* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quantum Mechanics
Quantum teleportation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Tight binding wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Elementary particle wikipedia , lookup
Atomic orbital wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Identical particles wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Coherent states wikipedia , lookup
Canonical quantization wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Renormalization wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum state wikipedia , lookup
Path integral formulation wikipedia , lookup
Atomic theory wikipedia , lookup
Hidden variable theory wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Dirac equation wikipedia , lookup
EPR paradox wikipedia , lookup
Renormalization group wikipedia , lookup
Schrödinger equation wikipedia , lookup
Hydrogen atom wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Double-slit experiment wikipedia , lookup
Electron scattering wikipedia , lookup
Probability amplitude wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Wave function wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Particle in a box wikipedia , lookup
Wave–particle duality wikipedia , lookup
Matter wave wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Quantum Mechanics V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. [email protected] October 12, 2008 Who can use this? The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the first-semester (September 2008 - January 2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of studying this: this chapter is worth 20 marks in the final exam! Cheers ;-) Syllabus as prescribed by VTU Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schrödinger wave equation, Eigen values and eigen function, Application of Schrödinger wave equation : Energy eigen values for a free particle, Energy eigen values of a particle in a potential well of infinite depth. Reference • Arthur Beiser, Concepts of Modern Physics, 6th Edition, Tata McGraw-Hill Publishing Company Limited, ISBN- 0-07-049553-X. ———————————— This document is typeset in Free Software LATEX2e distributed under the terms of the GNU General Public License. 1 www.satheesh.bigbig.com/EnggPhy 1 2 Introduction Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to provide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric effect and Compton effect. The term quantum mechanics was first coined by Max Born in 1924. Within the field of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the electron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quantum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic. 2 Heisenberg’s uncertainty principle We know from the wave-particle duality that every particle has wave-like properties. These wave properties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4π. That is h . (1) ∆x · ∆px ≥ 4π This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; h , 4π h ∆t · ∆E ≥ . 4π ∆θ · ∆l ≥ Sometimes 2.1 h 2π (2) (3) is written as ~. In that case the right had side of the uncertainty relations will have ~2 . Explanation of uncertainty principle using gamma ray microscope We use a hypothetical experiment of observing an electron using gamma ray microscope to illustrate the uncertainty principle. Suppose, we look at an electron using light of wavelength λ. Each photon of this light has the momentum h/λ. When one of these photons bounces off the electron, the electron’s original momentum will be changed. The exact amount of the change ∆p cannot be predicted, but it will be of the same order of magnitude as the photon momentum h/λ. Hence ∆p ≈ h/λ. The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s momentum. www.satheesh.bigbig.com/EnggPhy 3 Because light is a wave phenomenon as well as a particle phenomenon, we cannot expect to determine the electron’s location with perfect accuracy regardless of the instrument used. A reasonable estimate of the minimum uncertainty in the measurement might be one photon wavelength, so that ∆x ≥ λ. The shorter the wavelength, the smaller the uncertainty in location. However, if we use light of short wavelength to increase the accuracy of the position measurement, there will be a corresponding decrease in the accuracy of the momentum measurement because the higher photon momentum will disturb the electron’s motion to a greater extent. Light of long wavelength will give a more accurate momentum but a less accurate position. Combining the above results gives us ∆x · ∆p ≥ λ. This agrees well with the uncertainty principle. 2.2 Physical significance of uncertainty principle The uncertainty principle is based on the assumption that a moving particle is associated with a wave packet, the extension of which in space accounts for the uncertainty in the position of the particle. The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of the finite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracy of measurement but due to the inherent uncertainties in determining the quantities involved. Even though, the uncertainty principle prevents us from knowing the precise position and momentum, we can define the position where the probability of finding the particle is maximum and also the most probable momentum of the particle. That means, the uncertainty principle introduces the probabilistic interpretation of the physical quantities. This is the major difference between the classical physics and quantum mechanics. 2.3 Application of uncertainty principle The uncertainty principle has far reaching implications. In fact, it has been very useful in explaining many observations which cannot be explained otherwise. An important one being the proof of the non-existence of an electron inside the nucleus. In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius of a typical atomic nucleus to be about 5.0 × 10−15 m. Assuming that the uncertainty in the position of the electron inside the nucleus to be of the same order, we have ∆x = 5.0 × 10−15 m. The corresponding uncertainty in the momentum is, ∆px ≥ 1 h · , 4π ∆x 6.63 × 10−34 1 · , 4 × 3.14 5.0 × 10−15 ∆px ≥ 1.1 × 10−20 kg ms−1 . ∆px ≥ If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable in magnitude. An electron with such a momentum has a kinetic energy, KE, many times greater than its rest energy (which is mc2 ). The kinetic energy of such particle is given by KE = pc ≥ (1.1 × 10−20 ) × (3 × 108 ) www.satheesh.bigbig.com/EnggPhy 4 KE ≥ 3.3 × 10−12 J KE ≥ 20 M eV This means that the kinetic energy of an electron must exceed 20 M eV if it is to be inside a nucleus. Experiments show that the electrons emitted by certain unstable nuclei never have more than a small fraction of this energy, from which we conclude that nuclei cannot contain electrons. The electron that an unstable nucleus may emit comes into being only at the moment the nucleus decays. 3 Wave function In quantum mechanics, because of the wave-particle duality, the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over all of space. This is called a wave function. The wave function is usually complex and is represented by Ψ. Since the wave function is complex, its direct measurement in any physical experiment is not possible. It is just mathematical function of x, t etc. Once the wave function corresponding to a system is known, the state of the system can be determined. The physical state of system is completely characterized by a wave function. 3.1 Physical significance of a wave function The wave function contains information about the system it represents. Even though the wave function itself is not directly an observable quantity, the square of the absolute value of the wave function gives the probability of finding the particle at a given space and time. This probabilistic interpretation of wave function was given by Max Born in 1926. If Ψ is the wave function associated with a particle, the |Ψ|2 is the probability per unit volume that the particle will be found at the given point. The probability density is given by |Ψ|2 = Ψ · Ψ∗ where Ψ∗ is the complex conjugate of Ψ. For a particle restricted to move only long x− axis, the probability of finding it between x1 and x2 is given by Z x2 |Ψ|2 dx. x1 Since the probability of finding a particle any where in a given voluve must be one, we have Z +∞ |Ψ|2 dV = 1. −∞ This condition is know as normalization. 3.2 Properties of a wave function A wave function has the following characteristics. 1. Ψ must be continuous and single-valued everywhere. 2. ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be continuous and single-valued everywhere. 3. Ψ must be normalizable. www.satheesh.bigbig.com/EnggPhy 4 5 Time independent Schrödinger wave equation in one dimension In quantum mechanics, the Schrdinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton’s laws are to classical mechanics. The equation is named after Erwin Schrdinger, who discovered it in 1926. Consider a wave function of an arbitrary particle Ψ(x, t) = Ae−i(ωt−kx) . (4) Using the definitions of ω and k, we write the following ω = 2πν and 2π . λ From Planck’s law we have E = hν and substituting in the ω equation k= ω = 2π E E E = = . h h/2π ~ From de Bbroglie’s equation, we have λ = h/p and substituting in the k equation k= p p 2π = = . h/p h/2π ~ Now, we substitute the new expressions for ω and k in the equation of the wave function. This gives us −i Ψ(x, t) = Ae ~ (Et−px) . We re-write the wave function with separate space and time parts Ψ(x, t) = Ae −iEt ~ Ψ(x, t) = φe where φ = Ae ·e −iEt ~ ipx ~ ipx ~ , , . (5) (6) Differentiating the function φ with respect to x twice, ipx ∂φ ip = · Ae ~ ∂x ~ ipx ∂2φ ip ip = · · Ae ~ , 2 ∂x ~ ~ that is ∂2ψ −p2 = φ, ∂x2 ~2 From here, we can write p2 φ = −~2 ∂2φ ∂x2 (7) www.satheesh.bigbig.com/EnggPhy 6 The total energy E of a particle of kinetic energy p2 2m and potential energy U is given by, p2 +U 2m We multiply Ψ to both the sides of the above equation, to get, E =T +U = EΨ = p2 Ψ + UΨ 2m Substituting for Ψ, we get −iEt p2 φ −iEt e ~ + U φe ~ 2m Now inserting for p2 φ from previous equations, we get Eφe Eφe −iEt ~ −iEt ~ = = −iEt −~2 ∂ 2 φ −iEt e ~ + U φe ~ 2 2m ∂x Taking all the terms to the left hand side Eφe −iEt ~ + −iEt ~2 ∂ 2 φ −iEt e ~ − U φe ~ = 0. 2 2m ∂x Rearranging the terms −iEt −iEt ~2 ∂ 2 φ −iEt e ~ + Eφe ~ − U φe ~ = 0, 2 2m ∂x −iEt ~2 ∂ 2 φ −iEt e ~ + (E − U )φe ~ = 0. 2 2m ∂x 2m Multiplying throughout by ~2 , we get −iEt ∂ 2 φ −iEt 2m e ~ + 2 (E − U )φe ~ = 0. 2 ∂x ~ Now we absorb e equation. −iEt ~ into the partial differential operator in the first term as it does not affect the ∂ 2 φe −iEt ~ ∂x2 Using the relation Ψ = φe −iEt ~ + −iEt 2m (E − U )φe ~ = 0. 2 ~ ,, we get ∂ 2 Ψ 2m + 2 (E − U )Ψ = 0. (8) ∂x2 ~ This is the time-independent form of the Schrödinger wave equation in one-dimension. This is also known as Schrödinger’s steady-state equation. 4.1 Eigenvalues and eigenfunctions Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as eigenvalues of the observable and the corresponding wave functions are called eigenfunctions. The eigenfunctions are those eigenfunctions which are definite and single valued. When something is in the condition of being definitely ‘pinned-down’, it is said to possess an eigenvalue. For example, if the position of an electron has been made definite, it is said to have an eigenvalue of position. The term eigen can be roughly translated from German as inherent or as a characteristic). The German word ”eigen” was first used in this context by the mathematician David Hilbert in 1904. www.satheesh.bigbig.com/EnggPhy 4.2 4.2.1 7 Applications of Schrödinger wave equation Energy eigen values for a free particle The time-independent form of the Schrödinger wave equation in one-dimension is given by, ∂ 2 Ψ 2m + 2 (E − U )Ψ = 0. ∂x2 ~ A free particle is defined as the one which is not acted upon by any external force that modifies its motion. Hence the potential energy U in the Schrödinger equation is taken to be zero. That is, ∂ 2 Ψ 2m + 2 EΨ = 0. ∂x2 ~ where E is the total energy of the particle and is purely in the form of kinetic energy. The general solution of such a differential equation is of the form ! ! √ √ 2mE 2mE x + B cos x Ψ = A sin ~ ~ Its difficult to solve for constants A and B as we cannot impose any boundary conditions on the free √ 2mE particle. Since the solution has not imposed any restriction on the constant ~ which we call k, the free particle is permitted to have any value of energy given by E= ~2 k 2 2m The Schrdinger equation, applied to the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. 4.2.2 Energy eigen values of a particle in a potential well of infinite depth The time-independent form of the Schrödinger wave equation in one-dimension is given by, ∂ 2 Ψ 2m + 2 (E − U )Ψ = 0. ∂x2 ~ www.satheesh.bigbig.com/EnggPhy 8 Consider a particle trapped in a potential well of infinite depth and width L. A particle in this potential is completely free i.e., potential energy is zero, except at the two ends (x = 0 and x = L), where an infinite force prevents it from escaping; U = 0 for 0 ≥ x ≥ L. But within the well the particle does not lose any energy when it collides with the walls and hence the total energy of the particle remains constant. Since the article cannot exist outside the box, we have Ψ = 0 for 0 ≤ x and x ≥ L. The Schrödinger equation for such case takes the form ∂ 2 Ψ 2m + 2 EΨ = 0. ∂x2 ~ where E is the total energy of the particle and is purely in the form of kinetic energy. The general solution of such a differential equation is of the form ! ! √ √ 2mE 2mE x + B cos x Ψ = A sin ~ ~ We use the boundary conditions to find out the constants A and B. Applying the condition Ψ = 0 for x = 0, the solution becomes 0 = A sin (0) + B cos (0) This implies B = 0. Then, the solution reduces to √ Ψ = A sin 2mE x ~ ! Now, we use the second boundary condition Ψ = 0 for Then, √ A sin x = L, 2mE L ~ ! =0 If A = 0, the wavefunction will become zero irrespective of the value of x. Hence, A cannot be taken as zero. Therefore, ! √ 2mE sin L =0 ~ or √ 2mE L = nπ where n = 1, 2, 3, ..... ~ Here, n cannot be zero as it leads to trivial solution. Hence, the energy eigenvalues may be written as En = n 2 π 2 ~2 where n = 1, 2, 3, ..... 2mL2 www.satheesh.bigbig.com/EnggPhy 9 From this equation, we infer that the energy of the particle is discrete as n can have integer values. In other words, the energy is quantized. We also note that n cannot be zero because in that case, the wave function as well as the probability of finding the particle becomes zero for all values of x. The lowest energy of the particle can possess corresponds to n = 1 is given by E1 = π 2 ~2 . 2mL2 This is called the ground state energy or zero point energy. The first and second excited energies are given by 4π 2 ~2 , E2 = 2mL2 and 9π 2 ~2 . E3 = 2mL2 The energy levels are like E1 , 4E1 , 9E1 , 16E1 ....which indicates that the energy levels are not equally spaced. The eigenfunctions corresponding to the above eigenvalues are given by √ 2mEn x Ψ = A sin ~ Substituting En = n2 π 2 ~2 2mL2 in the above equation, we get nπ Ψ = A sin x L We apply the normalization condition to fix the value of A, that is Z L |Ψ|2 dx = 1, 0 Z L nπ A sin x dx = 1 L 0 or Z L nπ 2 A sin2 x dx = 1. L 0 From standard integrals, we know that Z L L 2 nπ sin x dx = . L 2 0 2 2 Hence, the above integral becomes L =1 2 r 2 A= L A2 or Now the eigenfunction becomes √ 2 2mEn Ψ= sin x L ~ r nπx 2 Ψ= sin L L r or www.satheesh.bigbig.com/EnggPhy 10 The above figure shows the variation of the wavefunction inside the infinite potential well for different values of n. The probability density is given by nπx 2 |Ψ|2 = sin2 L L This figure shows the variation of the probability densities of finding the particle at different places inside the infinite potential well for different values of n. Thus, it suggests that the probability of finding any particle at the lowest energy level is maximum at the center of the box. ***