{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Standard Model overview\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/standard-model-overview). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "The Standard Model is the quantum-field framework for electromagnetic, weak,\n", "and strong interactions. It specifies elementary fields, their symmetries, and\n", "the allowed couplings among them. Its predictions have passed an enormous range\n", "of experimental tests, but it is not a theory of gravity or a complete\n", "description of nature.\n", "\n", "The familiar particle chart is only an inventory. The real model is a system of\n", "quantum fields constrained by local gauge symmetry, Lorentz symmetry, and\n", "quantum consistency.\n", "\n", "## Field content\n", "\n", "The elementary fields fall into three groups:\n", "\n", "- spin-one-half fermion fields: six quarks and six leptons;\n", "- spin-one gauge fields: gluons, photon, and the\n", " $W^\\pm$ and\n", " $Z^0$ bosons;\n", "- one spin-zero Higgs field, whose observed excitation is the Higgs boson.\n", "\n", "Fermions come in three generations with the same charge pattern but increasing\n", "masses. Ordinary stable matter uses mostly first-generation up and down quarks\n", "and electrons. Heavier generations are unstable and decay through allowed\n", "interactions.\n", "\n", "Each field has an antiparticle relation. Some neutral bosons are their own\n", "antiparticles, while a neutral composite such as the neutron has a distinct\n", "antiparticle because other quantum numbers differ.\n", "\n", "## Gauge symmetry\n", "\n", "The Standard Model gauge group is conventionally written\n", "\n", "\n", "$$\n", "SU(3)_{\\mathrm C}\\times SU(2)_{\\mathrm L}\\times U(1)_{\\mathrm Y}\n", "$$\n", "\n", "\n", "The factors organize:\n", "\n", "- color and the strong interaction;\n", "- weak isospin;\n", "- weak hypercharge.\n", "\n", "After electroweak symmetry breaking, combinations of the electroweak gauge\n", "fields appear as the photon,\n", "$W^+$,\n", "$W^-$, and\n", "$Z^0$. Electric charge is\n", "related to weak isospin and hypercharge by\n", "\n", "\n", "$$\n", "Q=T_3+\\frac{Y}{2}\n", "$$\n", "\n", "\n", "when hypercharge is normalized with this convention. References sometimes place\n", "the factor of one half inside the definition of\n", "$Y$, so conventions must be\n", "aligned before comparing assignments.\n", "\n", "Gauge symmetry is not merely a taxonomy. Requiring local symmetry determines\n", "the form of interactions and introduces gauge fields.\n", "\n", "## Quantum electrodynamics\n", "\n", "Electrically charged quarks and leptons couple to the photon. The strength is\n", "described by the electromagnetic coupling, often expressed through the\n", "fine-structure constant\n", "\n", "\n", "$$\n", "\\alpha=\\frac{e^2}{4\\pi\\epsilon_0\\hbar c}\\approx\\frac{1}{137}\n", "$$\n", "\n", "\n", "The displayed fraction is a low-energy approximation. Couplings run with energy\n", "scale because quantum fluctuations affect measured interaction strengths.\n", "\n", "Photons carry no electric charge, so quantum electrodynamics lacks a direct\n", "photon self-coupling of the same gauge type as gluon self-interaction. Higher\n", "order processes can still make photons scatter through charged virtual fields.\n", "\n", "## Quantum chromodynamics\n", "\n", "Quarks carry color charge and interact through eight gluon fields. Gluons also\n", "carry color, so they interact with one another. Two central behaviors are:\n", "\n", "- **confinement** at ordinary long-distance scales, so isolated colored quarks\n", " and gluons are not observed;\n", "- **asymptotic freedom** at sufficiently high momentum transfer, where the\n", " strong coupling becomes weaker and short-distance quark and gluon calculations\n", " become more perturbative.\n", "\n", "Observable hadrons are color singlets. Baryons, mesons, and more exotic hadrons\n", "contain valence constituents together with gluons and sea quark–antiquark\n", "contributions. Most visible hadron mass comes from QCD energy, not from simply\n", "adding valence-quark rest masses.\n", "\n", "## Weak interactions\n", "\n", "The weak interaction acts on quarks and leptons through the massive\n", "$W^\\pm$ and\n", "$Z^0$ bosons. Charged\n", "weak currents can change quark flavor and charged-lepton/neutrino identity. A\n", "beta-minus process is\n", "\n", "\n", "$$\n", "n\\rightarrow p+e^-+\\bar\\nu_e\n", "$$\n", "\n", "\n", "At the quark level, a down quark changes into an up quark. The neutral\n", "$Z^0$ current does not\n", "change electric charge.\n", "\n", "Weak interactions violate parity maximally in their charged-current coupling:\n", "left-handed fermion fields and right-handed antifermion fields participate in a\n", "chiral way. This is a precise field-theory statement, not a claim that ordinary\n", "macroscopic left and right are generally inequivalent.\n", "\n", "## Electroweak symmetry breaking\n", "\n", "The Higgs field has a nonzero vacuum expectation value. Its coupling to the\n", "electroweak fields gives masses to\n", "$W^\\pm$ and\n", "$Z^0$ while leaving the\n", "photon massless. Yukawa couplings give fermions their Standard Model mass terms.\n", "\n", "The Higgs mechanism does not account for all mass in everyday matter. Proton\n", "and neutron masses are dominated by QCD binding and field energy. It is more\n", "accurate to say that the Higgs field supplies elementary fermion rest-mass terms\n", "and electroweak gauge-boson masses.\n", "\n", "## Flavor mixing\n", "\n", "Weak-interaction quark states are mixtures of mass eigenstates, summarized by\n", "the Cabibbo–Kobayashi–Maskawa matrix:\n", "\n", "\n", "$$\n", "\\begin{pmatrix}d'\\\\s'\\\\b'\\end{pmatrix}=V_{\\mathrm{CKM}}\\begin{pmatrix}d\\\\s\\\\b\\end{pmatrix}\n", "$$\n", "\n", "\n", "The matrix is unitary in the three-generation Standard Model. Its complex phase\n", "permits CP violation in quark processes, but the measured amount is insufficient\n", "by itself to explain the cosmic matter–antimatter imbalance.\n", "\n", "Neutrino flavor states are mixtures of neutrino mass states through the PMNS\n", "matrix:\n", "\n", "\n", "$$\n", "|\\nu_\\alpha\\rangle=\\sum_iU_{\\alpha i}^*|\\nu_i\\rangle\n", "$$\n", "\n", "\n", "Neutrino oscillations establish nonzero neutrino mass differences. The minimal\n", "original Standard Model assumed massless neutrinos, so neutrino masses require\n", "an extension; their absolute mass scale and detailed origin remain active\n", "questions.\n", "\n", "## Conservation laws and selection rules\n", "\n", "Exact conservation of electric charge follows from unbroken electromagnetic\n", "gauge symmetry. Total energy, momentum, and angular momentum follow spacetime\n", "symmetries. Color is conserved even though individual color labels are\n", "basis-dependent and confined.\n", "\n", "Baryon and lepton numbers are extremely useful low-energy bookkeeping rules,\n", "but their theoretical status needs nuance. Individual flavors can change, and\n", "nonperturbative electroweak effects do not preserve baryon and lepton number\n", "separately in the same way as electric charge. Experimental searches strongly\n", "constrain processes such as proton decay." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "# Introductory classification of elementary fermions, not all interactions.\n", "fermions = [\n", " {\"name\": \"electron\", \"charge\": -1, \"kind\": \"lepton\"},\n", " {\"name\": \"electron neutrino\", \"charge\": 0, \"kind\": \"lepton\"},\n", " {\"name\": \"up quark\", \"charge\": 2 / 3, \"kind\": \"quark\"},\n", "]\n", "for particle in fermions:\n", " interactions = [\"weak\"]\n", " if particle[\"charge\"] != 0:\n", " interactions.append(\"electromagnetic\")\n", " if particle[\"kind\"] == \"quark\":\n", " interactions.append(\"strong\")\n", " print(particle[\"name\"], interactions)\n", "print(\"Gravity lies outside the Standard Model.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "This Python is introductory. Gluons participate in the strong interaction,\n", "gauge bosons have self-interactions determined by the theory, and Higgs and\n", "electroweak couplings require more than matter-particle tags.\n", "\n", "## From fields to predictions\n", "\n", "The Standard Model Lagrangian specifies propagating fields and interactions.\n", "Perturbation theory expands some observables in powers of couplings, often\n", "represented with Feynman diagrams. A diagram is a term or class of terms in a\n", "calculation, not a literal time-lapse photograph of hidden particle paths.\n", "\n", "Predictions combine:\n", "\n", "1. a process and initial state;\n", "2. quantum amplitudes from allowed interactions;\n", "3. interference among amplitudes;\n", "4. phase-space integration and parton structure when needed;\n", "5. detector acceptance and reconstruction for comparison with data.\n", "\n", "Probabilities come from squared total amplitudes, not from adding separate\n", "diagram probabilities before interference.\n", "\n", "Renormalization relates theory parameters to measured quantities at a chosen\n", "energy scale. The Standard Model is predictive because a finite set of measured\n", "parameters supports many independent predictions, not because every particle\n", "mass and mixing angle is derived from first principles.\n", "\n", "## Experimental evidence\n", "\n", "Evidence includes deep-inelastic scattering for quark substructure, jets for\n", "quarks and gluons, weak neutral currents, production and decay of\n", "$W$ and\n", "$Z$ bosons, precision\n", "electroweak measurements, neutrino oscillations, and observation of a Higgs\n", "boson with properties so far consistent with the Standard Model.\n", "\n", "Collider detectors reconstruct tracks, showers, missing transverse momentum,\n", "and invariant masses. A resonance claim requires a signal model, backgrounds,\n", "systematic uncertainties, and statistical assessment. The particle chart is a\n", "summary of this inference, not raw visual observation.\n", "\n", "## What the Standard Model does not explain\n", "\n", "The framework does not include a quantum theory of gravity. It does not identify\n", "the observed dark matter, explain dark energy, or fully account for the\n", "matter–antimatter asymmetry. It contains many measured parameters whose values\n", "are not predicted internally. The origin of neutrino masses also lies beyond\n", "its minimal original form.\n", "\n", "These limitations do not make its successful predictions approximate guesses.\n", "They define its scope and motivate extensions. Any proposed extension must\n", "recover the Standard Model's tested results in the regime where they apply.\n", "\n", "## Consistency checks for an overview\n", "\n", "A sound map should verify:\n", "\n", "- quarks carry color and fractional electric charge;\n", "- leptons carry no color;\n", "- charged fermions couple electromagnetically;\n", "- quarks and leptons participate in weak interactions;\n", "- gravity is labeled outside the Standard Model;\n", "- the Higgs is a scalar, not a gauge boson;\n", "- composite hadrons are not placed among elementary fields;\n", "- antiparticles and charge signs are explicit;\n", "- numerical properties cite a current, versioned source.\n", "\n", "The map remains an introduction. Full calculations require quantum field theory,\n", "group representations, gauge fixing, renormalization, nonperturbative QCD, and\n", "careful experimental modeling.\n", "\n", "## A worked Python notebook\n", "\n", "Illustrate the convention Q equals T3 plus Y divided by two for a fixed weak-isospin component. The question for this calculation is precise: how does **electric charge for T3=minus-one-half**\n", "change when we change **weak hypercharge**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "Hypercharge assignments are discrete properties of representations; the curve is an algebraic guide, not a continuum of new particles. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "-2 to 2 dimensionless. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for weak hypercharge, expressed in dimensionless. The output is\n", "electric charge for T3=minus-one-half, expressed in elementary-charge units. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\frac{q}{2} - \\frac{1}{2}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At -1 dimensionless, the expression gives -1 elementary-charge units. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"-1/2+q/2\", locals={\"q\": q})\n", "reference_input = -1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"dimensionless\")\n", "print(\n", " \"Predicted electric charge for T3=minus-one-half:\",\n", " reference_output,\n", " \"elementary-charge units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "-1.5 elementary-charge units. The midpoint, at 0 dimensionless, gives\n", "-0.5 elementary-charge units. The upper endpoint gives 0.5 elementary-charge units.\n", "Read these together with the assumptions: they describe different choices of weak hypercharge\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([-2.0, 0.0, 2.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(\n", " f\"{input_value:.6g} dimensionless -> \"\n", " f\"{output_value:.6g} elementary-charge units\"\n", " )\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows weak hypercharge; the vertical axis shows electric charge for T3=minus-one-half. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "input_grid = np.linspace(-2.0, 2.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Weak hypercharge (dimensionless)\")\n", "ax.set_ylabel(\"Electric charge for t3=minus-one-half (elementary-charge units)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "electric charge for T3=minus-one-half change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at -1 dimensionless is\n", "0.5 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = \\frac{1}{2}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = -0.98\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"elementary-charge units\")\n", "print(\"Full expression:\", full_prediction, \"elementary-charge units\")\n", "print(\n", " \"Approximation difference:\",\n", " full_prediction - linear_prediction,\n", " \"elementary-charge units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Here the changed input is -0.98 dimensionless. The tangent estimate is\n", "-0.99 elementary-charge units, while the full expression gives\n", "-0.99 elementary-charge units. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "input_span = 4.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "weak hypercharge inside the stated interval, predict whether electric charge for T3=minus-one-half will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: Hypercharge assignments are discrete properties of representations; the curve is an algebraic guide, not a continuum of new particles. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of elementary-charge units:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(2.0) - evaluate(-2.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-2.0, 2.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 3*, §11.2, “Particle Conservation Laws”](https://openstax.org/books/university-physics-volume-3/pages/11-2-particle-conservation-laws).\n", "- [OpenStax, *University Physics Volume 3*, §11.3, “Quarks”](https://openstax.org/books/university-physics-volume-3/pages/11-3-quarks).\n", "- [OpenStax, *University Physics Volume 3*, §11.5, “The Standard Model”](https://openstax.org/books/university-physics-volume-3/pages/11-5-the-standard-model).\n", "- [Particle Data Group, *Review of Particle Physics*](https://pdg.lbl.gov/).\n", "- [Tipler and Llewellyn, *Modern Physics*, particle physics and the Standard Model](https://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf).\n", "- [Benjamin Crowell, *Light and Matter*, particle physics](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, Nuclear and Particle Physics bookshelf](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "standard-model-overview.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/standard-model-overview", "source_sha256": "1540467c1e28feb03d01f65eb764b09b16e45f3cfeffa634057b709d51e5ee18" } }, "nbformat": 4, "nbformat_minor": 5 }