Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential functions. Classic Mechanics Semiconductor Physics and Devices Newton s law - Describe macro world. Quantum Mechanics Seong Jun Kang Department of Advanced Materials Engineering for Information and Electronics Laboratory for Advanced Nano Technologies Wave Mechanics - Describe atomic world. - Schrodinger s wave equation. Introduction to Quantum Mechanics Introduction to Quantum Mechanics Classical Mechanics vs. Quantum Mechanics SOLVAY CONFERENCE, 1927. Principles of Quantum Mechanics; wave-particle duality In our experiences, particles and waves are always different concepts. - Stone is a particle and sound is a wave. Therefore, particles and waves are treated as a separate components in classical physics. In the microscopic world, it is difficult to separate the concepts of particles and waves. That means the wave-particle duality. We regard electrons as particles because they have charge and mass and behave according to the laws of particle mechanics. However, sometimes moving electrons can be explained by wave. Electromagnetic waves are waves because it shows diffraction, interference and polarization. However, sometimes electromagnetic waves behave as though they are streams of particles. Principles of Quantum Mechanics; Young s experiment In 1801, Thomas Young demonstrated the interference of light waves by using a pair of slits, which were illuminated by monochromatic light from a single source. In his experiments, each slits make spread out secondary waves as though originating at the slit. (Diffraction of light) At the places on the screen where the path lengths from the two slits differ by an odd number of half wavelengths ( /2, 3 /2, 5 /2, ), destructive interference occurs and the result is a dark line. At the places on the screen where the path lengths from the two slits differ by a whole number of wavelength (, 2, 3, ), constructive interference occurs and the result is a bright line. Interference and diffraction are found only in waves. Therefore, Young s experiment is a proof that light consists of waves. Until the end of the nineteenth century, the light is the waves.
Principles of Quantum Mechanics; Blackbody radiation Blackbody radiation can be explained by using only the quantum theory of light. Principles of Quantum Mechanics; Blackbody radiation To explain the blackbody spectra, Rayleigh and Jeans made a formula. After the Hertz s & Young s experiments, scientists believed that the light is waves. However, after 12 years, they could not explain the radiation emitted from bodies of matter by using the wave theory. To understand the radiation emitted from materials, it is convenient to consider a blackbody. Blackbody is an ideal body, which absorbs all radiation incident upon it, regardless of frequency. In our experiences, the radiation should be proportional to the temperature and frequency. However, it is proportional only to the temperature. Based on the classical physics, Rayleigh and Jeans made a formula, which indicates that the energy density of radiation is proportional to the temperature and to the square root of frequency. As shown in the spectra, Rayleigh/Jeans formula could not explain the radiation of blackbody perfectly. Therefore, blackbody radiation can not explain by classical physics. We need to use quantum theory to explain the phenomena. Max Planck discovered that the radiation is emitted in quanta whose energy is h. From this fact, now we can guess that the light is not just a continuous wave. Principles of Quantum Mechanics; Blackbody radiation In 1900, Max Planck made another formula to explain the blackbody radiation by using a concept of quantum physics and Planck s constant. Principles of Quantum Mechanics; Photoelectric effect The energies of electrons liberated by light depend on the frequency of the incident light. Electrons are emitted when the frequency of the incident light is sufficiently high. The phenomena is known as the photoelectric effect and the emitted electron are called photoelectrons. At high frequency region, the energy density is going to zero due to the exponential component. (Good approximation for the experiments at high frequency.) Also, at low frequency region, the Planck s formula becomes the Rayleigh-Jean s formula due to the exponential component in the equation. Based on his formula, Max Planck found the oscillators, which can not have a continuous distribution of energies but must have only the specific energies. An oscillator emits radiation of frequency when it drops from one energy state to the next lower one, and it jumps to the next higher state when it absorbs radiation of frequency. From the photoelectric effect, we know that light waves carry energy. However, it is very difficult to explain the photoelectric effect by using classical physics. Einstein explained the photoelectric effect and won the Nobel prize. (He didn t get the Nobel prize by the theory of relativity.) Each discrete bundle of energy h is called a quantum. Therefore, we can guess that light is a bundle of discrete energy. Principles of Quantum Mechanics; Photoelectric effect There are three experimental findings, which makes difficult to explain the photoelectric effect by the classical physics. (that means if we consider the wave and particle as a separate concept.) 1. There is no time interval between the arrival of light at a metal surface and the emission of photoelectrons. 2. A bright light yields more photoelectrons but the electron energies remain the same. The light intensity increase the photoelectron current. However, the photoelectron energy is the same for all intensities of light of the same frequency. 3. Photoelectrons have a higher energy, if the frequency of the incident light is higher. The photoelectron energy depends on the frequency of light. Principles of Quantum Mechanics; Photoelectric effect In 1905, Einstein realized that the photoelectric effect could be understood if the energy in light is not spread out over wavefronts but is concentrated in small packets PHOTONS. Photon, with a light of frequency, has the energy h, which is same with the Planck s quantum energy. From the concept of photons, Einstein enables to explain three experimental findings, which is difficult to explain the photoelectric effect by using wave theory (or classical physics). 1. Because of EM wave energy is concentrated in photons, there should be no delay in the emission of photoelectrons. 2. All photons of frequency have the same energy. So changing the intensity of monochromatic light beam will change the number of photoelectrons but not their energies. 3. The higher the frequency, the greater the photon energy h. Therefore the photoelectrons have more energy with the higher frequency. Therefore, we can guess that light is packets of energy. PHOTON
Principles of Quantum Mechanics; Work function There must be a minimum energy for an electron to escape from a particular metal surface. This energy is called work function of the metal, and related to the critical frequency 0, below which no photoelectrons are emitted. Principles of Quantum Mechanics; Work function If the energy h 0 is needed to remove an electron from a metal surface, the maximum electron kinetic energy will be h -h 0 when light of frequency is directed at the surface. The greater the work function of a metal, the more energy is needed for an electron to leave the surface. And also, the higher the critical frequency is needed. According to the Einstein, the photoelectric effect in a given metal should obey the equation. We can express the photon energy E=h in terms of electronvolts (ev). where, h is the photon energy, KE max is the maximum photoelectron energy, and is the minimum energy needed for an electron to leave the metal. Principles of Quantum Mechanics; De Broglie waves A moving body behaves as it has a wave nature. Principles of Quantum Mechanics; Wave formula The wave formula can be changed as below. A photon of light of frequency has the momentum. Therefore the wavelength of a photon is specified by its momentum according to the relation below. De Broglie suggest the equation could be applied to the materials with mass as well as a photon. The momentum of a particle of mass m and velocity v is p= mv. Therefore, we can define De Broglie wavelength. is defined as angular frequency and k is wave number. Therefore, the wave formula can be written as below. Using these equation we will learn the phase velocity and group velocity of waves. is the relativistic factor. If the speed of mass m is very small compare to the speed of light, the relativistic factor should be 1. De Broglie wavelength shows that the particles with a larger momentum has the shorter De Broglie wavelength. Principles of Quantum Mechanics; Phase and group velocities A group of waves do not need to have the same velocity as the waves themselves. Phase velocity is the velocity of waves itself. Group velocity is the velocity of wave packet or wave group. The wave group can be formed by superposition of the waves. Principles of Quantum Mechanics; Phase and group velocities The amplitude of the de Broglie waves that correspond to a moving body reflects the probability that it will be found at a particular place at a particular time. We can not use the simple harmonic wave formula to describe de Broglie waves, because the amplitude of simple harmonic wave is same everywhere. Therefore, we can expect a moving body can be represented by a wave packet (wave group). Using the wave group, the probability is different at a particular place and a particular time because the amplitude is different with x, t. Amplitude is same everywhere. Amplitude is different with the position or time.
Principles of Quantum Mechanics; Particle diffraction In 1927, several scientist confirmed de Broglie s suggestion by demonstrating that electron beams are diffracted when they are scattered by the regular atomic arrays of crystals. (De Broglie suggest the wavelength of a matter is specified by its momentum or energy such as a photon) Davisson and Germer were studying the scattering of electrons from a solid using an equipment like below. (Their equipment can change the energy of incident electron beam, angle of the beam and the position of detector. (similar with the equipment of X-ray diffraction) In the classical physics, we can expect that the scattered electrons will emerge in all directions with the angle dependence. (The intensity depends on the angle of the incident beam to the target, Bragg s equation) However, they found that if a single crystal materials are used for the experiment, the intensity of the scattered electrons are depended on the energy of the incident electrons as well as the angle. Principles of Quantum Mechanics; Particle in a box Let s try to understand why the energy of trapped particle is quantized. When a particle is restricted to a specific region of space instead of free space, the wave nature of a moving particle shows some remarkable phenomena. Let s consider the simplest case such as the figure. (a particle bounces back and forth between the walls of a box.) And if we assume the walls of the box are infinitely hard, so the particle does not lose energy by the collision. And the velocity is sufficiently small, so that we can ignore relativistic considerations. In this case, a particle trapped in a box is like a standing wave such as the image. The wave s displacement should be 0 at the wall, since the wave stop there. Therefore, the possible de Broglie wavelengths of the particle in a box are determined by the width L of the box. The longest wavelength is =2L, the next is =L and then =2L/3 and so forth. Therefore, de Broglie wavelengths of trapped particle is given by Principles of Quantum Mechanics; Particle in a box Principles of Quantum Mechanics; Particle in a box The restriction on de Broglie wavelength gives the limits on momentum of the particle and also on the kinetic energy. (mv=h/ ) The kinetic energy is shown below. The particle in a box does not have a potential energy in this model. Therefore the total energy of the particle in a box is given as below. From this equations, we can find that the energy is not continuous. Each permitted energy is called as an energy level. The integer n is called as quantum number. Therefore the energy of the particle in a box is quantized. From the equation, we can find some general conclusions. 1. A trapped particle can not have a continuous energy. The restriction of a particle limits the wave function. And the wave function limits the energy of the particles. It means the energy is quantized. 2. A trapped particle cannot have zero energy. From the =h/mv, a speed v=0 means an infinite wavelength. But we can not expect an infinite wavelength for the trapped particles. Because it is limited by the width L of the box. 3. Because Planck s constant is so small, quantization of energy can be occurred when m and L are also so small. This is why we can not aware of energy quantization in our own experience. electron shell Principles of Quantum Mechanics; Electromagnetic waves Principles of Quantum Mechanics; Electromagnetic frequency spectrum Electromagnetic (EM) waves are coupled electric and magnetic oscillations that move with the speed of light and exhibit a typical wave behavior. From the work of Faraday, Maxwell already knew the relation between magnetic field and current. - a changing magnetic field can induce a current in a wire loop. And a changing electric field has a magnetic field associate with it. Maxwell suggested that an accelerated electric charge generates linked electric and magnetic wave, which is electromagnetic waves. Maxwell shows that the speed of electromagnetic waves in free space is c, which is same with the speed of light. ε 0 is the electric permittivity of free space and μ 0 is magnetic permeability. Therefore in 1864, Maxwell concluded that light consists of electromagnetic waves.
Principles of Quantum Mechanics; Uncertainty principle If we consider a moving particle as a wave group, there are fundamental limits to the accuracy of measuring the particle properties as position and momentum. Principles of Quantum Mechanics; Uncertainty principle If the wave group is wide, we can define the wavelength more precisely. That means the momentum of the particle, which is described by the wave group, can be defined precisely. However, in the wide wave group, the range of particle existence is wide. So, we can not define the exact position of particle. Let s consider a wave group in 1-dimension. The particle corresponds to this wave group could be located anywhere within the group at a specific time. The probability density 2 is a maximum in the middle of the group, therefore the particle can be found there mostly. Also, the particle may be found anywhere that 2 is not zero. Therefore, if the wave group is narrow, we can decide the position of particle more precisely. However, the wavelength of the waves in a narrow wave group can not be well defined because there are not enough waves to measure accurately (or to describe the wave group). Since =h/p, if we can not define the accurately, the particle s momentum can not be defined precisely. Therefore, the uncertainty principle is It is impossible to know both the exact position and exact momentum of an object at the same time. The uncertainty principle was discovered by Werner Heisenberg in 1927 and it is one of the most significant of physics law. Principles of Quantum Mechanics; Uncertainty principle We can obtain the uncertainty principle from the point of view of wave properties of particles. To measure the position and momentum of an object at a certain moment, we should do something to measure such as expose a light and detect the reflection as shown in the image. Principles of Quantum Mechanics; Uncertainty principle If the wave group is wide, we can define the wavelength more precisely. That means the momentum of the particle, which is described by the wave group, can be defined precisely. In the mathematical formula, the uncertainty principle is described as below. The equation states that the product of the uncertainty x in the position of an object at some instant and the uncertainty of p in its momentum component in the x direction at the same instant is equal or greater than ħ/2. In the case of a narrow wave group, the uncertainty x is small and p will be large. If we reduce the uncertainty of p in some way, the uncertainty of x will be large. The action itself to measure can interfere the momentum or position of an object. By considering such interferences, we can also understand the uncertainty principle. These uncertainties are due to the nature of the quantities. Since we can not know exactly both where a particle is right now and what its momentum is at the same time, we don t know the present. However, we can say that the particle is more likely to be in one place than another and its momentum is more likely to have a certain value than another. Example for uncertainty principle If we measure the position of a proton with an accuracy of 1.00 10-11 m. Find the uncertainty in the proton s position after 1 sec. The speed of proton is much smaller than that of the light. Let s assume x 0 = 1.00 10-11 m. The uncertainty of its momentum at the same time is given as below. The uncertainty of the proton s velocity can be obtain as below, if we consider p= (mv)=m v. Principles of Quantum Mechanics; Energy and Time There is an uncertainty in energy and time. We can not measure the energy and time precisely at the same time. The equation states that the product of the uncertainty E in an energy measurement and the uncertainty t in the time at which the measurement is made is equal to or greater than ħ/2. We know that the frequency is the number of waves during a time interval (or unit time). And the minimum uncertainty in the number of waves we count is one wave during the time interval t. After a specific time t, the uncertainty of the position can be express as below. Also we know that E=h. By combining these two equation, we can obtain the uncertainty principle of energy and time. The wave group has spread more than 3.15 km from the original wave group. The reason is that the original position was quite accurate, so the uncertainty of momentum was large.
Principles of Quantum Mechanics; The Bohr atom Bohr and de Broglie found a successful atomic model. The de Broglie wavelength of electron in orbit around a hydrogen nucleus and the velocity of electron for a stable orbit are given as below. Principles of Quantum Mechanics; The Bohr atom The wavelengths of the electron wave should be always fit an integer number of times of the circumference of the electron orbit. By combining these equations, we can obtain the orbital electron wavelength as a function of r, the radius of the electron orbit. The radius of the electron orbit of hydrogen is 5.3 10-11 m. Therefore, the electron wavelength can be calculated as below. The electron wavelength is exactly same with the circumference of the electron orbit. If a fractional number of wavelengths is not same to the circumference of the electron orbit, destructive interference will occur as the waves travel around the loop, and the vibrations will disappear. Therefore, an electron can circle a nucleus only if its orbit contains an integral number of de Broglie wavelengths. This statement combines both the particle and wave characters of the electron because the electron wavelength depends on the orbital velocity needed to balance the pull of the nucleus. The orbit of the electron corresponds to one complete electron wave. Principles of Quantum Mechanics; The Bohr atom The condition for orbit stability can be written as below. r n is the radius of the orbit that contain n wavelengths. The integer n is called as the quantum number of the orbit. Principles of Quantum Mechanics; Energy levels and spectra The various permitted orbits according to the Bohr s atomic model related with the different electron energies. The electron energy E n is given in terms of the orbit radius r n. Therefore, the orbital radii in Bohr atom is The radius of the innermost orbit is called as the Bohr radius of the hydrogen atom. By combination those two equations, we can obtain the energy level formula. Here, the lowest energy level E 1 is called the ground state of the atom, and the higher levels E 2, E 3, E 4, are called excited states. As the quantum number n increases, the energy level approaches to 0. And the electron is no longer bound to the nucleus. Principles of Quantum Mechanics; Quantum Mechanics The fundamental difference between Classical mechanics and Quantum mechanics lies in what they describe. In classical mechanics, the future of a particle is completely determined by its initial position and momentum with the force that act upon it. In quantum mechanics, the uncertainty principle suggests that the nature of an observable quantity is different in the atomic-scale. In quantum mechanics, certainty about the future is impossible because the initial state of a particle can not be determined with sufficient accuracy. That means, the more we know about the position of a particle now, the less we know about its momentum. Therefore, we don t know its future position. Principles of Quantum Mechanics; Quantum Mechanics The quantities, that quantum mechanics describe, are probabilities. For example, the radius of the electron s orbit in a ground state hydrogen is 5.3 10-11 m. However quantum mechanics states that this is the most probable radius. That means in the experiment, most trial will give a different value, either larger or smaller, but the value most likely to be found at 5.3 10-11 m. From the De Broglie electron wave, we can say the electron can be exist at some place with some probability. It is impossible to know both the exact position and exact momentum of an object at the same time.
Principles of Quantum Mechanics; Quantum Mechanics WAVE FUNCTION The quantity with which quantum mechanics is concerned is the wave function of a body. While itself has no physical interpretation, the square of its absolute magnitude 2, evaluated at a particular place and time, is proportional to the probability of finding the body there at that time. The quantum mechanics is about to determine for a body when its freedom of motion is limited by the external forces. The wave functions are usually complex with both real and imaginary parts. Principles of Quantum Mechanics; Quantum Mechanics Normalization Because 2 is proportional to the probability density P of finding the body described by, the integral of 2 over all space must be finite. That means the body should be somewhere. If the integration of 2 is 0, the particle does not exist. 2 is the probability, therefore the integration should be 1 to have a meaning. However, a probability must be a positive real quantity. The probability density 2 for a complex is taken as the product *. * is called as complex conjugate of. A wave function that obeys this relation is said to be normalized. Every acceptable wave function can be normalized by multiplying it by an appropriate constant. Therefore, 2 is always a positive real quantity. Besides being normalizable, must be single-valued, since probability can have only one value at a particular place and time, and continuous. Also the partial derivatives of should be finite, continuous, and single valued. Principles of Quantum Mechanics; Quantum Mechanics Therefore, wave function should be satisfied the followings to be a meaningful wave function. 1. must be continuous and single-valued everywhere. 2. / x, / y, / z must be continuous and single-valued everywhere. 3. must be normalizable, which means that must go to 0 as x, y, z in order that 2 dv over all space be a finite constant. By given a normalized and acceptable wave function, the probability that the particle describes by the wave function will be found in a certain region is simply the integral of the probability density 2 over that region. Principles of Quantum Mechanics; The wave equation Schrödinger s equation is a wave equation in the variable. Schrödinger s equation is the fundamental equation of quantum mechanics in the same sense that the second law of motion (F=ma) is the fundamental equation of classical mechanics. Let s consider the wave equation below. The wave equation describes a wave whose variable quantity is y that propagates in the x direction with the speed v. Therefore, for a particle restricted to motion in the x direction, the probability of finding it between x 1 and x 2 is given as below. Waves in the xy plane traveling in the x direction along a stretched string lying on the x axis. Principles of Quantum Mechanics; The wave equation Schrödinger s equation: Time-dependent form The wave function for a particle moving freely in the +x direction is specified as below. Therefore, the wave equation above has the solution in the form as below, where F is any function that can be differentiated. By replacing ω in the formula by 2 and v by, The solution F(t - x/v) represent waves traveling in the +x direction, and the solution F(t + x/v) represent waves traveling in the x direction. Also, by using the relation between total energy E and momentum p with the and, we can obtain the wave function for a free particle in terms of Energy and Momentum. The equation describes the wave of a free particle of total energy E and momentum p moving in the +x direction. Let s consider the wave of a free particle. The wave can be described by the general solution of the wave equation for a constant amplitude A, constant angular frequency ω in the +x direction as below. To obtain the fundamental differential equation of, which we can solve for in a specific situation, we have to use differential formula.
Schrödinger s equation: Time-dependent form By differentiating two times with respect to x, Schrödinger s equation: Time-dependent form The time dependent form of Schrödinger s equation in three dimensions are shown. By differentiating once with respect to t, Any restrictions that may be present on the particle s motion will affect the potential energy U. Therefore, once we know U, by solving the Schrödinger s equation, we can obtain the wave function and the probability density 2 in terms of x, y, z, t. Schrödinger s equation is a basic principle of physics itself. The total energy is given as below, and we can obtain the time dependent form of Schrödinger s equation. Linearity and superposition An important property of Schrödinger's equation is that it is linear in the wave function. That means a linear combination of solutions of Schrödinger's equation for a given system is also a solution. If 1 and 2 are two solutions, the sum is also a solution. a 1 and a 2 are constants. Therefore, we can say that superposition principle is also valid for the wave function, which is the solution of Schrödinger's equation. Let s consider the diffraction of an electron beam. Expectation values The solution of Schrödinger's equation contains all the information as a probability about the particle that is permitted by the uncertainty principle. Let s try to obtain the expectation value x of the position of a particle confined to the x axis that is described by the wave function (x, t). This is the value of x we would obtain if we measure the positions of many particles described by the same wave function at the specific time t. If there are N 1 particles at x 1, N 2 particles at x 2, and so on, the average position can be written as below. When we are consider a single particle, we just need to replace the number N i of particles at x i by the probability P i that the particle be found in an interval dx at x i. i is the particle wave function evaluated at x=x i. Expectation values Therefore, the expectation value for position can be written as below. Operators To evaluate p and E, the wave function of free particle should be differentiated with respect to x and t. Example A particle limited to the x axis has the wave function = ax between x = 0 and x = 1. = 0 elsewhere. From these relations, we can obtain the operators for momentum and energy. (a) Find the probability that the particle can be found between x = 0.45 and x = 0.55. (b) Find the expectation value x of the particle s position. And, the expectation value for momentum and energy define as the equation below.
Examples of potential function and the corresponding wave solutions Finite potential well vs. Infinite potential well - Particle can penetrate the finite potential well.