Lecture Notes in Quantum Mechanics

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Quantum Mechanics Made Simple: Lecture Notes 1 Weng Cho CHEW October 5, 2012 1 The author is with U of Illinois, Urbana-Champaign. He works part time at Hong Kong U this summer.Chapter 1 Introduction 1.1 Introduction Quantum mechanics is an important intellectual achievement of the 20th century. It is one of the more sophisticated eld in physics that has a ected our understanding of nano-meter length scale systems important for chemistry, materials, optics, and electronics. The existence of orbitals and energy levels in atoms can only be explained by quantum mechanics. Quantum mechanics can explain the behaviors of insulators, conductors, semi-conductors, and giant magneto-resistance. It can explain the quantization of light and its particle nature in addition to its wave nature. Quantum mechanics can also explain the radiation of hot body, and its change of color with respect to temperature. It explains the presence of holes and the transport of holes and electrons in electronic devices. Quantum mechanics has played an important role in photonics, quantum electronics, and micro-electronics. But many more emerging technologies require the understanding of quantum mechanics; and hence, it is important that scientists and engineers understand quantum mechanics better. One area is nano-technologies due to the recent advent of nano- fabrication techniques. Consequently, nano-meter size systems are more common place. In electronics, as transistor devices become smaller, how the electrons move through the device is quite di erent from when the devices are bigger: nano-electronic transport is quite di erent from micro-electronic transport. The quantization of electromagnetic eld is important in the area of nano-optics and quantum optics. It explains how photons interact with atomic systems or materials. It also allows the use of electromagnetic or optical eld to carry quantum information. Moreover, quantum mechanics is also needed to understand the interaction of photons with materials in solar cells, as well as many topics in material science. When two objects are placed close together, they experience a force called the Casimir force that can only be explained by quantum mechanics. This is important for the un- derstanding of micro/nano-electromechanical sensor systems (M/NEMS). Moreover, the un- derstanding of spins is important in spintronics, another emerging technology where giant magneto-resistance, tunneling magneto-resistance, and spin transfer torque are being used. Quantum mechanics is also giving rise to the areas of quantum information, quantum 12 Quantum Mechanics Made Simple communication, quantum cryptography, and quantum computing. It is seen that the richness of quantum physics will greatly a ect the future generation technologies in many aspects. 1.2 Quantum Mechanics is Bizarre The development of quantum mechanicsis a great intellectual achievement, but at the same time, it is bizarre. The reason is that quantum mechanics is quite di erent from classical physics. The development of quantum mechanics is likened to watching two players having a game of chess, but the watchers have not a clue as to what the rules of the game are. By observations, and conjectures, nally the rules of the game are outlined. Often, equations are conjectured like conjurors pulling tricks out of a hat to match experimental observations. It is the interpretations of these equations that can be quite bizarre. Quantum mechanics equations were postulated to explain experimental observations, but the deeper meanings of the equations often confused even the most gifted. Even though Einstein received the Nobel prize for his work on the photo-electric e ect that con rmed that light energy is quantized, he himself was not totally at ease with the development of quantum mechanicsas charted by the younger physicists. He was never comfortable with the probabilistic interpretation of quantum mechanics by Born and the Heisenberg uncertainty principle: \God doesn't play dice," was his statement assailing the probabilistic interpreta- tion. He proposed \hidden variables" to explain the random nature of many experimental observations. He was thought of as the \old fool" by the younger physicists during his time. Schr odinger came up with the bizarre \Schr odinger cat paradox" that showed the struggle that physicists had with quantum mechanics's interpretation. But with today's understanding of quantum mechanics, the paradox is a thing of yesteryear. The latest twist to the interpretation in quantum mechanics is the parallel universe view that explains the multitude of outcomes of the prediction of quantum mechanics. All outcomes are possible, but with each outcome occurring in di erent universes that exist in parallel with respect to each other. 1.3 The Wave Nature of a ParticleWave Particle Dual- ity The quantized nature of the energy of light was rst proposed by Planck in 1900 to successfully explain the black body radiation. Einstein's explanation of the photoelectric e ect further 1 asserts the quantized nature of light, or light as a photon. However, it is well known that light is a wave since it can be shown to interfere as waves in the Newton ring experiment as far back as 1717. The wave nature of an electron is revealed by the fact that when electrons pass through a crystal, they produce a di raction pattern. That can only be explained by the wave nature 1 In the photoelectric e ect, it was observed that electrons can be knocked o a piece of metal only if the light exceeded a certain frequency. Above that frequency, the electron gained some kinetic energy proportional to the excess frequency. Einstein then concluded that a packet of energy was associated with a photon that is proportional to its frequency.Introduction 3 of an electron. This experiment was done by Davisson and Germer in 1927. De Broglie hypothesized that the wavelength of an electron, when it behaves like a wave, is h  = (1.3.1) p 2 where h is the Planck's constant, p is the electron momentum, and 34 h 6:626 10 Joule second (1.3.2) When an electron manifests as a wave, it is described by (z)/ exp(ikz) (1.3.3) 3 where k = 2=. Such a wave is a solution to 2 2 =k (1.3.4) 2 z A generalization of this to three dimensions yields 2 2 r (r) =k (r) (1.3.5) We can de ne p = k (1.3.6) 4 where =h=(2). Consequently, we arrive at an equation 2 2 p 2 r (r) = (r) (1.3.7) 2m 2m 0 0 where 31 m  9:11 10 kg (1.3.8) 0 2 The expression p =(2m ) is the kinetic energy of an electron. Hence, the above can be 0 considered an energy conservation equation. The Schr odinger equation is motivated by further energy balance that total energy is equal to the sum of potential energy and kinetic energy. De ning the potential energy to be V (r), the energy balance equation becomes   2 2 r +V (r) (r) =E (r) (1.3.9) 2m 0 2 Typical electron wavelengths are of the order of nanometers. Compared to 400 nm of wavelength of blue light, they are much smaller. Energetic electrons can have even smaller wavelengths. Hence, electron waves can be used to make electron microscope whose resolution is much higher than optical microscope. 3 The wavefunction can be thought of as a \halo" that an electron carries that determine its underlying physical properties and how it interact with other systems. 4 This is also called Dirac constant sometimes.4 Quantum Mechanics Made Simple where E is the total energy of the system. The above is the time-independent Schr odinger equation. The ad hoc manner at which the above equation is arrived at usually bothers a beginner in the eld. However, it predicts many experimental outcomes, as well as predicting the existence of electron orbitals inside an atom, and how electron would interact with other particles. One can further modify the above equation in an ad hoc manner by noticing that other experimental nding shows that the energy of a photon is E = . Hence, if we let i (r;t) =E (r;t) (1.3.10) t then it (r;t) =e (r;t) (1.3.11) Then we arrive at the time-dependent Schr odinger equation:   2 2 r +V (r) (r;t) =i (r;t) (1.3.12) 2m t 0 Another disquieting fact about the above equation is that it is written in complex functions and numbers. In our prior experience with classical laws, they can all be written in real functions and numbers. We will later learn the reason for this. Mind you, in the above, the frequency is not unique. We know that in classical physics, the potential V is not unique, and we can add a constant to it, and yet, the physics of the problem does not change. So, we can add a constant to both sides of the time-independent Schr odinger equation (1.3.10), and yet, the physics should not change. Then the total E on the right-hand side would change, and that would change the frequency we have arrived at in the time-dependent Schr odinger equation. We will explain how to resolve this dilemma later on. Just like potentials, in quantum mechanics, it is the di erence of frequencies that matters in the nal comparison with experiments, not the absolute frequencies. The setting during which Schrodinger  equation was postulated was replete with knowledge of classical mechanics. It will be prudent to review some classical mechanics knowledge next.Chapter 2 Classical Mechanics 2.1 Introduction Quantum mechanics cannot be derived from classical mechanics, but classical mechanics can inspire quantum mechanics. Quantum mechanics is richer and more sophisticated than classical mechanics. Quantum mechanics was developed during the period when physicists had rich knowledge of classical mechanics. In order to better understand how quantum mechanics was developed in this environment, it is better to understand some fundamental concepts in classical mechanics. Classical mechanics can be considered as a special case of quantum mechanics. We will review some classical mechanics concepts here. 1 In classical mechanics, a particle moving in the presence of potential V (q) will experience a force given by dV (q) F (q) = (2.1.1) dq where q represents the coordinate or the position of the particle. Hence, the particle can be described by the equations of motion dp dV (q) dq =F (q) = ; =p=m (2.1.2) dt dq dt For example, when a particle is attached to a spring and moves along a frictionless surface, the force the particle experiences is F (q) =kq where k is the spring constant. Then the equations of motion of this particle are dp dq =p _ =kq; =q_ =p=m (2.1.3) dt dt Givenp andq at some initial timet , one can integrate (2.1.2) or (2.1.3) to obtainp andq for 0 all later time. A numerical analysist can think of that (2.1.2) or (2.1.3) can be solved by the 1 The potential here refers to potential energy. 78 Quantum Mechanics Made Simple V(q) q q Figure 2.1: The left side shows a potential well in which a particle can be trapped. The right side shows a particle attached to a spring. The particle is subject to the force due to the spring, but it can also be described by the force due to a potential well. nite di erence method, where time-stepping can be used to nd p and q for all later time. For instance, we can write the equations of motion more compactly as dV = F(V) (2.1.4) dt t where V = p;q , and F is a general vector function of V. It can be nonlinear or linear; in the event if it is linear, then F(V) = A V. Using nite di erence approximation, we can rewrite the above as V(t + t) V(t) = tF(V(t)); V(t + t) = tF(V(t)) + V(t) (2.1.5) The above can be used for time marching to derive the future values of V from past values. The above equations of motion are essentially derived using Newton's law. However, there exist other methods of deriving these equations of motion. Notice that only two variables p and q are sucient to describe the state of a particle. 2.2 Lagrangian Formulation Another way to derive the equations of motion for classical mechanics is via the use of the Lagrangian and the principle of least action. A Lagrangian is usually de ned as the di erence between the kinetic energy and the potential energy, i.e., L(q_;q) =TV (2.2.1) where q_ is the velocity. For a xed t, q and q_ are independent variables, since q_ cannot be derived fromq if it is only known at one givent. The equations of motion is derived from the principle of least action which says that q(t) that satis es the equations of motion between two times t and t should minimize the action integral 1 2 Z t 2 S = L(q_(t);q(t))dt (2.2.2) t 1Classical Mechanics 9 Assuming that q(t ) and q(t ) are xed, then the function q(t) between t and t should 1 2 1 2 minimize S, the action. In other words, a rst order perturbation in q from the optimal answer that minimizes S should give rise to second order error in S. Hence, taking the rst variation of (2.2.2), we have   Z Z Z t t t 2 2 2 L L S = L(q_;q)dt = L(q_;q)dt = q_ +q dt = 0 (2.2.3) q_ q t t t 1 1 1 In order to take the variation into the integrand, we have to assume that L(q_;q) is taken with constant time. At constant time, q _ and q are independent variables; hence, the partial derivatives in the next equality above follow. Using integration by parts on the rst term, we have   Z Z t 2 t t 2 2 L d L L S =q q dt + q dt = 0 (2.2.4) q_ dt q_ q t t t 1 1 1 The rst term vanishes because q(t ) =q(t ) = 0 because q(t ) and q(t ) are xed. Since 1 2 1 2 q(t) is arbitrary between t and t , we have 1 2   d L L = 0 (2.2.5) dt q_ q The above is called the Lagrange equation, from which the equation of motion of a particle can be derived. The derivative of the Lagrangian with respect to the velocity q _ is the momentum L p = (2.2.6) q_ The derivative of the Lagrangian with respect to the coordinate q is the force. Hence L F = (2.2.7) q The above equation of motion is then p _ =F (2.2.8) Equation (2.2.6) can be inverted to express q _ as a function of p and q, namely q_ =f(p;q) (2.2.9) The above two equations can be solved in tandem to nd the time evolution of p and q. For example, the kinetic energy T of a particle is given by 1 2 T = mq_ (2.2.10) 2 Then from (2.2.1), and the fact that V is independent of q_, L T p = = =mq_ (2.2.11) q_ q_10 Quantum Mechanics Made Simple or p q_ = (2.2.12) m Also, from (2.2.1), (2.2.7), and (2.2.8), we have V p _ = (2.2.13) q The above pair, (2.2.12) and (2.2.13), form the equations of motion for this problem. The above can be generalized to multidimensional problems. For example, for a one particle system in three dimensions, q has three degrees of freedom, and i = 1; 2; 3. (The q i i can represent x;y;z in Cartesian coordinates, but r;; in spherical coordinates.) But for N particles in three dimensions, there are 3N degrees of freedom, and i = 1;:::; 3N. The formulation can also be applied to particles constraint in motion. For instance, forN particles in three dimensions, q may run from i = 1;:::; 3Nk, representing k constraints on the i motion of the particles. This can happen, for example, if the particles are constraint to move in a manifold (surface), or a line (ring) embedded in a three dimensional space. Going through similar derivation, we arrive at the equation of motion   d L L = 0 (2.2.14) dt q_ q i i In general,q may not have a dimension of length, and it is called the generalized coordinate i (also called conjugate coordinate). Also, q_ may not have a dimension of velocity, and it is i called the generalized velocity. The derivative of the Lagrangian with respect to the generalized velocity is the generalized momentum (also called conjugate momentum), namely, L p = (2.2.15) i q_ i The generalized momentum may not have a dimension of momentum. Hence, the equation of motion (2.2.14) can be written as L p _ = (2.2.16) i q i Equation (2.2.15) can be inverted to yield an equation for q _ as a function of the other i variables. This equation can be used in tandem (2.2.16) as time-marching equations of motion. 2.3 Hamiltonian Formulation For a multi-dimensional system, or a many particle system in multi-dimensions, the total time derivative of L is   X dL L L = q_ + q  (2.3.1) i i dt q q_ i i iClassical Mechanics 11 d Since L=q = (L=q_ ) from the Lagrange equation, we have i i dt       X X dL d L L d L = q_ + q  = q_ (2.3.2) i i i dt dt q_ q_ dt q_ i i i i i or X d L q_ L = 0 (2.3.3) i dt q_ i i The quantity X L H = q_ L (2.3.4) i q_ i i is known as the Hamiltonian of the system, and is a constant of motion, namely, dH=dt = 0. As shall be shown, the Hamiltonian represents the total energy of a system. It is a constant of motion because of the conservation of energy. The Hamiltonian of the system, (2.3.4), can also be written, after using (2.2.15), as X H = q_p L (2.3.5) i i i where p =L=q_ is the generalized momentum. The rst term has a dimension of energy, i i and in Cartesian coordinates, for a simple particle motion, it is easily seen that it is twice the kinetic energy. Hence, the above indicates that the Hamiltonian H =T +V (2.3.6) The total variation of the Hamiltonian is X H = p q_ L i i i   X X L L = (q_p +p q_ ) q + q_ (2.3.7) i i i i i i q q_ i i i i Using (2.2.15) and (2.2.16), we have X X H = (q_p +p q_ ) (p _ q +p q_ ) i i i i i i i i i i X = (q_p p _ q ) (2.3.8) i i i i i From the above, we gather that the Hamiltonian is a function of p and q . Taking the rst i i variation of the Hamiltonian with respect to these variables, we have   X H H H = p + q (2.3.9) i i p q i i i12 Quantum Mechanics Made Simple Comparing the above with (2.3.8), we gather that H q_ = (2.3.10) i p i H p _ = (2.3.11) i q i These are the equations of motion known as the Hamiltonian equations. The (2.3.4) is also known as the Legendre transformation. The original function L is a function ofq_ ,q . Hence,L depends on bothq_ andq . After the Legendre transformation, i i i i H depends on the di erential p and q as indicated by (2.3.8). This implies that H is a i i function ofp andq . The equations of motion then can be written as in (2.3.10) and (2.3.11). i i 2.4 More on Hamiltonian The Hamiltonian of a particle in classical mechanics is given by (2.3.6), and it is a function of p and q . For a non-relativistic particle in three dimensions, the kinetic energy i i p p T = (2.4.1) 2m and the potential energy V is a function of q. Hence, the Hamiltonian can be expressed as p p H = +V (q) (2.4.2) 2m in three dimensions. When an electromagnetic eld is present, the Hamiltonian for an electron can be derived by letting the generalized momentum p =mq_ +eA (2.4.3) i i i wheree is the electron charge and A is component of the vector potential A. Consequently, i the Hamiltonian of an electron in the presence of an electromagnetic eld is (peA) (peA) H = +e(q) (2.4.4) 2m The equation of motion of an electron in an electromagnetic eld is governed by the Lorentz force law, which can be derived from the above Hamiltonian using the equations of motion provided by (2.3.10) and (2.3.11). 2.5 Poisson Bracket Yet another way of expressing equations of motion in classical mechanics is via the use of Poisson bracket. This is interesting because Poisson bracket has a close quantum mechanics analogue. A Poisson bracket of two scalar variables u and v that are functions of q and p is de ned as uv vu fu;vg = (2.5.1) q p q pClassical Mechanics 13 In this notation, using (2.3.10) and (2.3.11), du u dq u dp uH uH = + = dt q dt p dt q p p q =fu;Hg (2.5.2) which is valid for any variable u that is a function of p and q. Hence, we have the equations of motion as q_ =fq;Hg; p _ =fp;Hg (2.5.3) in the Poisson bracket notation. As we shall see later, similar equations will appear in quantum mechanics. The algebraic properties of Poisson bracket are fu;vg =fv;ug (2.5.4) fu +v;wg =fu;wg +fv;wg (2.5.5) fuv;wg =fu;wgv +ufv;wg (2.5.6) fu;vwg =fu;vgw +vfu;wg (2.5.7) ffu;vg;wg +ffv;wg;ug +ffw;ug;vg = 0 (2.5.8) These properties are antisymmetry, distributivity, associativity and Jacobi's identity. If we de ne a commutator operation between two noncommuting operator u and v as u; v =u v v u; (2.5.9) it can be shown that the above commutator have the same algebraic properties as the Pois- son bracket. An operator in quantum mechanics can be a matrix operator or a di erential operator. In general, operators do not commute unless under very special circumstances.Chapter 3 Quantum MechanicsSome Preliminaries 3.1 Introduction With some background in classical mechanics, we may motivate the Schr odinger equation in a more sanguine fashion. Experimental evidence indicated that small particles such as electrons behave quite strangely and cannot be described by classical mechanics alone. In classical mechanics, once we knowp andq and their time derivatives (orp _,q_) of a particle at time t , one can integrate the equations of motion 0 p _ =F; q_ =p=m (3.1.1) or use the nite di erence method to nd p and q at t + t, and at all subsequent times. 0 In quantum mechanics, the use of two variablesp andq and their derivatives is insucient to describe the state of a particle and derive its future states. The state of a particle has to be more richly endowed and described by a wavefunction or state function (q;t). The state function (also known as a state vector) is a vector in the in nite dimensional space. At this juncture, the state function or vector is analogous to when we study the control theory of a highly complex system. In the state variable approach, the state of a control system is described by the state vector, whose elements are variables that we think are important to capture the state of the system. For example, the state vector V, describing the state of the factory, can contain variables that represent the number of people in a factory, the number of machines, the temperature of the rooms, the inventory in each room, etc. The state equation of this factory can then be written as d V(t) = A V(t) (3.1.2) dt It describes the time evolution of the factory. The matrix A causes the coupling between state variables as they evolve. It bears strong similarity to the time-dependent Schr odinger 1516 Quantum Mechanics Made Simple Figure 3.1: The state of a particle in quantum mechanics is described by a state function, which has in nitely many degrees of freedom. equation, which is used to describe the time-evolution of the state function or the wavefunction of an electron. In the wavefunction, a complex number is assigned to each location in space. In the Schr odinger equation, the wavefunction (q;t) is a continuous function of of the position variable q at any time instant t; hence, it is described by in nitely many numbers, and has in nite degrees of freedom. The time evolution of the wavefunction (q;t) is governed by the Schr odinger equation. It was motivated by experimental evidence and the works of many others such as Planck, Einstein, and de Broglie, who were aware of the wave nature of a particle and the dual wave-particle nature of light. 3.2 Probabilistic Interpretation of the wavefunction The wavefunction of the Schr odinger equation has de ed an acceptable interpretation for many years even though the Schr odinger equation was known to predict experimental out- comes. Some thought that it represented an electron cloud, and that perhaps, an electron, at the atomistic level, behaved like a charge cloud, and hence not a particle. The nal, most accepted interpretation of this wavefunction (one that also agrees with experiments) is that its magnitude squared corresponds to the probabilistic density function. In other words, the probability of nding an electron in an interval x;x + x is equal to 2 j (x;t)j x (3.2.1) For the 3D case, the probability of nding an electron in a small volume V in the vicinity of the point r is given by 2 j (r;t)j V (3.2.2) Since the magnitude squared of the wavefunction represents a probability density function, it must satisfy the normalization condition of a probability density function, viz., Z 2 dVj (r;t)j = 1 (3.2.3)Quantum MechanicsSome Preliminaries 17 with its counterparts in 1D and 2D. The magnitude squared of this wavefunction is like some kind of \energy" that cannot be destroyed. Electrons cannot be destroyed and hence, charge conservation is upheld by the Schr odinger equation. Motivated by the conservation of the \energy" of the wavefunction, we shall consider an \energy" conserving system where the classical Hamiltonian will be a constant of motion. In this case, there is no \energy" loss from the system. The Schr odinger equation that governs the time evolution of the wavefunction is d H =i (3.2.4) dt 1 where H is the Hamiltonian operator. One can solve (3.2.4) formally to obtain H i t (t) =e (t = 0) (3.2.5) Since the above is a function of an operator, it has meaning only if this function acts on the eigenvectors of the operator H. It can be shown easily that if A V = V , i i i exp(A) V = exp( )V (3.2.6) i i i If H is a Hermitian operator, then there exists eigenfunctions, or special wavefunctions, , such that n H =E (3.2.7) n n n where E is purely real. In this case, the time evolution of from (3.2.5) is n n E n i t i t n (t) =e (t = 0) =e (t = 0) (3.2.8) n n n In the above, E = , or the energy E is related to frequency via the reduced Planck n n n n constant . The reduced Planck constant is related to the Planck constant by = h=(2) 34 and h = 6:626068 10 J s. Scalar variables that are measurable in classical mechanics, such as p andq, are known as observables in quantum mechanics. They are elevated from scalar variables to operators in quantum mechanics, denoted by a \" symbol here. In classical mechanics, for a one particle system, the Hamiltonian is given by 2 p H =T +V = +V (3.2.9) 2m The Hamiltonian contains the information from which the equations of motion for the particle can be derived. But in quantum mechanics, this is not sucient, andH becomes an operator 2 p H = +V (3.2.10) 2m d 1 Rightfully, one should use the bra and ket notation to write this equation as Hj i =i j i. In the less dt rigorous notation in (3.2.4), we will assume that H is in the representation in which the state vector is in. That is if is in coordinate space representation, H is also in coordinates space representation.18 Quantum Mechanics Made Simple This operator works in tandem with a wavefunction to describe the state of the particle. The operator acts on a wavefunction (t), where in the coordinateq representation, is (q;t). When (q;t) is an eigenfunction with energy E , it can be expressed as n i t n (q;t) = (q)e (3.2.11) n n 2 where E = . The Schr odinger equation for (q) then becomes n n n   2 p H (q) = +V (q) =E (q) (3.2.12) n n n n 2m For simplicity, we consider an electron moving in free space where it has only a constant kinetic energy but not in uenced by any potential energy. In other words, there is no force acting on the electron. In this case, V = 0, and this equation becomes 2 p (q) =E (q) (3.2.13) n n n 2m It has been observed by de Broglie that the momentum of a particle, such as an electron which behaves like a wave, has a momentum p = k (3.2.14) where k = 2= is the wavenumber of the wavefunction. This motivates that the operator p can be expressed by d p =i (3.2.15) dq in the coordinate space representation. This is chosen so that if an electron is described by a ikq state function (q) =c e , thenp (q) = k (q). The above motivation for the form of the 1 operator p is highly heuristic. We will see other reasons for the form of p when we study the correspondence principle and the Heisenberg picture. Equation (3.2.13) for a free particle is then 2 2 d (q) =E (q) (3.2.16) n n n 2 2mdq Since this is a constant coecient ordinary di erential equation, the solution is of the form ikq (q) =e (3.2.17) n which when used in (3.2.16), yields 2 2 k =E (3.2.18) n 2m Namely, the kinetic energy T of the particle is given by 2 2 k T = (3.2.19) 2m 2 Forn the Schr odinger equation in coordinate space, V turns out to be a scalar operator.Quantum MechanicsSome Preliminaries 19 where p = k is in agreement with de Broglie's nding. In many problems, the operator V is a scalar operator in coordinate space representation which is a scalar function of position V (q). This potential traps the particle within it acting as a potential well. In general, the Schr odinger equation for a particle becomes   2 2 +V (q) (q;t) =i (q;t) (3.2.20) 2 2mq t For a particular eigenstate with energy E as indicated by (3.2.11), it becomes n   2 2 d +V (q) (q) =E (q) (3.2.21) n n n 2 2mdq The above is an eigenvalue problem with eigenvalue E and eigenfunction (q). These n n eigenstates are also known as stationary states, because they have a time dependence indicated 2 by (3.2.11). Hence, their probability density functionsj (q;t)j are time independent. n These eigenfunctions correspond to trapped modes in the potential well de ned by V (q) very much like trapped guided modes in a dielectric waveguide. These modes are usually countable and they can be indexed by the index n. In the special case of a particle in free space, or the absence of the potential well, the particle or electron is not trapped and it is free to assume any energy or momentum indexed by the continuous variable k. In (3.2.18), the index for the energy should rightfully be k and the eigenfunctions are uncountably in nite. Moreover, the above can be generalized to two and three dimensional cases. 3.3 Simple Examples of Time Independent Schr odinger Equation At this juncture, we have enough knowledge to study some simple solutions of time-independent Schr odinger equation such as a particle in a box, a particle impinging on a potential barrier, and a particle in a nite potential well. 3.3.1 Particle in a 1D Box Consider the Schr odinger equation for the 1D case where the potential V (x) is de ned to be a function with zero value for 0xa (inside the box) and in nite value outside this range. The Schr odinger equation is given by   2 2 d +V (x) (x) =E (x) (3.3.1) 2 2mdx where we have replaced q withx. SinceV (x) is in nite outside the box, (x) has to be zero. Inside the well, V (x) = 0 and the above equation has a general solution of the form (x) =A sin(kx) +B cos(kx) (3.3.2)20 Quantum Mechanics Made Simple The boundary conditions are that (x = 0) = 0 and (x =a) = 0. For this reason, a viable solution for (x) is (x) =A sin(kx) (3.3.3) where k = n=a, n = 1;:::;1. There are in nitely many eigensolutions for this problem. For each chosen n, the corresponding energy of the solution is 2 (n=a) E = (3.3.4) n 2m These energy values are the eigenvalues of the problem, with the corresponding eigenfunctions given by (3.3.3) with the appropriate k. It is seen that the more energetic the electron is (high E values), the larger the number of oscillations the wavefunction has inside the box. n The solutions that are highly oscillatory have higher k values, and hence, higher momentum or higher kinetic energy. The solutions which are even about the center of the box are said to have even parity, while those that are odd have odd parity. One other thing to be noted is that the magnitude squared of the wavefunction above represents the probability density function. Hence, it has to be normalized. The normalized version of the wavefunction is p (x) = 2=a sin(nx=a) (3.3.5) Moreover, these eigenfunctions are orthonormal to each other, viz., Z a  dx (x) (x) = (3.3.6) m nm n 0 The orthogonality is the generalization of the fact that for a Hermitian matrix system, where the eigenvectors are given by H V = V (3.3.7) i i i then it can be proven easily that y V  V =C  (3.3.8) i j ij j Moreover, the eigenvalues are real. Figure 3.2: The wavefunctions of an electron trapped in a 1D box (from DAB Miller).Quantum MechanicsSome Preliminaries 21 3.3.2 Particle Scattering by a Barrier In the previous example, it is manifestly an eigenvalue problem since the solution can be found only at discrete values of E . The electron is trapped inside the box. However, in an n open region problem where the electron is free to roam, the energy of the electron E can be arbitrary. We can assume that the potential pro le is such that V (x) = 0 for x 0 while V (x) =V for x 0. The energy of the electron is such that 0 E V . On the left side, 0 0 we assume an electron coming in from1 with the wavefunction described by A exp(ik x). 1 When this wavefunction hits the potential barrier, a re ected wave will be present, and the general solution on the left side of the barrier is given by ik x ik x 1 1 (x) =A e +B e (3.3.9) 1 1 1 2 where (k ) =(2m) = E is the kinetic energy of the incident electron. On the right side, 1 however, the Schr odinger equation to be satis ed is   2 2 d (x) = (EV ) (x) (3.3.10) 2 0 2 2 2mdx The solution of the transmitted wave on the right is ik x 2 (x) =A e (3.3.11) 2 2 where p k = 2m(EV )= (3.3.12) 2 0 Given the known incident wave amplitudeA , we can match the boundary conditions atx = 0 1 to nd the re ected wave amplitude B and the transmitted wave amplitude A . By eye- 1 2 balling the Schr odinger equation (3.3.1), we can arrive at the requisite boundary conditions are that and =x are continuous at x = 0. SinceE V ,k is pure imaginary, and the wave is evanescent and decays when x1. 0 2 This e ect is known as tunneling. The electron as a nonzero probability of being found inside the barrier, albeit with decreasing probability into the barrier. The larger V is compared to 0 E, the more rapidly decaying is the wavefunction into the barrier. However, if the electron is energetic enough so that E V , k becomes real, and then 0 2 the wavefunction is no more evanescent. It penetrates into the barrier; it can be found even a long way from the boundary. It is to be noted that the wavefunction in this case cannot be normalized as the above represents a ctitious situation of an electron roaming over in nite space. The above example illustrates the wave physics at the barrier. 3.3.3 Particle in a Potential Well If the potential pro le is such that 8 V if xa=2, region 1 1 V (x) = V = 0 ifjxja=2, region 2 (3.3.13) 2 : V if xa=2, region 3 322 Quantum Mechanics Made Simple Figure 3.3: Scattering of the electron wavefunction by a 1D barrier (from DAB Miller). then there can be trapped modes (or states) inside the well represented by standing waves, whereas outside the well, the waves are evanescent for eigenmodes for which E V and 1 E V . 3 The wavefunction forjxja=2 can be expressed as (x) =A sin(k x) +B cos(k x) (3.3.14) 2 2 2 2 2 p where k = 2mE=. In region 1 to the left, the wavefunction is 2 x 1 (x) =A e (3.3.15) 1 1 p where = 2m(V E)=. The wave has to decay in the left direction. Similar, in region 1 1 3 to the right, the wavefunction is x 3 (x) =B e (3.3.16) 3 3 p where = 2m(V E)=. It has to decay in the right direction. Four boundary conditions 3 3 can be imposed at x =a=2 to eliminate the four unknowns A , A , B , and B . However, 1 2 2 3 non-trivial eigensolutions can only exist at selected values of E which are the eigenvalues of the Schr odinger equation. The eigenequation from which the eigenvalues can be derived is a transcendental equation. To illustrate this point, we impose that is continuous at x =a=2 to arrive at the following two equations: a=2 1 A e =A sin(k a=2) +B cos(k a=2) (3.3.17) 1 2 2 2 2 a=2 3 A e =A sin(k a=2) +B cos(k a=2) (3.3.18) 3 2 2 2 2 We further impose that =x is continuous at x =a to arrive at the following two equa- tions: a=2 1 A e =k A cos(k a=2) +k B sin(k a=2) (3.3.19) 1 1 2 2 2 2 2 2 a=2 3 A e =k A cos(k a=2)k B sin(k a=2) (3.3.20) 3 3 2 2 2 2 2 2

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