T (t) 2 t ickt ickt T (t) = B e +B e + (x;t) = A cos(k(xct) + ) + R R + A cos(k(x +ct) + ) L L Notice that we have two solutions for the wave equation (as expected from a second order equation): one with argument k(xct) the other with k(x +ct). The rst corresponds to a wave moving right with velocity c, the second to a wave moving left. The boundary conditions determine the value of the constantsA ;A ; ; . The argument of the cos is called the phase R L R L of the wave. What about the units ? The argument of the cos must be dimensionless, so 1 if x is in units of length, then k is in units of length . So we can make the notation 2 Wave-number k = (1.6)  so that when x x + then kx kx + 2 and the cos shifts by exactly 2 which means nothing changes. This is the periodic structure of the wave1.2 Waves - a brief summary. 11 solution. We call the wavelength andk the wavenumber. Let us make the following notation: =ck (1.7) Linear dispersion 1 Again, sincek is in units of length andc is in units of length/time then 1 is in units of time which is (angular) frequency. De ning 2 = = 2 (1.8) T we see that takingtt+T means thattt+2 and again we've shifted the cos by exactly 2 which means nothing changes. T is called the period of the wave and its angular frequency. We can also de ne the frequency 1  = = . 2 T The relation between the wave frequency (k) and its wavenumber k is called the dispersion relation and in the linear wave equation we have linear dispersion. 1.2.3 Superposition 2 The wave equation is linear (no (x;t) or any of its derivatives) and there- fore if and are solutions to the equation, then =A +A is also 1 2 1 1 2 2 a solution, with A ;A arbitrary constants. This is the same principle of 1 2 superposition which appears in Maxwell's equations (which are also linear). Therefore, if there is more than one source generating waves, then the re- sulting wave is the addition (superposition) of the waves generated by each source. Look at Fig. 1.2, there are places where the two waves add to each other in a positive way - \constructive interference", and there are places where the two waves cancel each other \destructive interference". We will see later that the same e ects play a crucial role in quantum mechanics.12 Chapter 1 Experimental background Constructive Destructive Out In Figure 1.2 Interference. (Wikipedia) 1.3 Double-slit experiment: a qualitative descrip- tion. Let us rst consider when we shine a light wave through a narrow constriction which we call a slit. See Fig. 1.3. In both cases, we see a demonstration of the Huygens principle (Wikipedia) (proposed back in 1678 ) which states that every point light reaches becomes a source of a spherical wave. So we Figure 1.3 Single slit can treat the narrow slit (of width the wavelength) as a point generating a spherical wave. Now we can graduate to the famous double-slit experiment. The double-slit experiment captures well what is sometimes called the \wave-particle" duality of quantum mechanics. A coherent light source such as a laser beam illuminates a thin plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. See Fig. 1.4. What happens in the experiment ? The two slits are positioned so that they are the same distance from the light source (left). According to the Huygens principle, both slits act as sources of spherical waves. The wave nature of light causes the spherical waves emitted from the two slits to interfere, producing bright and dark bands on the screen a result that would not be expected if light consisted strictly of classical particles. If light was Figure 1.4 Double slit a classical particle (eg. tennis ball) then logic says it cannot interfere like a1.3 Double-slit experiment: a qualitative description. 13 wave. Since we see interference, light must be a wave. But is our analysis correct ? Let's now start dimming the light source so that it produces less and less light. If light is purely a wave, then all we are doing is decreasing the intensity of the light, which means we should see the same interference pattern on the screen, only weaker. Let's take it a step further. Today we have the technology to produce single photons, single particles of light. Say we shine them one at a time. Each time, a single photon shoots through the slits (which one?) and is registered on the screen as a dot. The interference pattern disappears Light, when we shoot it one photon at a time, behaves as we expect particles to. Surely, a particle must pass through either the top slit or the bottom slit ? We repeat the experiment many times, each time noting the location the particle hits the screen. Finally, when we look at the sum of all the experiments, we see the same interference pattern. A single particle shows the same interference with itself as waves do. This means that a single particle essentially passes through both holes, as a wave will do. Light has both wave-like properties (interference) and particle- like properties (produces a dot and not an interference pattern when single photons are shot). We will see that this observation is not limited to light and will include "matter" as well. This is sometimes called the \wave-particle duality". In a wave description, the superposition happens at the level of the electric eld E =E +E (1.9) Superposition 1 2 The electric eld vectors come from the same source and are in the same direction (polarization, see below). The light intensity I is proportional to the amplitude squared 2 2   2 2   I/jEj =jE +Ej = (E +E )(E +E ) =jEj +jEj +E E +E E 1 2 1 2 1 2 1 2 1 2 2 1 (1.10) The two free waves have phases  (x) and  (x) at point x, meaning that 1 214 Chapter 1 Experimental background the electric elds of the two waves are, i (x) 1 E = A e (1.11) 1 1 i (x) 2 E = A e 2 2 give intensity 2 2 2 I/jE +Ej = A +A + 2A A cos( (x) (x)) (1.12) 1 2 1 2 1 2 1 2 1 1 2 A =A = A jAj (1 + cos (x)) 1 2 2 2 (x) 2 2 = jAj cos 2 this oscillates in the x direction with a stripe pattern like the one produced by the two slit experiment. Note that at (x) = 0 which is (x) = (x), 1 2 the intensity is maximal, as we would expect from a superposition of equal phases. What happens when we shoot the photons one at a time? Each shot has dx dx a probabilityP (x)dx to land on the target screen in location x ;x+ . Christiaan Huygens (1629-1695) was 2 2 a prominent Dutch mathematician and After ring many shots, we accumulate a statistic, hI(x)i, which is very natural philosopher. He is known par- close to the I(x) we have derived from the wave description. As we shoot ticularly as an astronomer, physicist, probabilist and horologist. His work in- more and more photons,hI(x)i converges to the wave I(x). This is because cluded early telescopic studies of the quantum mechanics is non-deterministic: a given situation can have several rings of Saturn and the discovery of its di erent outcomes, each having a di erent probability. By repeating the moon Titan, the invention of the pen- dulum clock and other investigations experiment many times, we gather statistics which converge to a probability in timekeeping. He published major distribution. studies of mechanics and optics, and a pioneer work on games of chance. Huy- gens is remembered especially for his 1.4 Linear Polarization wave theory of light, which relies on the principle that the speed of light is nite. (Wikipedia) Take a linearly polarized light beam in thee direction (pointing in thexy p plane), its electric eld is a solution to the wave equation i(kzt) E(z;t) =E e Ree (1.13) 0 p

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