, contents
preface vii
Why Mathematica?...................................................................................................................................... viii
Mathematica source code...................................................................................................................viii
outline of discussed topics............................................................................................................................ viii
1 Wolfram language overview 1
1.1 introduction.............................................................................................................................................2
1.1.1 exercises....................................................................................................................................2
1.2 variables and assignments...................................................................................................................... 3
1.2.1 immediate and delayed assignments........................................................................................ 4
1.2.2 exercises....................................................................................................................................4
1.3 four kinds of bracketing........................................................................................................................ 5
1.4 prefix and postfix.................................................................................................................................... 5
1.4.1 exercises....................................................................................................................................6
1.5 programming constructs........................................................................................................................6
1.5.1 procedural programming.......................................................................................................... 6
1.5.2 exercises....................................................................................................................................7
1.5.3 functional programming........................................................................................................... 8
1.5.4 exercises....................................................................................................................................9
1.6 function definitions................................................................................................................................ 9
1.6.1 immediate function definitions..................................................................................................9
1.6.2 delayed function definitions..................................................................................................... 10
1.6.3 memoization: functions that remember their results...........................................................10
1.6.4 functions with conditions on their arguments...................................................................... 11
1.6.5 functions with optional arguments........................................................................................12
1.7 rules and replacements........................................................................................................................12
1.7.1 immediate and delayed rules...................................................................................................13
1.7.2 repeated rule replacement...................................................................................................... 14
1.8 debugging and finding out how Mathematica expressions are evaluated........................................ 14
1.8.1 exercises................................................................................................................................. 15
1.9 many ways to define the factorial function........................................................................................16
1.9.1 exercises................................................................................................................................. 18
1.10 vectors, matrices, tensors................................................................................................................... 18
1.10.1 vectors..................................................................................................................................... 18
1.10.2 matrices................................................................................................................................... 19
1.10.3 sparse vectors and matrices.....................................................................................................19
1.10.4 matrix diagonalization............................................................................................................20
1.10.5 tensor operations.................................................................................................................... 22
1.10.6 exercises................................................................................................................................. 23
1.11 complex numbers.................................................................................................................................24
1.12 units.......................................................................................................................................................24
2 quantum mechanics: states and operators 27
2.1 basis sets and representations.............................................................................................................. 28
iii
, 2.1.1 incomplete basis sets.............................................................................................................. 28
2.1.2 exercises................................................................................................................................. 29
2.2 time-independent Schrödinger equation..............................................................................................29
2.2.1 diagonalization........................................................................................................................30
2.2.2 exercises................................................................................................................................. 30
2.3 time-dependent Schrödinger equation.................................................................................................30
2.3.1 time-independent basis........................................................................................................... 31
2.3.2 time-dependent basis: interaction picture............................................................................. 32
# $
2.3.3 special case: Ĥ(t ), Ĥ(t J ) = 0 6(t, tJ)..................................................................................32
2.3.4 special case: time-independent Hamiltonian...........................................................................33
2.3.5 exercises................................................................................................................................. 33
2.4 basis construction................................................................................................................................ 33
2.4.1 description of a single degree of freedom............................................................................. 33
2.4.2 description of coupled degrees of freedom............................................................................34
2.4.3 reduced density matrices......................................................................................................... 36
2.4.4 exercises................................................................................................................................. 38
3 spin and angular momentum 39
3.1 quantum-mechanical spin and angular momentum operators..........................................................40
3.1.1 exercises................................................................................................................................. 41
3.2 spin-1/2 electron in a dc magnetic field............................................................................................ 41
3.2.1 time-independent Schrödinger equation................................................................................. 43
3.2.2 exercises................................................................................................................................. 43
3.3 coupled spin systems: 87Rb hyperfine structure.................................................................................43
3.3.1 eigenstate analysis..................................................................................................................45
3.3.2 “magic” magnetic field.............................................................................................................47
3.3.3 coupling to an oscillating magnetic field............................................................................... 47
3.3.4 exercises................................................................................................................................. 52
3.4 coupled spin systems: Ising model in a transverse field.................................................................... 52
3.4.1 basis set....................................................................................................................................53
3.4.2 asymptotic ground states....................................................................................................... 54
3.4.3 Hamiltonian diagonalization...................................................................................................54
3.4.4 analysis of the ground state................................................................................................... 55
3.4.5 exercises................................................................................................................................. 61
3.5 coupled spin systems: quantum circuits............................................................................................. 62
3.5.1 quantum gates........................................................................................................................ 62
3.5.2 a simple quantum circuit.........................................................................................................65
3.5.3 application: the Quantum Fourier Transform...................................................................... 66
3.5.4 application: quantum phase estimation................................................................................. 68
3.5.5 exercises................................................................................................................................. 70
4 quantum motion in real space 71
4.1 one particle in one dimension............................................................................................................. 72
4.1.1 units..........................................................................................................................................72
4.1.2 computational basis functions................................................................................................ 73
4.1.3 the position operator...............................................................................................................77
4.1.4 the potential-energy operator................................................................................................ 78
4.1.5 the kinetic-energy operator.................................................................................................... 78
4.1.6 the momentum operator........................................................................................................ 79
4.1.7 example: gravity well.............................................................................................................. 79
4.1.8 the Wigner quasi-probability distribution............................................................................... 83
4.1.9 1D dynamics in the square well..............................................................................................86
4.1.10 1D dynamics in a time-dependent potential..........................................................................89
4.2 non-linear Schrödinger equation...........................................................................................................91
4.2.1 ground state of the non-linear Schrödinger equation...........................................................93
4.3 several particles in one dimension: interactions................................................................................. 95
, iv CONTENTS
4.3.1 two identical particles in one dimension with contact interaction.......................................95
4.3.2 two particles in one dimension with arbitrary interaction................................................... 101
4.4 one particle in several dimensions.................................................................................................... 102
4.4.1 exercises............................................................................................................................... 105
5 combining spatial motion and spin 107
5.1 one particle in 1D with spin..............................................................................................................108
5.1.1 separable Hamiltonian...........................................................................................................108
5.1.2 non-separable Hamiltonian..................................................................................................... 108
5.1.3 exercises............................................................................................................................... 113
5.2 one particle in 2D with spin: Rashba coupling............................................................................... 113
5.2.1 exercises............................................................................................................................... 115
5.3 phase-space dynamics in the Jaynes–Cummings model.................................................................. 115
5.3.1 exercises............................................................................................................................... 119
list of attached notebooks 121
index 123
solutions to exercises 127
preface vii
Why Mathematica?...................................................................................................................................... viii
Mathematica source code...................................................................................................................viii
outline of discussed topics............................................................................................................................ viii
1 Wolfram language overview 1
1.1 introduction.............................................................................................................................................2
1.1.1 exercises....................................................................................................................................2
1.2 variables and assignments...................................................................................................................... 3
1.2.1 immediate and delayed assignments........................................................................................ 4
1.2.2 exercises....................................................................................................................................4
1.3 four kinds of bracketing........................................................................................................................ 5
1.4 prefix and postfix.................................................................................................................................... 5
1.4.1 exercises....................................................................................................................................6
1.5 programming constructs........................................................................................................................6
1.5.1 procedural programming.......................................................................................................... 6
1.5.2 exercises....................................................................................................................................7
1.5.3 functional programming........................................................................................................... 8
1.5.4 exercises....................................................................................................................................9
1.6 function definitions................................................................................................................................ 9
1.6.1 immediate function definitions..................................................................................................9
1.6.2 delayed function definitions..................................................................................................... 10
1.6.3 memoization: functions that remember their results...........................................................10
1.6.4 functions with conditions on their arguments...................................................................... 11
1.6.5 functions with optional arguments........................................................................................12
1.7 rules and replacements........................................................................................................................12
1.7.1 immediate and delayed rules...................................................................................................13
1.7.2 repeated rule replacement...................................................................................................... 14
1.8 debugging and finding out how Mathematica expressions are evaluated........................................ 14
1.8.1 exercises................................................................................................................................. 15
1.9 many ways to define the factorial function........................................................................................16
1.9.1 exercises................................................................................................................................. 18
1.10 vectors, matrices, tensors................................................................................................................... 18
1.10.1 vectors..................................................................................................................................... 18
1.10.2 matrices................................................................................................................................... 19
1.10.3 sparse vectors and matrices.....................................................................................................19
1.10.4 matrix diagonalization............................................................................................................20
1.10.5 tensor operations.................................................................................................................... 22
1.10.6 exercises................................................................................................................................. 23
1.11 complex numbers.................................................................................................................................24
1.12 units.......................................................................................................................................................24
2 quantum mechanics: states and operators 27
2.1 basis sets and representations.............................................................................................................. 28
iii
, 2.1.1 incomplete basis sets.............................................................................................................. 28
2.1.2 exercises................................................................................................................................. 29
2.2 time-independent Schrödinger equation..............................................................................................29
2.2.1 diagonalization........................................................................................................................30
2.2.2 exercises................................................................................................................................. 30
2.3 time-dependent Schrödinger equation.................................................................................................30
2.3.1 time-independent basis........................................................................................................... 31
2.3.2 time-dependent basis: interaction picture............................................................................. 32
# $
2.3.3 special case: Ĥ(t ), Ĥ(t J ) = 0 6(t, tJ)..................................................................................32
2.3.4 special case: time-independent Hamiltonian...........................................................................33
2.3.5 exercises................................................................................................................................. 33
2.4 basis construction................................................................................................................................ 33
2.4.1 description of a single degree of freedom............................................................................. 33
2.4.2 description of coupled degrees of freedom............................................................................34
2.4.3 reduced density matrices......................................................................................................... 36
2.4.4 exercises................................................................................................................................. 38
3 spin and angular momentum 39
3.1 quantum-mechanical spin and angular momentum operators..........................................................40
3.1.1 exercises................................................................................................................................. 41
3.2 spin-1/2 electron in a dc magnetic field............................................................................................ 41
3.2.1 time-independent Schrödinger equation................................................................................. 43
3.2.2 exercises................................................................................................................................. 43
3.3 coupled spin systems: 87Rb hyperfine structure.................................................................................43
3.3.1 eigenstate analysis..................................................................................................................45
3.3.2 “magic” magnetic field.............................................................................................................47
3.3.3 coupling to an oscillating magnetic field............................................................................... 47
3.3.4 exercises................................................................................................................................. 52
3.4 coupled spin systems: Ising model in a transverse field.................................................................... 52
3.4.1 basis set....................................................................................................................................53
3.4.2 asymptotic ground states....................................................................................................... 54
3.4.3 Hamiltonian diagonalization...................................................................................................54
3.4.4 analysis of the ground state................................................................................................... 55
3.4.5 exercises................................................................................................................................. 61
3.5 coupled spin systems: quantum circuits............................................................................................. 62
3.5.1 quantum gates........................................................................................................................ 62
3.5.2 a simple quantum circuit.........................................................................................................65
3.5.3 application: the Quantum Fourier Transform...................................................................... 66
3.5.4 application: quantum phase estimation................................................................................. 68
3.5.5 exercises................................................................................................................................. 70
4 quantum motion in real space 71
4.1 one particle in one dimension............................................................................................................. 72
4.1.1 units..........................................................................................................................................72
4.1.2 computational basis functions................................................................................................ 73
4.1.3 the position operator...............................................................................................................77
4.1.4 the potential-energy operator................................................................................................ 78
4.1.5 the kinetic-energy operator.................................................................................................... 78
4.1.6 the momentum operator........................................................................................................ 79
4.1.7 example: gravity well.............................................................................................................. 79
4.1.8 the Wigner quasi-probability distribution............................................................................... 83
4.1.9 1D dynamics in the square well..............................................................................................86
4.1.10 1D dynamics in a time-dependent potential..........................................................................89
4.2 non-linear Schrödinger equation...........................................................................................................91
4.2.1 ground state of the non-linear Schrödinger equation...........................................................93
4.3 several particles in one dimension: interactions................................................................................. 95
, iv CONTENTS
4.3.1 two identical particles in one dimension with contact interaction.......................................95
4.3.2 two particles in one dimension with arbitrary interaction................................................... 101
4.4 one particle in several dimensions.................................................................................................... 102
4.4.1 exercises............................................................................................................................... 105
5 combining spatial motion and spin 107
5.1 one particle in 1D with spin..............................................................................................................108
5.1.1 separable Hamiltonian...........................................................................................................108
5.1.2 non-separable Hamiltonian..................................................................................................... 108
5.1.3 exercises............................................................................................................................... 113
5.2 one particle in 2D with spin: Rashba coupling............................................................................... 113
5.2.1 exercises............................................................................................................................... 115
5.3 phase-space dynamics in the Jaynes–Cummings model.................................................................. 115
5.3.1 exercises............................................................................................................................... 119
list of attached notebooks 121
index 123
solutions to exercises 127
