C H A P T E R 2 FUNDAMENTAL QUANTUM CONCEPTS 1.1 The Bohr model The results of emission spectra experiments led Niels Bohr (1913) to construct a model for the hydrogen atom, based on the mathematics of planetary systems. If the electron in the hydrogen atom has a series of planetary-type orbits available to it, it can be excited to an outer orbit and then can fall to any one of the inner orbits, giving off energy corresponding to one of the lines of Fig. 2.1.To develop the model, Bohr made several postulates è Electrons exist in certain stable, circular orbits about the nucleus. This assumption implies that the orbiting electron does not give off radiation as classical electromagnetic theory would normally require of a charge experiencing angular acceleration; otherwise, the electron would not be stable in the orbit but would spiral into the nucleus as it lost energy by radiation è The electron may shift to an orbit of higher or lower energy, thereby gaining or losing energy equal to the difference in the energy levels (by absorption or emission of a photon of energy h w). è The angular momentum p q of the electron in an orbit is always an integral multiple of Planck’s constant divided by 2p (h/2p is often abbreviated Ñ for convenience), p q = n Ñ, (n = 1, 2, 3, 4, ). Although Bohr proposed this ad hoc relationship simply to explain the data, one can see that this is equivalent to having an integer number of de Broglie wavelengths fit within the circumference of the electron orbit (Prob. 2.2). These were called pilot waves, guiding the motion of the electrons around the nucleus. The de Broglie wave concept provided the inspiration for the Schrödinger wave equation in quantum mechanics discussed later. Fig. 2.1. Electron orbits and transitions in the Bohr model of the hydrogen atom. Orbit spacing is not drawn to scale. Bohr supposed that the electron in a stable orbit of radius r about the proton of the hydrogen atom, we can equate the electrostatic force between the charges to the centripetal force: (1.1) - 1 4 pe 0 q 2 r 2 =- mv 2 r where m is the mass of the electron, and v is its velocity. From assumption 3 we have (1.2) p q = mvr = nÑ Since n takes on integral values, r should be denoted by r n to indicate the nth orbit. Then Eq. (2-1) can be written (1.3) m 2 v 2 = n 2 Ñ 2 r n 2 Substituting Eq. (2.3) into Eq. (2.1), we find that the n-th radius is (1.4) r n = 1 4 pe 0 n 2 Ñ 2 mq 2 = n 2 r 1 , n = 1, 2, 3, 4,  therefore r 1 = a 0 = Ñ 2 mq 2 º 0.529 Þ has the name as the first Bohr radius. We noted that the allowed radii are r n = r 1 , 4 r 1 , 9 r 1 , 16 r 1 , The electron velocity in each of its orbits can be estimated from (2.2) - (2.4) (1.5) v n = n Ñ mr n = 1 n v 1 with v 1 = 1 4 pe 0 q 2 Ñ º 2.2 ä 10 6 m/s. From (2.3) we have the total energy of the electron as the summ of the kinetic and potential ones, as following, (1.6) E n = E d n + E t n = mv n 2 2 + - 1 4 pe 0 q 2 r n =- mq 4 4 pe 0  2 2 Ñ 2 1 n 2 =- 1 n 2 E 1 with the ionization energy for hydrogen atom E 1 º - 13.6 eV. The transitions in hydrogen emission spectra have been explained (see Fig, 2.1), for example, between the orbits n 1 and n 2 (1.7) h n 21 = Ñ w 21 =  mq 4 4 pe 0  2 2 Ñ 2 hc  1 n 1 2 - 1 n 2 2 = E 1 hc 1 n 1 2 - 1 n 2 2 = R 1 n 1 2 - 1 n 2 2 with R is the Rydberg constant, which is frequently serving as an unity of energy in atomic physics. Although the Bohr model was immensely successful in explaining the hydrogen spectra, numer- ous attempts to extend the "semi-classical" Bohr analysis to more complex atoms such as hclium proved to be futile. Success along these lines had to await further development of the quantum mechanical formalism. Nevertheless, the Bohr analysis reinforced the concept of energy quantiza- tion and the attendant failure of classical mechanics in dealing with systems on an atomic scale. Moreover, the quantization of angular momentum in the Bohr model clearly extended the quantum concept, seemingly suggesting a general quantization of atomic-scale observables. 2 SSED Le Tuan Chapters 2.nb 1.2 Black body radiaton In this part we will very briefly recall some important moments in hystory of physics at the end of the XIX century. We know that heat transfer takes place through the processes of conduction, convection and radiation. The first two processes can take place only in a medium, while the last does not need a medium .Robert Kirchhoff studied (1860) the problem of radiation and the proper- ties of perfect absorb materials. He pointed out that when such a material is heated, it will emit radiations of all wavelengths i.e. it will be a perfect emitter. Such a body is referred to as a black body.For instance, a hollow body whose walls are at the same temperature behaves like a black body. They pointed out that it will be a perfect emitter emitting through a tiny hole in its surface, radiations of all wavelengths. Further, any radiation entering such a cavity will undergo infinitely many reflections inside and will lose energy at every reflection. Thus no incident radiation will emerge out of the tiny hole. In other words, it is also a perfect absorber. Thus such a cavity, which can be easily constructed , is almost a perfect black body. In 1879, the Austrian physicist Josef Stefan reassessed the work of Dulong and Petit. He made the corrections and calculated the pure radiation component of heat transfer and the went to the law for the total emission power of black body (1.8) E =sT 4 with the Stefan constant s = 5.70 ä 10 -8 W m -2 K -4 . Then, Wien (1893) calculated the energy u( l , T) emitted per unit volume per sec in unit wavelength interval. (1.9) ul, T = f l, T  l 5 ïl p T = b where b = 2.898 ä 10 -3 m.K is the Wien constant anf the nature of the function f( l , T) still is unknown. Based on Newtonian physics, by very adequate calculation of the emission within the black body cavity to be made up of a series of standing waves, Ragleigh and Jeans came to the famous formula: (1.10 ) ul, T = 8 p k  T l 4 Fig. 2.2. The black body emission spectra. The outstanding violet catastrophe with Rayleigh-Jeans formula led Planck to wor k out the concept of quantization of energy transfer. However, it completely fails near the low wavelength end where it diverges to an infinite value for u( l , T). This is often referred to as the Ultra-violet Catastrophe. SSED Le Tuan Chapters 2.nb 3 The important step was taken by Max Planck around 1900. Planck suggested an interpolation formula by taking for Wien's function `f(l, T)' the form: (1.11 ) f l, T = 8 p k b Exp b l T - 1 where k as the Boltzmann constant and b as an adjustable parameter. At that time, the physicists thought that the atoms of a material which absorbed or emitted radiation as harmonic oscillators. Hence, oscillators were in equilibrium with radiation exchanging energy with it. Planck knew that radiations are electromagnetic waves. Hence the cavity was filled with these waves. These waves were absorbed and emitted by the oscillators that behaved like classical `pendulums'. Planck believed in the electromagnetic nature of radiation. But the oscillator mechanism of absorption and emission of electromagnetic waves was more a theoretical model. He arriveed at his own formula for the black-body spectrum, suggesting that a harmonic oscillator can not have any energy but only in integral multiples of a quantum of energy e 0 = h n , where n is the natural frequency of the oscillator and h a constant to be determined. In other words, Planck quan- tized the permitted energies of an oscillator. Thus, they were strictly non-classical in nature with energies e = n h n , n being an integer. These oscillators are in thermal equilibrium at any tempera- ture. Therefore, Planck invoked the Boltzmann distribution to describe them. Accordingly, the average oscillator energy e of the system of oscillators becomes: (1.12 ) e= h n  Exp h n kT - 1 with the Planck constant h = 6.626 ä 10 -34 J.s. Then, the total energy of radiation per unit volum e at l per unit wavelength interval is: (1.13 ) ul, T = 8 p hc Exp hc l kT - 1l 5  It is worth to say that at the limit T Ø ¶ we can return again to the well-known Ragleigh - Jeans formula. 4 SSED Le Tuan Chapters 2.nb 1.3 Basic Formalism 1.3.1 The five postulates of Quantum Mechanism The formulation of quantum mechanics, also called wave mechanics focuses on the wave function, Y(x,y,z,t), which depends on the spatial coordinates x, y, z, and the time t. In the following sections we shall restrict ourselves to one spatial dimension x, so that the wave function depends solely on x. An extension to three spatial dimensions can be done easily. The wave function Y(x,t) and its complex conjugate Y * (x,t) are the focal point of quantum mechanics, because they provide a concrete meaning in the macroscopic physical world: The product Y * (x, t)dx is the probability to find a particle, for example an electron, within the interval x and x + dx. The particle is described quantum mechanically by the wave function Y(x,t). The product Y*(x,t)Y(x,t) is therefore called the window of quantum mechanics to the real world. Quantum mechanics further differs from classical mechanics by the employment of operators rather than the use of dynamical variables. Dynamical variables are used in classical mechanics, and they are variables such as position, momentum, or energy. Dynamical variables are contrasted with static variables such as the mass of a particle. Static variables do not change during typical physical processes considered here. In quantum mechanics, dynamical variables are replaced by operators which act on the wave function. Mathematical operators are mathematical expressions that act on an operand. For example, (d / dx) is the differential operator. In the expression (d / dx) Y(x,t), the differential operator acts on the wave function, Y(x,t), which is the operand. Such operands will be used to deduce the quantum mechanical wave equation or Schrödinger equation. The postulates of quantum mechanics cannot be proven or deduced. The postulates are hypothe- ses, and, if no violation with nature (experiments) is found, they are called axioms, i. e. non- provable, true statements. Postulate 1 The wave function (x,y,z,t) describes the temporal and spatial evolution of a quantummechani- cal particle. The wave function (x,t) describes a particle with one degree of freedom of motion. Postulate 2 The product Y*(x,t)Y(x,t) is the probability density function of a quantum-mechanical particle. Y*(x,t)Y(x,t)dx is the probability to find the particle in the interval between x and x + dx. Therefore, (1.14 )  -¶ +¶ Y*x, tY x, t„ x = 1 If a wave function (x, t) fulfills Eq. (2.14), then (x, t) is called a normalized wave function. Equation (2.14) is the normalization condition and implies the fact that the particle must be located somewhere on the x axis. Postulate 3 The wave function (x, t) and its derivative (/ x) (x, t) are continuous in an isotropic medium. (1.15 ) Limit  x, t, x Ø x 0 =  x 0 , t (1.16 ) Limit ∑ ∑x  x, t, x Ø x 0 = ∑ ∑x  x, t x=x 0 SSED Le Tuan Chapters 2.nb 5 In other words, (x, t) is a continuous and continuously differentiable function throughout isotropic media. Furthermore, the wave function has to be finite and single valued throughout position space (for the one-dimensional case, this applies to all values of x). Postulate 4 Operators are substituted for dynamical variables. The operators act an the wave function (x, t). In classical mechanics, variables such as the position, momentum, or energy are called dynami- cal variables. In quantum mechanics operators rather than dynamical variables are employed. Table 2.1 shows common dynamical variables and their corresponding quantummechanical operators Table 2.1. Common physical dynamic variables and quantummechanic operatotors Dynamic variables Operators in positionspace in momentumspace Position, x x  — i  p x  i—  p x Function of position, f x poteial energy, U x f x;Ux f  — i  p x   f i—  p x  Momentum, p x i—  x  — i  x p x Function of momentum, f p x  f  i—  x   f  — i  x  f p x  Kinetic energy, 1 2 mv 2  — 2 2m  2 x 2 p x 2 2m Total energy, E — i  t — i  t Total energy, E  — 2 2m  2 x 2 U x p x 2 2m  U  — i  p x  Postulate 5 The expectation value, , of any dynamical variable , is calculated from the wave function according to (1.17 ) =  -¶ +¶x Y * x, tx ` Y x, t„ x =<Y * Ï x ` ÏY> 6 SSED Le Tuan Chapters 2.nb 1.3.2 Some properties of quantummechanical operators Eigenfunctions and eigenvalues Any mathematical rule which changes one function into some other function is called an opera- tion. Such an operation requires an operator, which provides the mathematical rule for the opera- tion, and an operand which is the initial function that will be changed under the operation. Quan- tum mechanical operators act on the wave function Y(x, t). Thus, the wave function Y(x, t) is the operand. Examples for operators are the differential operator (d / dx) or the integral operator  …dx. In the following sections we shall use the symbol x ` for an operator and the symbol f(x) for an operand. The definition of the eigenfunction and the eigenvalue of an operator is as follows: If the effect of an operator x ` operating on a function f(x) is that the function f(x) is modified only by the multiplication with a scalar, then the function f(x) is called the eigenfunction of the operator x ` , that is (1.18 ) x ` Yx, t =l s Yx, t where l s is a scalar (constant). l s is called the eigenvalue of the eigenfunction. For example, the eigenfunctions of the differential operator are exponential functions, because (1.19 ) „ „ x e l s x =l s e l s x where l s is the eigenvalue of the exponential function and the differential operator. Linear operators and commutation law Virtually all operators in quantum mechanics are linear operators. An operator is a linear operator if (1.20 ) x ` c Yx, t = c x ` Yx, t where c is a constant. For example d / dx is a linear operator, since the constant c can be exchanged with the operator d / dx. On the other hand, the logarithmic operator (log) is not a linear operator, as can be easily verified. In classical mechanics, dynamical variables obey the commuta- tion law. For example, the product of the two variables position and momentum commutes, that is (1.21 ) xp x = p x x However, in quantum mechanics the two linear operators, which correspond to x and p, do not commute, as can be easily shown. One obtains (1.22 ) x ` p ` Yx, t = x  Ñ i ∑ ∑x Yx, t and alternatively (1.23 ) p ` x ` Yx, t = Ñ i ∑ ∑ x x Yx, t  = Ñ i Yx, t+ Ñ i x ∑ ∑ x Yx, t Linear operators do not commute, since the result of Eqs. (2.22) and (2.23) are different. SSED Le Tuan Chapters 2.nb 7 Hermitian operators In addition to linearity, most of the operators in quantum mechanics possess a property which is known as hermiticity. Such operators are hermitian operators, which will be defined in this section. The expectation value of a dynamical variable is given by the 5th Postulate according to (2.17). The expectation value is now assumed to be a physically observable quantity such as posi- tion or momentum. Thus, the dynamical variable is real, and is identical to its complex conjugate. (1.24 ) =  -¶ +¶x Y * x, tx ` Y x, t„ x =<Y * Ï x ` ÏY> <YÏ x ` * ÏY * >=  -¶ +¶x Y x, t x ` * Y * x, t„ x =  *  Operators x ` and x ` * , which satisfy Eq. (2.24) are called hermitian operators. The definition of an hermitian operator is in fact more general than given above. In general, hermitian operators satisfy the condition (1.25 )  -¶ +¶x Y 1 * x, tx ` Y 2 x, t„ x =  -¶ +¶x Y 2 x, t x ` * Y 1 * x, t„ x where y 1 (x) and y 2 (x) may be different functions. If y 1 1(x) and y 2 (x) are identical, Eq. (2.23) simplifies into Eq. (2.24) There are a number of consequences and implications resulting from the hermiticity of an operator. Two more properties of hermitian operators will explicitly mentioned. First, eigenvalues of hermitian operators are real. To prove this, suppose x ` is an hermitian operator with eigenfunc- tion y(x) and eigenvalue l. Then and also due to hermiticity of the operator Since Eqs. (4.17) and (4.19) are identical, therefore l = l*, which is only true if l is real. Thus, eigenvalues of Hermitian operators are real. Second, eigenfunctions corresponding to two unequal eigenvalues of an hermitian operator are orthogonal to each other. This is, if x ` is an hermitian operator and y 1 (x) and y 2 (x) are eigenfunc- tions of this operator and l 1 and l 2 are eigenvalues of this operator then Time-Independent Formulation If the particle in the system under analysis has a fixed total energy E, the quantum mechanical formulation of the problem is significantly simplified. Consider the general expression for the energy expectation value in the total volume space ý as deduced from Table 2.1 and Postulate 5: (1.26 ) < E >=  ý Y * r  , t - Ñ i ∑Yr  , t ∑t „ý By the normalization requirements, one can find that <E> = E = constant, so (1.27 ) - Ñ i ∑Yr  , t ∑t = E Yr  , t Indeed, if we make a direct substitution of the last expression into the above, the desired result will be obtained 8 SSED Le Tuan Chapters 2.nb (1.28 ) < E >=  ý Y * - Ñ i ∑Y ∑t „ý = E  ý Y * Y„ý = E = constant In the result, we can have a general solusion of the form: (1.29 ) Yr  , t =Yx, y, z, t =yx, y, ze -i Et Ñ for so-called , reminiscing about the main Newtonian equation in classical mechanics: (1.30 ) p 2 2 m + Ur   = E total ó H ` Yr  , t = t ó - Ñ 2 2 m “+Ur   Yr  , t =- Ñ i ∑Yr  , t ∑t where by definition (1.31 ) “= ∑ 2 ∑ x 2 + ∑ 2 ∑ y 2 + ∑ 2 ∑z 2 After substitution the general solution into the time-dependent , we can be dealed now with the time-independent Schrödiger equation for a steady processes: (1.32 ) - Ñ 2 2 m “+Ur   yr   = E yr   where eigen value E is total energy and eigen function y(r  ) for the patrticle in the steady state. The function y(r  ) must be finite, continuous, and single-valued for all values of x, y, z and has the similar statistical meaning like Yr  , t does. 1.4 Simple Quantum Mechanics Problem Solutions 1.4.1 The infinite square-shaped quantum well The infinite square-shaped well potential is the simplest of all possible potential wells. The one- dimension infinite square well potential is illustrated in Fig. 2.3(a) and is defined as Fig. 2.3. a) Schematic illustration of the 1-D infinite square quantum well (QW). The solutions of this QW are shown in the terms of b) eigen-functions Y n xand eigen-state energies E n , and c) probability densities Y n * xY n x. SSED Le Tuan Chapters 2.nb 9 To find the stationary solutions for y n (x) and E n, we must find functions for y n (x), which satisfy the Schrödinger equation. The time-independent Schrödinger equation contains only the differential operator d / dx, whose eigenfunctions are exponential or sinusoidal functions. Since the Schrödinger equation has the form of an eigenvalue equation, it is reasonable to try only eigenfunc- tions of the differential operator. Furthermore, we assume that y n (x) = 0 for | x | > L / 2, because the potential energy is infinitely high in the barrier regions. Since the 3rd Postulate requires that the wave function be continuous, the wave function must have zero amplitude at the two potential discontinuities, that is y n (x = ±L / 2) = 0. We therefore employ sinusoidal functions and differenti- ate between states of even and odd symmetry in well region, like that (1.33 ) y n x= A cos n+1p L , n = 0, 2, 4, and x § L 2 A sin n+1p L , n = 1, 3, 5, and x § L 2 and vanishes in the outside (1.34 ) y n x= 0,n = 0, 1, 2, and x > L 2 The condition of continuity of wave functions yields the value of the constant A (1.35 ) A = 2  L By inspection of the Schrödinger equation with the determined above wave functions, one can deduce the eigen values - energy levels for the QW: (1.36 ) E n = Ñ 2 2 m  n + 1p L  2 , n = 0, 1, 2, The lowest value has the name as ground state energy (1.37 ) E 0 = Ñ 2 p 2 2 mL 2 when other levels are n-th excited state energies with n = 1, 2, 3, , respectively. The spacing between two adjacent energy levels, that is E n – E n-1 , is proportional to n. Thus, the energetic spacing between states increases with energy. The eigenstate energies are, as already mentioned, expectation values of the total energy of the respective state. Its easily to show, based on Postulate 5, that the energy of a particle in an infinite square well is purely kinetic. The particle has no potential energy. 1.4.2 The 1-D asymmetric and symmetric finite square-shaped quantum well In contrast to the infinite square well, the finite square well has barriers of finite height. The potential of a finite square well is shown in Fig. 2.4. The two barriers of the well have a different height and therefore, the structure is denoted asymmetric square well. The potential energy is constant within the three regions I, II, and III, as shown in Fig. 2.4. In order to obtain the solutions to the Schrödinger equation for the square well potential, the solutions in a constant p otential will be considered first. 10 SSED Le Tuan Chapters 2.nb [...]... the result and find the wavelength limit for each of the three series? 2 6 The de Broglie wavelength of a particle l = h/m V describes the wave-particle duality for small particles such as electrons What is the de Broglie wavelength (in Å) of an electron at 150 eV?The same question for electrons at 10 keV, which is typical of electron microscope? What are the advantage of electron microscopes compairing... (1.60 ) SSED Le Tuan Chapters 2.nb 15 Problems 2.1 Point A is at an eletrostatic potential of + 1 V relative to point B in vacuum An electron initially at rest at B moves to A What energy (expressed in J and eV) does the electron have at A? What's its velocity in m/s? 2 2 Sketch an experimental setup in which a silver electrode (work function 4.73 eV) is sealed in vacuum envelope with a second electrode... single-valley band, (ii) an anisotropic band, (iii) a band with multiple valleys, and (iv) the density of states in a semiconductor with reduced degrees of freedom such as quantum wells, quantum wires, and quantum boxes Finally the effective density of states will be calculated 1.1.1 Single-valley, spherical, and parabolic band The simplest band structure of a semiconductor consists of a single valley... which results in continuous excitation of electrons from the valence band to the conduction band, and leaving an equal number of holes in the valence band This process is balanced by recombination of the electrons in the conduction band with holes in the valence band At steady state, the net result is n = p = ni , where ni is the intrinsic carrier density The Fermi level for an intrinsic semiconductor (which... four valence electrons with the four neighboring atoms, forming four covalent bonds Figure 4 8b shows an n-type silicon, where a substitutional phosphorous atom with five valence electrons has replaced a silicon atom, and a negative-charged electron is donated to the lattice in the conduction band The phosphorous atom is called a donor Figure 4 8c similarly shows that when a boron atom with three valence... energy levels are introduced that usually lie within the energy gap A donor impurity has a donor level which is defined as being neutral if filled by an electron, and positive if empty Conversely, an acceptor level is neutral if empty and negative if filled by an electron The ionization energy for a donor (Ec - ED ) in a lattice can be obtained by replacing mo by the conductivity effective mass of electrons... D F D B where h is the number of electrons that can physically occupy the level E, and g is the number of electrons that can be accepted by the level, also called the groundstate degeneracy of the donor impurity level (g = 2) ] 4 8 Calculate the average kinetic energy of electrons in the conduction band of an n-type nondegenerate semiconductor 4 9 If a silicon sample is doped with 1016 phosphorous... the density-of-states effective mass 1.1.3 Multiple valleys At several points of the Brillouin zone, several equivalent minima occur For example, eight equivalent minima occur at the L-point as schematically shown in Fig 4.4 Each of the valleys can accommodate carriers, since the minima occur at different kx , k y , and kz values, i e the Pauli principle is not violated The density of states is thus... wave-like nature of the electrons, a diffraction pattern characteristic of the first few atomic layers is observed on the screen if the surface is flat and the material is crystalline With a distance between atomic planes of d = 5 Å, a glancing angle of 1o , and an operating de Broglie wavelength for the electrons of 2dsinU, compute the electron energy employed in the technique 16 SSED Le Tuan Chapters 2.nb... valence electrons substitutes for a silicon atom, a positivecharged hole is created in the valence band, and an additional electron will be accepted to form four covalent bonds around the boron This is p-type, and the boron is an acceptor Fig 4 8 Three basic bond pictures of a semiconductor: a) intrisic Si with no impurity; b) n-type Si with donor (P); c) p-type Si with acceptor (B) 12 SSED Le Tuan Chapter . scale. Moreover, the quantization of angular momentum in the Bohr model clearly extended the quantum concept, seemingly suggesting a general quantization of atomic-scale observables. 2 SSED Le Tuan. and find the wavelength limit for each of the three series? 2. 6 The de Broglie wavelength of a particle l = h/m V describes the wave-particle duality for small particles such as electrons. What. of d = 5 Å, a glancing angle of 1 o , and an operating de Broglie wavelength for the electrons of 2dsinU, compute the electron energy employed in the technique. SSED Le Tuan Chapters 2.nb 15 2.14