Appunti introduzione Meccanica quantistica e tecnologie

Introduction to quantum mechanics and quantum technologies

Notes from Lectures iatrSaordaccRi University of PaviaDepartment of Industrial and Information EngineeringMaster’s Degree in ElectronicsSpecialization in Space Communications and Sensing (LM-29)© Riccardo Satriano, 2025

Preface

This document has been written with the intention of summarizing the content of Prof. Daniele Bajoni’s lectures for the Introduction to quantum mechanics and quantum technologies course, part of the Space Communications and Sensing program within the Facultyof Electronic Engineering at the University of Pavia. The aim is to provide a concise yettrdiscursive overview of the material.

It is important to note that this document is intended solely as a personal study aid andis neither official nor endorsed by the professor. Any use of this document for purposesSaother than individual study is beyond my control and responsibility.

If you notice any inaccuracies or errors, I welcome your feedback. I hope you find thismaterial helpful and wish you success in studying this fascinating course!Riccardo oPavia, June 19, 2025rda

Technical details

cc Overleaf,This document was written using an online LaTeX editor. The Over-Notesleaf compiler used is pdfLaTex with the TexLive version 2024. The file name isQMQT, rev_08.and the current revision isRi

Contacts

• Riccardo Satriano
• Master’s student in Electronic Engineering, specializing in Space Communications andSensing, at the Faculty of Engineering of the University of Pavia.
• Mobile: +39 348 378 6916
• Email 1: satrianoriccardo@gmail.com
• Email 2: riccardo.satriano01@universitadipavia.iti

Contents

oList of Figures vinList of Equations ixiaI Introduction to Quantum Mechanicstr

• 1 The Crisis of Classical Physics 2
• 1.1 History of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
• 1.2 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Sa
• 1.3 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
• 2 Schrödinger Equation and Wavefunction 10
• 2.1 From De Broglie to Schrödinger . . . . . . . . . . . . . . . . . . . . . . . . 10
• 2.2 Linearity and Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . 11o
• 2.3 Energy and Plane Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . 12
• 2.4 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15rd
• 3 Statistical Distributions 17
• 3.1 Review of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
• 3.2 Statistics in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 19a
• 3.2.1 Operators in Quantum Mechanics . . . . . . . . . . . . . . . . . . . 22cc
• 4 Simple systems in 1D 25
• 4.1 Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
• 4.2 Quantum Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
• 4.3 Potential barrier and tunnelling . . . . . . . . . . . . . . . . . . . . . . . . 36Ri
• 4.4 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
• 5 System in 3D 49
• 5.1 Schrödinger equation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 49
• 5.2 The Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
• 6 Dirac formalism and operator and Time evolution 61
• 6.1 Dirac formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
• 6.2 Hermitian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
• 6.3 Properties of Hermitian operators . . . . . . . . . . . . . . . . . . . . . . . 64ii
• 6.4 New Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
• 6.5 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
• 7 Heisenberg Uncertainty Principle 78
• 7.1 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 78
• 7.2 Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . 78
• 7.2.1 Collapse of the Wavefunction . . . . . . . . . . . . . . . . . . . . 79
• 7.2.2 Wavefunction with Minimum Uncertainty . . . . . . . . . . . . . 82o
• 7.2.3 Physical Interpretations and Consequences . . . . . . . . . . . . . 83n
• 8 Crystals and Bloch Theorem 85
• 8.1 Introduction to Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
• 8.2 Periodic Potential and Schrödinger Equation . . . . . . . . . . . . . . . . . 85ia
• 8.3 Tight Binding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
• 8.4 Band Structure and Conductivity . . . . . . . . . . . . . . . . . . . . . . . 92tr

II Quantum technologies

• Sa9 Quantum bits 100
• 9.1 Introduction to Qbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
• 9.2 Mathematical representation of Qbits . . . . . . . . . . . . . . . . . . . . . 102
• 9.3 The Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
• 9.4 Operations on Qbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
• 9.5 Fundamental operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108o
• 10 Black body radiation 111rd
• 10.1 Introduction to the Problem and the Ultraviolet Catastrophe . . . . . . . . 111
• 10.2 Blackbody Cavity and Standing Waves . . . . . . . . . . . . . . . . . . . . 112
• 10.3 Quantum Treatment and Bose-Einstein Statistics . . . . . . . . . . . . . . 116
• 10.4 Planck Law of Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . 118a
• 10.5 Photon Interpretation and Frequency Domains . . . . . . . . . . . . . . . . 120cc
• 11 Quantum photonic 123
• 11.1 Photonic Qubits and Polarization . . . . . . . . . . . . . . . . . . . . . . . 123
• 11.2 Integrated Waveguides and Dual Rail Encoding . . . . . . . . . . . . . . . 125
• 11.2.1 Beam Splitters and Interference Effects . . . . . . . . . . . . . . . . 126Ri
• 11.2.2 Mach-Zehnder Interferometer and Quantum Gates . . . . . . . . . 127
• 11.3 Time-Bin Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
• 11.4 Other Quantum Photonic Platforms . . . . . . . . . . . . . . . . . . . . . . 131
• 11.4.1 Ion Trap Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
• 11.4.2 Superconducting Qubits . . . . . . . . . . . . . . . . . . . . . . . 131
• 12 Entanglement 134
• 12.1 Introduction to Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 134
• 12.2 Bell States and Measurement Outcomes . . . . . . . . . . . . . . . . . . . 136
• 12.3 Half-wave Plates and Polarizers . . . . . . . . . . . . . . . . . . . . . . . . 138iii
• 12.4 EPR Paradox and Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . 140
• 12.5 No Cloning Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
• 12.6 Applications of Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 144
• 13 Quantum communication protocols 146
• 13.1 Introduction to Secrecy and Classical Limitations . . . . . . . . . . . . . . 146
• 13.2 Quantum Key Distribution (QKD) . . . . . . . . . . . . . . . . . . . . . . 147
• 13.2.1 The BB84 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 147o
• 13.2.2 The Ekert Protocol (E91) . . . . . . . . . . . . . . . . . . . . . . 148
• 13.3 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148n
• 13.4 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151ia
• 14 Quantum computing 154
• 14.1 Introduction to Quantum Computing . . . . . . . . . . . . . . . . . . . . . 154
• 14.1.1 DiVincenzo’s Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 154
• 14.1.2 Challenges in Practical Implementation . . . . . . . . . . . . . . . 155tr
• 14.2 Quantum Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
• 14.2.1 Visual Representation of a Qubit . . . . . . . . . . . . . . . . . . 155
• 14.2.2 Basic Operations and Gates . . . . . . . . . . . . . . . . . . . . . 156Sa
• 14.2.3 Measurement in the Computational Basis . . . . . . . . . . . . . . 156
• 14.3 Multi-Qubit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
• 14.3.1 Hilbert Space and Computational Basis . . . . . . . . . . . . . . . 157
• 14.3.2 Tensor Product and State Construction . . . . . . . . . . . . . . . 158
• 14.3.3 Tensor Product with General States . . . . . . . . . . . . . . . . . 158o
• 14.3.4 Extension to Three and Four Qubits . . . . . . . . . . . . . . . . 159
• 14.3.5 Exponential Growth and Registers . . . . . . . . . . . . . . . . . 159rd
• 14.4 Operators on Multi-Qubit States . . . . . . . . . . . . . . . . . . . . . . . 160
• 14.4.1 Tensor Product of Operators . . . . . . . . . . . . . . . . . . . . . 160
• 14.4.2 Two-Qubit Operations via Tensor Products . . . . . . . . . . . . 161
• 14.4.3 Single-Qubit Operation in a Multi-Qubit System . . . . . . . . . . 161a
• 14.4.4 Parallel Hadamard Transformations . . . . . . . . . . . . . . . . . 162
• 14.4.5 Quantum Superposition and Decoherence . . . . . . . . . . . . . . 162cc
• 14.5 Entanglement and Controlled Gates . . . . . . . . . . . . . . . . . . . . . . 163
• 14.5.1 Separable vs Entangled States . . . . . . . . . . . . . . . . . . . . 163
• 14.5.2 Controlled-NOT (CNOT) Gate . . . . . . . . . . . . . . . . . . . . 163
• 14.5.3 Generation of Bell States . . . . . . . . . . . . . . . . . . . . . . . 164Ri
• 14.5.4 Matrix Representation of the Bell Transformation . . . . . . . . . 164
• 14.5.5 Reverse Transformation . . . . . . . . . . . . . . . . . . . . . . . 166
• 14.6 Additional Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
• 14.6.1 The Pauli-Z Gate and the Controlled-Z Gate . . . . . . . . . . . . 166
• 14.6.2 Controlled Unitary Gates . . . . . . . . . . . . . . . . . . . . . . 167
• 14.6.3 Universal Quantum Gate Set . . . . . . . . . . . . . . . . . . . . 167
• 14.6.4 The SWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
• 14.6.5 The Toffoli Gate (ccX̂) . . . . . . . . . . . . . . . . . . . . . . . . 168
• 15 Quantum algorithms 171iv
• 15.1 Deutsch–Jozsa Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
• 15.2 Grover’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
• 15.3 Bernstein–Vazirani Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 175oniatrSaordaccRi v

List of figures

• 1.1 Schematic of the black body cavity experiment. The inner walls absorbradiation. Heating the cavity increases the temperature, and radiationnescapes from the hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
• 1.2 Comparison between Planck’s Law and Rayleigh-Jeans Law for black bodyiaradiation. Both curves start with the same initial slope, but the classicalprediction diverges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
• 1.3 Photoelectric effect: kinetic energy of ejected electrons as a function oftrphoton frequency. No electrons are emitted below the threshold frequency. 7
• 3.1 Discrete distribution of height values with average and visual represen-⟨h⟩Satation of variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192σ
• 4.1 Potential (x) for an infinite quantum well. The particle is confined in theVregion 0 where the potential is zero. Outside this region, the< x < L,potential is infinite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
• 4.2 Quantum tunnelling through a potential barrier. . . . . . . . . . . . . . . . 37o
• 4.3 Transmission probability as a function of energy showing quantumT E,tunnelling for and resonance effects for . . . . . . . . . . . . 41E < V E > Vrd0 02 for the hydrogen atom. . . . . . . . . . . 52
• 5.1 Coulomb potential (r) = e−V r4πε0
• 5.2 Graphical interpretation of as the probability of finding the par-2|ψ(x)| dxticle in the interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56dx.a
• 5.3 Comparison between the radial probability density and the total2|R(r)|probability density = for the hydrogen atom in the ground2 2|R(r)|ρ(r) rccstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
• 5.4 Discrete energy levels of the hydrogen atom for different quantum num-E nbers The levels get closer as increases, approaching the ionizationn. nRilimit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
• 5.5 Spectral lines of hydrogen: electron transitions to = 1 produce the ultra-nviolet Lyman series, while transitions to = 2 result in the visible Balmernseries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
• 8.1 1D periodic crystal structure. Atoms are located at integer multiples ofthe lattice constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85a.
• 8.2 Discrete energy levels of an isolated atom. Each level corresponds to{E }na bound electronic state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
• 9.1 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104vi
• 10.1 Schematic of the black body cavity experiment. When the system is heated,radiation escapes through the small hole. . . . . . . . . . . . . . . . . . . . 111
• 10.2 Comparison between Quantum Mechanics and Classical Mechanics for blackbody radiation. Both curves start with the same initial slope, but the clas-sical prediction diverges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
• 10.3 Quantized energy levels of a quantum harmonic oscillator. Each level cor- responds to = + Adding one photon increases the energy by1E n ℏω.n 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117ℏω. o
• 10.4 Blackbody radiation curves for different temperatures. The peak shifts tohigher frequencies for increasing . . . . . . . . . . . . . . . . . . . . . . . 119T n
• 11.1 Measurement of a photon in the state using a polarizer and two de-|+⟩tectors. The polarizer separates diagonal polarization into horizontal andiavertical components. Only one of the two single-photon detectors will click,indicating the result of the measurement. . . . . . . . . . . . . . . . . . . . 124
• 11.2 Simplified 3D view of an optical waveguide. Depth points into the pagetr(top-right), representing the direction of light propagation. The waveguidehas confined width and height in the transverse plane. . . . . . . . . . . . . 125
• 11.3 Dual rail encoding of a photonic qubit. Each waveguide represents oneSalogical state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
• 11.4 Phase shifter acting on the lower rail in dual rail encoding. . . . . . . . . . 126
• 11.5 Smooth integrated beam splitter: two input waveguides curve and crossthrough an evanescent coupling region, leading to symmetric output paths. 127
• 11.6 Mach-Zehnder interferometer implemented using two curved beam splittersoand a phase shifter on the lower path. The device enables full single-qubitgate control in dual rail encoding. . . . . . . . . . . . . . . . . . . . . . . . 128rd
• 11.7 Time-bin encoding: a single photon is encoded based on its arrival time inone of two distinct time slots, separated by ∆t. . . . . . . . . . . . . . . . 130
• 11.8 Unbalanced interferometer used for time-bin qubits. The lower path in-cludes a delay line ∆t, enabling interference between early and late photonaarrivals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
• 11.9 Series LC circuit with a Josephson junction, forming the core of a super-ccconducting qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
• 12.1 A photon in state enters a polarizer. It is either transmitted as or|0⟩ |0⟩deflected as each with some probability depending on alignment. . . . 138Ri |1⟩,
• 12.2 Measurement in the rotated basis using a plate rotated at{|+⟩, |−⟩} λ/222.5 before the polarizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139◦
• 12.3 Measurement of entangled photons in the diagonal basis using two λ/2plates at 22.5 and two polarizers. The setup is symmetric for both photons.139◦
• 12.4 Experimental setup for testing the Bell inequality. Two entangled photonsare measured in a rotated basis using adjustable plates set at anglesλ/2and , followed by polarizers and detectors. . . . . . . . . . . . . . . . 140θ θA B
• 13.1 BB84 photon preparation setup. A single photon passes through a λ/2plate and a polarizer, producing one of four possible states. . . . . . . . . . 147vii
• 13.2 Entanglement-based QKD (Ekert protocol). A source emits entangled pho-tons, each sent to Alice and Bob through a plate and a polarizer. . . . 148λ/2
• 13.3 Quantum teleportation: the unknown state of photon is destroyed in aaBell measurement with photon and then reconstructed on photon usingb, cclassical information and a unitary correction. . . . . . . . . . . . . . . . . 149
• 13.4 Entanglement swapping: a Bell measurement on photons and projectsb cphotons and into an entangled state, even if they never interacted. . . . 151a d o
• 14.1 Circuit representation of a single qubit . . . . . . . . . . . . . . . . . . . . 155
• 14.2 Quantum circuit: state initialization and basic gate sequence . . . . . . . . 156n
• 14.3 Quantum register composed of = 5 qubits . . . . . . . . . . . . . . . . . 160n
• 14.4 CNOT gate: control on , target on . . . . . . . . . . . . . . . . . . . . 164q q0 1
• 14.5 SWAP gate: exchanges the states of two qubits . . . . . . . . . . . . . . . 168ia
• 14.6 Toffoli gate (CCNOT): flips third qubit if both control