Quantum Mechanics I Babis Anastasiou Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland E-mail: [email protected] August 7, 2017 Contents Contents 1 Bibliography 5 1 Quantum behavior 9 1.1 Atomic mechanics . 9 1.2 Basic two slit experiments . 9 1.2.1 An experiment with bullets . 9 1.2.2 An experiment with waves . 10 1.2.3 An experiment with electrons . 11 1.3 Watching electrons . 12 1.4 Back to the bullet experiment . 14 1.5 First principles of quantum mechanics . 14 1.6 Heisenberg's uncertainty principle . 16 1.6.1 Estimating the size of the atoms . 17 1.7 The laws for combining amplitudes . 18 1.7.1 Revisiting the two-slit experiment . 19 1.7.2 A subatomic travel from Zurich to China? . 20 2 Quantum measurement and quantum states 23 2.1 Stern-Gerlach experiment . 23 2.2 The Stern-Gerlach filter . 24 2.2.1 Experimental setup . 24 2.2.2 Successive aligned Stern-Gerlach filters . 25 2.2.3 Successive rotated Stern-Gerlach filters . 27 2.2.4 Three filters . 28 2.3 Base states . 30 2.4 Rules of quantum mechanics at work . 31 3 Quantum Mechanics and Linear Algebra 33 3.1 Ket-space . 33 3.2 Operators . 34 3.3 Dual space . 34 1 Contents 3.4 Inner product . 34 3.5 Properties of operators and their duals . 35 3.6 Hermitian operators . 36 3.7 Matrix representation of states and operators . 40 3.8 Compatible and incompatible observables . 42 3.9 Expectation value and uncertainty of Hermitian operators . 43 3.10 The uncertainty principle . 45 3.11 Change of basis . 47 3.11.1 Transformation matrix . 48 3.11.2 Trace of operators . 49 3.12 Eigenstates and eigenvalues . 50 3.13 Unitary equivalent observables . 51 4 Time evolution 53 4.1 Schr¨odingerequation . 54 4.2 Eigenstates of the Hamiltonian operator . 55 4.3 Spin precession . 57 4.4 Heisenberg and Schr¨odingerpicture of time evolution . 59 4.4.1 Shift in picture: . 60 4.4.2 Recapitulation . 61 5 Two-state systems 63 5.1 The ammonia molecule . 64 5.2 The ammonia molecule inside an electric field . 67 5.3 Ammonia in an oscillating electric field . 69 5.3.1 Transitions at resonance . 71 5.3.2 Microwave amplification with stimulated emission of ra- diation . 72 5.3.3 Transition off resonance . 73 6 Position and momentum 75 6.1 Continuous spectra . 75 6.2 Position operator and eigenstates . 76 6.3 Translation operator . 76 6.3.1 Properties of the translation operator . 77 6.3.2 Generator of translations . 78 6.3.3 Momentum operator . 79 6.3.4 Translations of arbitrary length . 80 6.3.5 Commutator of momentum operators . 81 6.3.6 Commutation relations . 82 6.4 Wave-function . 82 6.5 Representation of momentum in position space . 83 2 Contents 6.6 Eigenstates of the momentum operator . 84 6.6.1 Position and momentum wave-functions . 84 6.7 Ehrenfest theorem . 86 6.7.1 Free particles in the Heisenberg picture . 86 6.7.2 Particles in potential . 86 7 The harmonic oscillator 89 7.1 Quantization . 89 7.2 Time evolution . 94 7.3 Coherent states . 96 8 Schr¨odinger'swave equation 101 8.1 Probability density and probability current . 102 8.2 Quantized energy levels . 103 8.3 Semiclassical approximation . 105 8.4 Linear potential . 109 9 Particle in a constant electromagnetic field 111 9.1 Wave-function for a particle in an electromagnetic field . 112 9.2 Aharonov-Bohm effect . 114 10 Symmetries in Quantum Mechanics 117 10.1 Continuous symmetry transformations . 119 10.2 Lie algebra and generators . 121 10.3 Symmetry and degeneracy . 124 10.4 Rotations and translations . 125 10.4.1 Generators . 126 10.4.2 Lie algebra . 128 11 Representations of angular momentum 129 11.1 Eigenstates of angular momentum . 129 1 11.2 spin- 2 representation of angular momentum . 131 11.3 Orbital angular momentum . 134 11.3.1 Spherical coordinates . 135 11.3.2 Orbital angular momentum operators and the Laplacian 136 11.3.3 Spherical harmonics . 137 11.3.4 Space-representations of rotations . 142 11.4 Potentials with spherical symmetry . 143 11.5 The hydrogen atom . 144 11.5.1 SO(4) symmetry of hydrogen atom . 148 12 Addition of angular momenta 149 3 Contents 12.1 Addition of angular momenta . 149 12.2 Application: Hydrogen atom . 153 12.3 Wigner-Eckart theorem . 154 13 Discrete symmetries 159 13.1 Parity . 159 13.2 Time reversal . 161 4 Bibliography [1] Modern Quantum Mechanics, Sakurai [2] The Feynman Lectures in Physics, Feynman [3] Lectures on Quantum Mechanics, Weinberg 5 Preface This script is based on the Feynman Lectures, Sakurai's Modern Quantum Mechanics and Weinberg's Lectures on Quantum Mechanics. These books are great and students are encouraged to study them as a first priority. The purpose of the present notes, which do not contain original material, is to facilitate the author for his lecture presentations. Students are of course welcome to read the script if they find it useful. However, the script should not replace the study of the recommended and other literature. 7 Chapter 1 Quantum behavior 1.1 Atomic mechanics For a long time, light was thought to behave like waves and electrons were thought to behave like particles. There are phenomena which violate these main \rules". So they behave neither like particles nor like waves. However, it is true that light, electrons, protons and all subatomic particles behave the same. This common subatomic behavior is very different from our intuition of large objects. We will demonstrate the difference of the quantum and classic behavior by a thought \two-slit" experiment first with macroscopic bullets, second with waves and third with subatomic particles such as electrons. 1.2 Basic two slit experiments 1.2.1 An experiment with bullets We have a gun firing very hard indestructible bullets towards a wall. The wall has two slits which are at a close distance and are big enough so that they can pass through. Behind the wall there is a \detector", a material which can stop the bullets. We can take a look at it afterwards and see how the bullets are distributed in space after they have gone through the holes. We perform the experiment in three stages: • First, we leave open \Hole 1" and cover up the second hole. We find that the bullets are distributed according to P1, a Gaussian centered around the Hole 1. • Then, we leave open \Hole 2" while we cover up \Hole 1". We find a similar Gaussian P2 centered around Hole 2. • Finally, we leave open both holes. We find a distribution which P12 = P1 + P2 . (1.1) 9 1. Quantum behavior P1 slit 1 gun slit 2 P2 P1 + P2 detector Figure 1.1: A two slit experiment with bullets. The probability that a bullet goes through the holes is the sum of the proba- bilities that it passes either through hole 1 or through hole 2. 1.2.2 An experiment with waves We now perform a different experiment where we create a spherical wave at some distance in front of the wall with the two holes. The detector measures the intensity of the wave arriving at it, which is proportional to the square of the amplitude: I ∼ jAj2 : We perform the experiment in three stages: • First, we leave open \Hole 1" and cover up the second the second hole. We find that the intensity I1 is a function centered around the Hole 1. • Then, we leave open \Hole 2" while we cover up \Hole 1". We find a similar function intensity I2 centered around Hole 2. • Finally, we leave open both holes. We find a distribution I12 6= I1 + I2. There is a simple explanation why the intensity I12 is not equal to the sum of the intensities when either one of the two holes is covered. The amplitude 10 1.2. Basic two slit experiments I1 wave source I2 I1 + I2 6= I12 detector interference pattern Figure 1.2: A two slit experiment with waves. of the wave is given by the sum of the amplitudes, A12 = A1 + A2 where A1 and A2 can be both positive or negative. The total intensity, 2 2 I12 = jA1j + jA2j + 2 jA1j jA2j cos δ (1.2) = I1 + I2 + 2 jA1j jA2j cos δ includes an interference term, giving the characteristic interference pattern of Fig 1.2. 1.2.3 An experiment with electrons We now shoot electrons at the wall. What happens is pictured in Fig 1.3. This is a surprising result. If an electron went through either hole 1 or hole 2, we would not have found P12 6= P1 + P2. Are electrons waves? Well, no. We always detect one \full" electron at the time at the detector. For example, we can reconstruct the full energy of an electron as it dissipates it in the detector, or if the detector is inside a magnetic field, the electron track has the curvature of charge one, etc. 11 1. Quantum behavior P1 electron source P2 P1 + P2 6= P12 detector interference pattern Figure 1.3: A two slit experiment with electrons with wave behavior. So, the electron arrives at the detector as a particle and the probability of arrival is distributed like the intensity of a wave. 1.3 Watching electrons Let's put a light source behind the two holes. Deflected light from the vicinity of hole 1 or hole 2 can tell us where the electron passed through.