Appendix E - Elements of Quantum Mechanics
|
|
- Juniper Ellis
- 7 years ago
- Views:
Transcription
1 1 Appendix E - Elements of Quantum Mechanics Quantum mechanics provides a correct description of phenomena on the atomic or sub- atomic scale, where the ideas of classical mechanics are not generally applicable. As we describe nuclear phenomena, we will use many results and concepts from quantum mechanics. While it is our goal not to have the reader, in general, perform detailed quantum mechanical calculation, it is important that the reader understand the basis for many of the descriptive statements made in the text. Therefore, we present, in this Appendix, a brief summary of the essential features of quantum mechanics that we shall use. For more detailed discussion of these features, we refer the reader to the references at the end of this Appendix. E- 1 Wave Functions All the knowable information about a physical system (i.e., energy, angular momentum, etc.) is contained in the wave function of the system. We shall restrict our discussion to one- body systems for the present. (We could easily generalize to many body systems). The wave function can be expressed in terms of space coordinates and time or momenta and time. In the former notation we write, ψ (x, y, z, t) or just ψ (E- 1)
2 2 These wave functions, must be well- behaved, i.e., they (and their derivatives with respect to the space coordinates), must be continuous, finite and single- valued. The functions Ψ are solutions to a second order differential equation called the Schrödinger equation (see below). The probability of finding a particle within a volume element dxdydz, W dxdydz, is given by W dx dy dz = ψ* ψ dx dy dz (E- 2) where ψ* is the complex conjugate of ψ. (To form the complex conjugate of any complex number, replace all occurrences of i (where i = ) with - i. Real numbers are their own complex conjugates. 6-5i is the complex conjugate of 6+5i. So (a+ib)*(a+ib) = (a- ib)(a+ib) = a 2 + b 2.) The probability per unit volume (the probability density) is W = ψ*ψ. If we look everywhere in the system, we must find the particle so that ψ* ψ dτ = 1 (E- 3) where dτ is a volume element dx dy dz. Wave functions possessing this numerical property are said to be normalized. If the value of some physical quantity P is a function of the position coordinates, the average or expectation value of P is given by < P> = ψ*pψdτ (E- 4)
3 3 This expectation value represents the average outcome of a large number of measurements. E- 2 Operators Often we must compute values of quantities that are not simple functions of the space coordinates, such as the y component of the momentum, py, where equation E- 4 is not applicable. To get around this, we say that corresponding to every classical variable, there is a quantum mechanical operator. An operator is a symbol that directs us to do some mathematical operation. For example, the momentum operators are (E- 5) while the total energy operator is given as (E- 6) Thus, to calculate the expectation value of the x- component of the momentum, px, we write (E- 7)
4 4 Similarly, the classical expression for the kinetic energy is T=p 2 /2m (E- 8) which, translated to quantum mechanics terms, means the kinetic energy operator,, is, in Cartesian coordinates, (E- 9) or, using the Laplacian operator, 2 (E- 10) where 2 is (E- 11) E- 3 The Schrödinger Equation In 1926, Schrödinger found that behavior on the atomic or subatomic scale was correctly described by a differential equation of the form
5 5 (E- 12) where V represents the potential energy and ψ the wave function of the system. Substituting from equation (E- 6), we can write (E- 13) This equation is an example of a general class of equations called eigenvalue equations of the form Ωψ = ωψ where Ω is an operator and ω is the value of an observable corresponding to that operator. (The mathematical expression ψ is referred to as an eigenfunction of the operator Ω). To use the Schrödinger equation to gain information about a physical system, we must perform a set of steps that are as follows: (a) Specify the potential energy function of the system, i.e., specify the forces acting (Section 1.6.1). (b) Find a mathematical function, ψ, which is a solution to the differential equation, the Schrödinger equation. (c) Of the many functions that satisfy the equation, reject those that do not conform to certain physical constraints on the system, known as boundary conditions.
6 6 Before illustrating this procedure for several cases of interest to nuclear chemists, we can point out another important property of the Schrödinger equation. If the potential energy V is independent of time, we can separate the space and time variables in the Schrödinger equation by setting Ψ(x,y,z,t) = ψ(x,y,z) τ(t) (E- 13b) Substituting this expression into equation E- 13, and simplifying, we have (E- 14) The only way this equation can be true is for both sides to equal a constant. If we call this separation constant E, we can write (E- 15) and (E- 16) Equation E- 15 is the time independent Schrodinger equation. The solution to equation E- 16 is (E- 17)
7 7 Using the Euler relation (e iθ = cos θ + i sin θ), we can write τ(t) = cos ωt - i sin ωt (E- 18) where τ(t) is a periodic function with angular frequency ω = E/h. The separation constant E can be shown to be the total energy, i.e. the sum of the kinetic and potential energies, T + V. E- 4 The Free Particle To illustrate how the Schrödinger equation might be applied to a familiar situation, consider the case of a free particle, i.e., a particle moving along at a constant velocity with no force acting on the particle (V=0). (Figure E- 1) For simplicity, let us consider motion in one dimension, the x- direction. For the time independent Schrödinger equation, we have (E- 19) or (E- 20)
8 8 where the constant k is given by (E- 21) The allowed values of the energy, E, are (Equation E- 21) (E- 22) where k can assume any value (E is not quantized). Since V=0, E is the kinetic energy of a particle with momentum p = hk. From de Broglie, we know that (E- 23) so that we can make the association that The solution for the Schrödinger equation, including the time- dependent part is (E- 24)
9 9 where k and ω are given (E- 21) as (E- 25) (E- 26) This solution is the equation for a wave traveling to the right (+x direction, the first term) and to the left (- x direction, second term). We can impose a boundary condition, namely, we can specify the particle is traveling in the +x direction. Then we have (E- 27) We can now calculate the values of any observable. For example, to calculate the value of the momentum p, we write (see equation E- 7) (E- 28) which agrees, of course, with the classical result. E- 5 Particle in a Box (One Dimension)
10 10 Continuing our survey of some simple applications of wave mechanics to problems of interest to the nuclear chemist, let us consider the problem of a particle confined to a one- dimensional box (Figure E- 2). This potential is flat across the bottom of the box and then rises at the walls. This can be expressed as: V(x) = 0 0 x L (E- 29) V(x) = x < 0, x > L The particle moves freely between 0 and L but is excluded from x < 0 and x> L. Inside the box, the Schrödinger equation has the form of equation E- 19 (the free particle). The time independent solution can be written ψ(x) = A sin kx + B cos kx (E- 30) But we know that ψ(x) = 0 at x = 0 and x = L. Thus B must be 0 and A sin kl = 0 (E- 31) To have sin kl = 0, we must have kl = nπ n = 1, 2, 3 (E- 32) and, using the result (E- 22), we have
11 11 (E- 33) In this case, the energy is quantized. Only certain values of the energy are allowed. One can show the normalization condition is satisfied if (E- 34) The allowed energy levels, the probability densities and the wave functions are shown for the first few levels of this potential in Figure E- 3. Sample Problem: Suppose a neutron is confined to a box that is the size of a nucleus, m. (a) What is the energy of the first excited state? (b) What is the probability of finding the neutron within a region corresponding to 20% of the width of the box, i.e., between 0.4 x m and 0.6 x m in the fourth excited state? Solution: (a) Eo (the energy of the ground state) = = 3.3 x J= 2.0 MeV
12 12 The energy of the first excited state, n=2, will be 4Eo and the energy spacing between the first excited state and the ground state will be 3Eo = 6 MeV. (b) Probability = which is the result obtained by inspection of the ψ 2 curve in Figure E- 3. E- 6 The Linear Harmonic Oscillator (One Dimension) One of the classic problems of quantum mechanics that is very important for our study of nuclei is the harmonic oscillator. For a simple harmonic oscillator, the restoring force is proportional to the distance from the center, i.e., F = - kx, so that V(x) = kx 2 /2. The Schrödinger equation is (E- 35) The solution of this equation is mathematically complicated and leads to wave functions of the form where (E- 36)
13 13 (E- 37) (the oscillator frequency) with a normalization constant of (E- 38) The expression Hn (β) is the nth Hermite polynomial (which can be found in handbooks of mathematical functions). The energy eigenvalues can be shown to be (E- 39) where m = 0,1,2,3... Thus the energy levels are equally spaced starting with the zero point energy hυ0. (Figure E- 4). Note the solutions have the property that there is some probability of finding the particle in classically forbidden regions, i.e., the particle penetrates into the walls.
14 14 E- 7 Barrier Penetration (One Dimension). Another important quantum mechanical problem of interest to nuclear chemists is the penetration of a one- dimensional potential barrier by a beam of particles. The results of solving this problem (and more complicated variations of the problem) will be used in our study of nuclear α- decay and nuclear reactions. The situation is shown in figure E- 5. A beam of particles originating at - is incident on a barrier of thickness L and height Vo that extends from x=0 to x=l. Each particle has a total energy E. (Classically, we would expect if E < Vo, the particles would bounce off the barrier while if E > Vo, the particles would pass by the barrier with no change in their properties. Both conclusions are altered significantly in quantum mechanics). It is conventional to divide the space into three regions I, II, and III, shown in Figure E- 5. In regions I and III, we have the free particle problem treated in E- 4. In region I, we have particles moving to the left (the incident particles) and particles moving to the right (reflected particles). So we expect a wave function of the form E- 24, whose time independent part can be written (E- 40) where. In region III, we have no particles incident from +, so, at best, we can only have particles moving in the +x direction (b=0). Thus (E- 41)
15 15 In region II, the time independent Schrödinger equation is (E- 42) where k2 = [2m(Vo - E)] ½ /, assuming Vo > E. The solution is (E- 43) Notice that the wave length λ is the same in regions I and III, but the amplitude of the wave beyond the barrier is much less then in front of the barrier. It can be shown that the probability of transmitting particles through the barrier is (E- 44) where V is the particle speed. To determine the value of aiii / ai, we eliminate the other constants bi, aii, bii by applying the conditions that ψ and dψ/dx must be continuous through all space. After much algebra (see, for example, the textbook by Evans), we have (E- 45)
16 16 For nuclear applications, the barriers are quite thick (k2l>>1), in which case,, thus (E- 46) The dominant term in this expression is the exponential. For a 6 MeVα- particle, Vo = 20 MeV, L = m, we have 5.1 x m - 1 Thus and T = 16 x 8/20 x (1-8/20)(5.1 x ) = 1.9 x So we ignore the pre- exponential term, and write T e - 2G (E- 47)
17 17 where 2G = 2k2L = 2[2m(Vo - E)] ½ /. For an arbitrarily shaped potential that would be more pertinent to nuclear α- decay, one can show (E- 48) where x1 and x2 are the points where E = V(x). What about the case where E > Vo. In regions I and III, the situation is the same. In region II, the wave functions will be given as (E- 49) where (E- 50) Since the wave length, we can note by comparing equations that λ2 > λ1, and the momentum (p (= (2mk2) ½ )) becomes less. In other words, the particle is scattered. E- 8 The Schrödinger Equation in Spherical Coordinates
18 18 Many problems in nuclear physics and chemistry involve potentials, such as the Coulomb potential, that are spherically symmetric. In these cases, it is advantageous to express the time- independent Schrödinger equation in spherical coordinates (Figure E- 6). The familiar transformations from a Cartesian coordinate system (x, y, z) to spherical coordinates (r, θ, φ) are (Figure E- 6) x = r sinθcosφ y = r sinθsinφ z = r cosθ (E- 51) The time independent Schrödinger equation becomes (E- 52) When the potential is spherically symmetric, v=v(r), then the wave function can be written as ψ(r, θ, φ) = R(r) Ylm (θ, φ) where Ylm are the spherical harmonic functions.
19 19 If we substitute this wave function in equation (E- 52) and collect terms, we find that all function of r can be separated from the functions of θ and φ. (E- 53) Setting both sides of the equation equal to a separation constant, l (l+ 1), where l= 0, 1, 2..., we have (E- 54) and (E- 55) Working on the equation E- 54, it is convenient to change variables (E- 56) (E- 57)
20 20 This is called the radial wave equation. Apart from the term involving l, it is the same as the one- dimensional time independent Schrödinger equation, a fact that will be useful in its solution. The last term is referred to as the centrifugal potential, i.e., a potential whose first derivative with respect to r gives the centrifugal force. It is important to note that equation E- 55 does not contain the potential energy term, and thus once we have solved it, the solutions will supply to all cases where V does not depend on Θ and φ, i.e., all so- called central potentials. The wave functions Ylm (θ, φ) are known as the spherical harmonic functions and are tabulated. The indices l and m are related to the orbital angular momentum, L, of the particle relative to the origin. The magnitude of L is [l (l+1)] ½ h and its 2l+1 possible projections on the z axis are equal to m (m = 0, ±1, ±2...±l)*. l is called the orbital angular momentum quantum number while m is the magnetic quantum number, in reference to the different energies of the m states in a magnetic field (the Zeeman effect). It follows, therefore, that the specification of a particular spherical harmonic function (as a solution to the angular equation) uniquely specifies the particle s orbital angular momentum and its z- component. *In more formal language, l 2 = 2 l (l + 1) lz = m
21 21 E- 9 The Infinite Spherical Well As an application of the Schrödinger equation, expressed in spherical coordinates, to a problem of interest in nuclear chemistry, let us consider the problem of a particle in an infinite spherical well (Figure E- 7). This potential can be defined as V(r) = 0 r < a (E- 58) V(r) = r > a Following our discussion in section E- 8, we expect the solution of the Schrödinger equation to be (E- 59) where the radial wave function Rl(r) is a solution to the equation (E- 60) inside the well. The solutions of this equation are the spherical Bessel functions (E- 61)
22 22 where. The boundary conditions require ψ = 0 at r = 0, and r = a. This will happen for values of ka that make the Bessel functions have a value of 0 (the zeros of these functions). (Each l value will have its own set of zeros). These resulting values of k can be used to calculate the allowed energy levels (Figure E- 8). Each level is labeled with a number (1, 2, 3...) and a letter (s, p, d, e, etc.). The letter follows the usual spectroscopic notation of l (l = 0, s; l= 1, p, etc.) while the number designates how many times that letter has occurred (the first d level is 1d; the second 2d, etc.). E- 10 Angular Momentum Classically the angular momentum of a particle can be written as = x. (Section 1.6.2). From this classical expression, we can write down the classical components of the vectorl_; lx = ypz - zpx ly = zpx - xpz (E- 62) lz = xpy - ypx
23 23 These classical expressions can be converted to the operator language of quantum mechanics by substitutions (such as x x, px i ( / x), etc.) (E- 63) As remarked earlier (Section E- 9), the expectation values of lz and l 2 for a central potential are lz = m m = 0, ±1, ±2...±l (E- 64) and l 2 = l(l+1) 2 (E- 65)
24 24 We can give these results a pictorial interpretation that is worth noting. Consider a state of definite orbital angular momentum l. Then The z component of l may have any value up to ±l. The possible values of lz can be represented as the projection of a vector of length l on the z axis (figure E- 9). This situation is referred to as spatial quantization. Only certain values of lz are allowed. Due to the Uncertainty Principle, the values of lx and ly are completely uncertain. In the language of Figure E- 9, the vector representing l is rotating about the z axis, so that l and lz are fixed, but lx and ly are continuously changing. In chemistry, we found that to describe the complete quantum state of an electron in an atom, we had to introduce another quantum number, the intrinsic angular momentum or spin. This quantum number is designated as s. By analogy to the orbital angular momentum quantum number l, we have s 2 = s(s + 1) 2 s2 = ms ms= ±½ (E- 66) Nucleons also have values of the spin quantum number of s = ½, like electrons. The total angular momentum of a nucleon j can be written as
25 25 = + (E- 67) The usual quantum mechanical rules apply to j, i.e., j = j (j+1) 2 jz = mj = lz + sz where mj = - j, - j j- 1, j Thus we have mj = m + ms = m ± ½. Since ml is always an integer, then mj must have a half integer and j must be a half integer, either j = l - ½ or j = l + ½. Alternatively, for a given l value, we have two possible values of j, j = l - ½ or j = l + ½. For example, for l = 1 (p state), we have j = l - 1/2 = ½ or j = l + ½ = 3/2. We designate these states as p1/2 and p3/2, respectively. E- 11 Parity
26 26 A wave function has positive (or even) parity if it does not change sign by reflection through the origin. ψ (- x, - y, z) = ψ(x, y, z) positive parity, π = + (E- 69) Alternatively if reflection through the origin produces a change of sign, the parity of the wave function is negative (- ). ψ (- x, - y, z) = - ψ(x, y, z) negative parity, π = - (E- 70) When ψ is expressed in spherical coordinates as ψ(r, θ, φ), then reflection through the origin is accomplished by replacing θ, and φ by (π- θ) and (π + φ), respectively. (r cannot change sign as it is just a distance). In other words, the parity of the wave function is determined only by its angular part. For spherically symmetric potentials, the value of l uniquely determines the parity as π = (- 1) l (E- 71) A corollary of this in that for a system of particles, the parity is even if the sum of the individual orbital angular momentum quantum numbers Σli is even; the parity is odd if Σli is odd. Thus the parity of each level depends on its wave function. An excited state of a nucleus need not have the same parity as the ground state.
27 27 Parity will be valuable to us in our discussion of nuclei because it is conserved in beta decay which will tell us that a different force, the weak interaction, is acting in beta decay compared to nuclear reactions. Also the rates of the γ- ray transitions between nuclear excited states depend on the changes in parity and can be used to determine the parity of nuclear states.
28 28 E- 12 Quantum Statistics The parity of a system is related to the symmetry properties of the spatial portion of the wave function. Another important quantum mechanical property of a system of two or more identical particles is the effect on the wave function of exchanging the coordinates of two particles. If no change in the wave function occurs when the spatial and spin coordinates are exchanged, we say the wave function is symmetric and the particles obey Bose- Einstein statistics. If, upon exchange of the spatial and spin coordinates of the two particles the wave function changes sign, the wave function is said to be antisymmetric and the particles obey Fermi- Dirac statistics. The statistics these particles followed, profoundly affects the property of an assembly of such particles. Particles with half- integer spins, such as neutrons, protons, and electrons, are fermions, and obey Fermi- Dirac statistics, have antisymmetric wave functions, and as a consequence, obey the Pauli principle. (No two particles can have identical values of the quantum numbers, m, l, ml, s, and ms). Photons, or other particles with integer spins, such as the π meson, are bosons, obey Bose- Einstein statistics, have symmetric wave functions and do not obey the Pauli principle. This difference between fermions and bosons is reflected in how they occupy a set of states, especially as a function of temperature. Consider the system shown in Figure E- 10. At zero temperature (T = 0), the bosons will try to occupy the lowest energy state (a Bose- Einstein coordinate) while for the fermions, the occupancy will be one per quantum
29 29 state. At high temperatures the distributions are similar and approach the Maxwell Boltzman distribution. The Fermi- Dirac distribution can be described by the equation (E- 72) where ffd is the number of particles per quantum state, k is Boltzman s constant and EF is the Fermi energy. At T = 0, all energy levels up to EF are occupied (ffd = 1) and all energy levels above EF are empty (ffd = 0). As T increases, some levels above EF become occupied at the expense of levels below EF. References K.S. Krane, Modern Physics (Wiley, New York, 1983) A well written introductory treatment of quantum physics. M. Scharff, Elementary Quantum Mechanics (Wiley, London, 1969) A very lucid, elementary treatment of quantum mechanics, emphasizing physical insight rather than formal theory.
30 30 L.I. Schiff, Quantum Mechanics (McGraw- Hill, New York, 1955) An old classic treatment that contains several applications of interest. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961) Another treatment with several nuclear physics applications. C. Cohen- Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977) An encyclopedic treatment. R.M. Eisberg, Fundamentals of Modern Physics (Wiley, New York, 1961). A comprehensive treatment of modern physics.
31 31 -x x E 0 Figure E- 1. The free particle problem. v = E x = 0 v = 0 x = L Figure E- 2. A schematic diagram of a particle in a one- dimensional box. The particle is free to move between x = 0 and x = L, but not allowed to have x < 0 or x > L.
32 32 Figure E- 3. The allowed energy levels of a particle in a one- dimensional box. The wave function is shown as a solid line for each level while the shaded area gives the probability density.
33 33 Figure E- 4. The low- lying levels and associated probability densities for the harmonic oscillator.
34 34 V 0 L Figure E- 5. A schematic diagram of a particle of energy E incident on a barrier of height V0 and thickness L. The wave function ψ is shown also.
35 35 Figure E- 6. Spherical polar coordinates.
36 36 v(r) 0 r = a R Figure E- 7. Schematic diagram of the infinite square well potential.
37 37 Figure E- 8a. Energy levels of an infinitely deep spherical square well. The radical probability density functions are shown for different values of ℓ
38 38 Figure E- 8b. The three- dimensional probability densities, n, ℓ, m (r,θ) for an infinitely deep three- dimensional square well.
39 39 Figure E- 9. The spatial orientation and z components of a vector with l = 2.
40 40 # particles/ level (a) (b) # particles/ level (c) Figure E- 10. (a) The Bose- Einstein distribution function. (b) The Fermi- Dirac distribution function.
41 41 (c) The filling of levels by fermions at T=0 and T=T1 > 0. The dashed line indicates the Fermi energies EF
The Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationBasic Nuclear Concepts
Section 7: In this section, we present a basic description of atomic nuclei, the stored energy contained within them, their occurrence and stability Basic Nuclear Concepts EARLY DISCOVERIES [see also Section
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More information2. Spin Chemistry and the Vector Model
2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More information2, 8, 20, 28, 50, 82, 126.
Chapter 5 Nuclear Shell Model 5.1 Magic Numbers The binding energies predicted by the Liquid Drop Model underestimate the actual binding energies of magic nuclei for which either the number of neutrons
More informationClassical Angular Momentum. The Physics of Rotational Motion.
3. Angular Momentum States. We now employ the vector model to enumerate the possible number of spin angular momentum states for several commonly encountered situations in photochemistry. We shall give
More informationFree Electron Fermi Gas (Kittel Ch. 6)
Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationBasic Concepts in Nuclear Physics
Basic Concepts in Nuclear Physics Paolo Finelli Corso di Teoria delle Forze Nucleari 2011 Literature/Bibliography Some useful texts are available at the Library: Wong, Nuclear Physics Krane, Introductory
More informationAtomic Structure Ron Robertson
Atomic Structure Ron Robertson r2 n:\files\courses\1110-20\2010 possible slides for web\atomicstructuretrans.doc I. What is Light? Debate in 1600's: Since waves or particles can transfer energy, what is
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationAtomic Structure: Chapter Problems
Atomic Structure: Chapter Problems Bohr Model Class Work 1. Describe the nuclear model of the atom. 2. Explain the problems with the nuclear model of the atom. 3. According to Niels Bohr, what does n stand
More informationChapter 18: The Structure of the Atom
Chapter 18: The Structure of the Atom 1. For most elements, an atom has A. no neutrons in the nucleus. B. more protons than electrons. C. less neutrons than electrons. D. just as many electrons as protons.
More informationAssessment Plan for Learning Outcomes for BA/BS in Physics
Department of Physics and Astronomy Goals and Learning Outcomes 1. Students know basic physics principles [BS, BA, MS] 1.1 Students can demonstrate an understanding of Newton s laws 1.2 Students can demonstrate
More informationElements in the periodic table are indicated by SYMBOLS. To the left of the symbol we find the atomic mass (A) at the upper corner, and the atomic num
. ATOMIC STRUCTURE FUNDAMENTALS LEARNING OBJECTIVES To review the basics concepts of atomic structure that have direct relevance to the fundamental concepts of organic chemistry. This material is essential
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan
More information5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM
5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,
More informationMain properties of atoms and nucleus
Main properties of atoms and nucleus. Atom Structure.... Structure of Nuclei... 3. Definition of Isotopes... 4. Energy Characteristics of Nuclei... 5. Laws of Radioactive Nuclei Transformation... 3. Atom
More informationName Date Class ELECTRONS IN ATOMS. Standard Curriculum Core content Extension topics
13 ELECTRONS IN ATOMS Conceptual Curriculum Concrete concepts More abstract concepts or math/problem-solving Standard Curriculum Core content Extension topics Honors Curriculum Core honors content Options
More informationNumerical analysis of Bose Einstein condensation in a three-dimensional harmonic oscillator potential
Numerical analysis of Bose Einstein condensation in a three-dimensional harmonic oscillator potential Martin Ligare Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837 Received 24
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationMASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)
MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced
More informationMasses in Atomic Units
Nuclear Composition - the forces binding protons and neutrons in the nucleus are much stronger (binding energy of MeV) than the forces binding electrons to the atom (binding energy of ev) - the constituents
More information3. Derive the partition function for the ideal monoatomic gas. Use Boltzmann statistics, and a quantum mechanical model for the gas.
Tentamen i Statistisk Fysik I den tjugosjunde februari 2009, under tiden 9.00-15.00. Lärare: Ingemar Bengtsson. Hjälpmedel: Penna, suddgummi och linjal. Bedömning: 3 poäng/uppgift. Betyg: 0-3 = F, 4-6
More informationDO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS
DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS Quantum Mechanics or wave mechanics is the best mathematical theory used today to describe and predict the behaviour of particles and waves.
More informationPrecession of spin and Precession of a top
6. Classical Precession of the Angular Momentum Vector A classical bar magnet (Figure 11) may lie motionless at a certain orientation in a magnetic field. However, if the bar magnet possesses angular momentum,
More information- particle with kinetic energy E strikes a barrier with height U 0 > E and width L. - classically the particle cannot overcome the barrier
Tunnel Effect: - particle with kinetic energy E strikes a barrier with height U 0 > E and width L - classically the particle cannot overcome the barrier - quantum mechanically the particle can penetrated
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationCHEM6085: Density Functional Theory Lecture 2. Hamiltonian operators for molecules
CHEM6085: Density Functional Theory Lecture 2 Hamiltonian operators for molecules C.-K. Skylaris 1 The (time-independent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction
More informationChapter NP-5. Nuclear Physics. Nuclear Reactions TABLE OF CONTENTS INTRODUCTION OBJECTIVES 1.0 NUCLEAR REACTIONS 2.0 NEUTRON INTERACTIONS
Chapter NP-5 Nuclear Physics Nuclear Reactions TABLE OF CONTENTS INTRODUCTION OBJECTIVES 1.0 2.0 NEUTRON INTERACTIONS 2.1 ELASTIC SCATTERING 2.2 INELASTIC SCATTERING 2.3 RADIATIVE CAPTURE 2.4 PARTICLE
More informationPhysical Principle of Formation and Essence of Radio Waves
Physical Principle of Formation and Essence of Radio Waves Anatoli Bedritsky Abstract. This article opens physical phenomena which occur at the formation of the radio waves, and opens the essence of the
More informationParticle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims
Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;
More informationMulti-electron atoms
Multi-electron atoms Today: Using hydrogen as a model. The Periodic Table HWK 13 available online. Please fill out the online participation survey. Worth 10points on HWK 13. Final Exam is Monday, Dec.
More informationSolutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 7
Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid April 21, 2002 Chapter 7 Problem 7.2 Obtain the Lorentz transformation in which the velocity is at an infinitesimal angle
More informationTheory of electrons and positrons
P AUL A. M. DIRAC Theory of electrons and positrons Nobel Lecture, December 12, 1933 Matter has been found by experimental physicists to be made up of small particles of various kinds, the particles of
More informationName Partners Date. Energy Diagrams I
Name Partners Date Visual Quantum Mechanics The Next Generation Energy Diagrams I Goal Changes in energy are a good way to describe an object s motion. Here you will construct energy diagrams for a toy
More information5.61 Fall 2012 Lecture #19 page 1
5.6 Fall 0 Lecture #9 page HYDROGEN ATOM Consider an arbitrary potential U(r) that only depends on the distance between two particles from the origin. We can write the Hamiltonian simply ħ + Ur ( ) H =
More informationPHYSICS TEST PRACTICE BOOK. Graduate Record Examinations. This practice book contains. Become familiar with. Visit GRE Online at www.gre.
This book is provided FREE with test registration by the Graduate Record Examinations Board. Graduate Record Examinations This practice book contains one actual full-length GRE Physics Test test-taking
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationd d Φ * Φdx T(B) Barrier (B ) : Vo = 5, a = 2 Well (W ) : Vo= -5, a = 2 0.0 0 2 4 6 8 10 12 14 16 18 20 ENERGY (E)
Quantum Mechanical Transmission with Absorption S. MAHADEVAN, A. UMA MAHESWARI, P. PREMA AND C. S. SHASTRY Physics Department, Amrita Vishwa Vidyapeetham, Coimbatore 641105 ABSTRACT Transmission and reflection
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationContents. Goldstone Bosons in 3He-A Soft Modes Dynamics and Lie Algebra of Group G:
... Vlll Contents 3. Textures and Supercurrents in Superfluid Phases of 3He 3.1. Textures, Gradient Energy and Rigidity 3.2. Why Superfuids are Superfluid 3.3. Superfluidity and Response to a Transverse
More informationObjectives 404 CHAPTER 9 RADIATION
Objectives Explain the difference between isotopes of the same element. Describe the force that holds nucleons together. Explain the relationship between mass and energy according to Einstein s theory
More informationAtoms and Elements. Outline Atoms Orbitals and Energy Levels Periodic Properties Homework
Atoms and the Periodic Table The very hot early universe was a plasma with cationic nuclei separated from negatively charged electrons. Plasmas exist today where the energy of the particles is very high,
More informationTIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points
TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points 1. Check your examination for completeness prior to starting.
More informationWave Function, ψ. Chapter 28 Atomic Physics. The Heisenberg Uncertainty Principle. Line Spectrum
Wave Function, ψ Chapter 28 Atomic Physics The Hydrogen Atom The Bohr Model Electron Waves in the Atom The value of Ψ 2 for a particular object at a certain place and time is proportional to the probability
More informationBasics of Nuclear Physics and Fission
Basics of Nuclear Physics and Fission A basic background in nuclear physics for those who want to start at the beginning. Some of the terms used in this factsheet can be found in IEER s on-line glossary.
More informationProf.M.Perucca CORSO DI APPROFONDIMENTO DI FISICA ATOMICA: (III-INCONTRO) RISONANZA MAGNETICA NUCLEARE
Prof.M.Perucca CORSO DI APPROFONDIMENTO DI FISICA ATOMICA: (III-INCONTRO) RISONANZA MAGNETICA NUCLEARE SUMMARY (I/II) Angular momentum and the spinning gyroscope stationary state equation Magnetic dipole
More informationPHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS
PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS 1. Photons 2. Photoelectric Effect 3. Experimental Set-up to study Photoelectric Effect 4. Effect of Intensity, Frequency, Potential on P.E.
More informationArrangement of Electrons in Atoms
CHAPTER 4 PRE-TEST Arrangement of Electrons in Atoms In the space provided, write the letter of the term that best completes each sentence or best answers each question. 1. Which of the following orbital
More informationReview of the isotope effect in the hydrogen spectrum
Review of the isotope effect in the hydrogen spectrum 1 Balmer and Rydberg Formulas By the middle of the 19th century it was well established that atoms emitted light at discrete wavelengths. This is in
More informationProblem Set V Solutions
Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationMASS DEFECT AND BINDING ENERGY
MASS DEFECT AND BINDING ENERGY The separate laws of Conservation of Mass and Conservation of Energy are not applied strictly on the nuclear level. It is possible to convert between mass and energy. Instead
More information1. Degenerate Pressure
. Degenerate Pressure We next consider a Fermion gas in quite a different context: the interior of a white dwarf star. Like other stars, white dwarfs have fully ionized plasma interiors. The positively
More informationPHYS 1624 University Physics I. PHYS 2644 University Physics II
PHYS 1624 Physics I An introduction to mechanics, heat, and wave motion. This is a calculus- based course for Scientists and Engineers. 4 hours (3 lecture/3 lab) Prerequisites: Credit for MATH 2413 (Calculus
More informationIntroduction to acoustic imaging
Introduction to acoustic imaging Contents 1 Propagation of acoustic waves 3 1.1 Wave types.......................................... 3 1.2 Mathematical formulation.................................. 4 1.3
More informationAP1 Electricity. 1. A student wearing shoes stands on a tile floor. The students shoes do not fall into the tile floor due to
1. A student wearing shoes stands on a tile floor. The students shoes do not fall into the tile floor due to (A) a force of repulsion between the shoes and the floor due to macroscopic gravitational forces.
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationQuestion: Do all electrons in the same level have the same energy?
Question: Do all electrons in the same level have the same energy? From the Shells Activity, one important conclusion we reached based on the first ionization energy experimental data is that electrons
More informationarxiv:cond-mat/9301024v1 20 Jan 1993 ABSTRACT
Anyons as Dirac Strings, the A x = 0 Gauge LPTB 93-1 John McCabe Laboratoire de Physique Théorique, 1 Université Bordeaux I 19 rue du Solarium, 33175 Gradignan FRANCE arxiv:cond-mat/930104v1 0 Jan 1993
More information3. Electronic Spectroscopy of Molecules I - Absorption Spectroscopy
3. Electronic Spectroscopy of Molecules I - Absorption Spectroscopy 3.1. Vibrational coarse structure of electronic spectra. The Born Oppenheimer Approximation introduced in the last chapter can be extended
More informationarxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More informationLevel 3 Achievement Scale
Unit 1: Atoms Level 3 Achievement Scale Can state the key results of the experiments associated with Dalton, Rutherford, Thomson, Chadwick, and Bohr and what this lead each to conclude. Can explain that
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationAlgebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information13- What is the maximum number of electrons that can occupy the subshell 3d? a) 1 b) 3 c) 5 d) 2
Assignment 06 A 1- What is the energy in joules of an electron undergoing a transition from n = 3 to n = 5 in a Bohr hydrogen atom? a) -3.48 x 10-17 J b) 2.18 x 10-19 J c) 1.55 x 10-19 J d) -2.56 x 10-19
More informationAP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion
S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates
More informationThe properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >> 1,
Chapter 3 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >>, which defines the degenerate Fermi gas. In
More informationAP1 Waves. (A) frequency (B) wavelength (C) speed (D) intensity. Answer: (A) and (D) frequency and intensity.
1. A fire truck is moving at a fairly high speed, with its siren emitting sound at a specific pitch. As the fire truck recedes from you which of the following characteristics of the sound wave from the
More informationLet s first see how precession works in quantitative detail. The system is illustrated below: ...
lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,
More information6 J - vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems
More informationSTRING THEORY: Past, Present, and Future
STRING THEORY: Past, Present, and Future John H. Schwarz Simons Center March 25, 2014 1 OUTLINE I) Early History and Basic Concepts II) String Theory for Unification III) Superstring Revolutions IV) Remaining
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More information0.33 d down 1 1. 0.33 c charm + 2 3. 0 0 1.5 s strange 1 3. 0 0 0.5 t top + 2 3. 0 0 172 b bottom 1 3
Chapter 16 Constituent Quark Model Quarks are fundamental spin- 1 particles from which all hadrons are made up. Baryons consist of three quarks, whereas mesons consist of a quark and an anti-quark. There
More informationCBE 6333, R. Levicky 1 Differential Balance Equations
CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,
More informationLecture 8. Generating a non-uniform probability distribution
Discrete outcomes Lecture 8 Generating a non-uniform probability distribution Last week we discussed generating a non-uniform probability distribution for the case of finite discrete outcomes. An algorithm
More informationPhysics 214 Waves and Quantum Physics. Lecture 1, p 1
Physics 214 Waves and Quantum Physics Lecture 1, p 1 Welcome to Physics 214 Faculty: Lectures A&B: Paul Kwiat Discussion: Nadya Mason Labs: Karin Dahmen All course information is on the web site. Read
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationCreating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
More informationConservation of Momentum and Energy
Conservation of Momentum and Energy OBJECTIVES to investigate simple elastic and inelastic collisions in one dimension to study the conservation of momentum and energy phenomena EQUIPMENT horizontal dynamics
More informationNanoelectronics. Chapter 2 Classical Particles, Classical Waves, and Quantum Particles. Q.Li@Physics.WHU@2015.3
Nanoelectronics Chapter 2 Classical Particles, Classical Waves, and Quantum Particles Q.Li@Physics.WHU@2015.3 1 Electron Double-Slit Experiment Q.Li@Physics.WHU@2015.3 2 2.1 Comparison of Classical and
More informationAn Introduction to Hartree-Fock Molecular Orbital Theory
An Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction Hartree-Fock theory is fundamental
More informationLab 2: Vector Analysis
Lab 2: Vector Analysis Objectives: to practice using graphical and analytical methods to add vectors in two dimensions Equipment: Meter stick Ruler Protractor Force table Ring Pulleys with attachments
More informationHistory of the Atom & Atomic Theory
Chapter 5 History of the Atom & Atomic Theory You re invited to a Thinking Inside the Box Conference Each group should nominate a: o Leader o Writer o Presenter You have 5 minutes to come up with observations
More informationAtomic and Nuclear Physics Laboratory (Physics 4780)
Gamma Ray Spectroscopy Week of September 27, 2010 Atomic and Nuclear Physics Laboratory (Physics 4780) The University of Toledo Instructor: Randy Ellingson Gamma Ray Production: Co 60 60 60 27Co28Ni *
More informationNMR - Basic principles
NMR - Basic principles Subatomic particles like electrons, protons and neutrons are associated with spin - a fundamental property like charge or mass. In the case of nuclei with even number of protons
More informationThree Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009
Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationPhysics 111 Homework Solutions Week #9 - Tuesday
Physics 111 Homework Solutions Week #9 - Tuesday Friday, February 25, 2011 Chapter 22 Questions - None Multiple-Choice 223 A 224 C 225 B 226 B 227 B 229 D Problems 227 In this double slit experiment we
More informationKinetic Theory of Gases. Chapter 33. Kinetic Theory of Gases
Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atoms or molecules. Atoms or molecules are considered as particles. This is based on
More information