An introduction to modern physics

A model can work very well and still have limits

Newton's laws allow us to study object motion in many everyday situations. Wave optics explains interference and diffraction. The success of these models does not mean that each describes all phenomena at atomic scales or at speeds close to that of light. A model is evaluated by its predictions and the conditions under which it agrees with experiments.

When experiments reveal a regularity incompatible with a model, we need to review its assumptions. That does not turn all its results into errors. Classical mechanics remains an excellent approximation for many macroscopic motions at speeds much smaller than c. A broader theory must explain why that approximation works in its domain.

In this lesson we will build an initial connection between three areas: quantum physics, atomic/nuclear physics and special relativity. They are not a collection of mysterious names: they answer concrete questions. Why do certain metals emit electrons only above a particular light frequency? Why do atoms produce spectral lines? Why is the energy of a massive particle not given by mv²/2 when v approaches c?

Review energy, frequency and microscopic scales

Energy is measured in joules, J. For atomic processes, a joule is a very large unit compared with the energies involved. We often use the electronvolt, eV: the energy corresponding to the magnitude of the elementary charge crossing a potential difference of one volt. The conversion is approximately 1 eV = 1,602 × 10⁻¹⁹ J. The eV is energy, not voltage.

A wave's frequency f is the number of cycles per second, in Hz. In a vacuum, electromagnetic radiation obeys c = λf, where λ is wavelength and c ≈ 3,00 × 10⁸ m/s. The smaller λ, the greater f. Do not confuse frequency with intensity: a beam's total energy can be increased without changing frequency by increasing the amount of radiation crossing an area per unit time.

In calculations we will use h ≈ 6,626 × 10⁻³⁴ J·s for Planck's constant. A constant with such a small value in SI units helps explain why quantum effects are more evident at microscopic scales. Powers of ten require care: 1 nm = 10⁻⁹ m and 1 μm = 10⁻⁶ m.

What quantization of radiation means

Electromagnetic radiation can exchange energy with matter in quanta called photons. The energy of a photon with frequency f is E = hf, or E = hc/λ in a vacuum. This relationship connects frequency with the energy of an individual interaction. The product J·s × s⁻¹ has units of joules.

A photon with higher frequency has more energy. Increasing only the intensity of a fixed-frequency beam means, in the model, increasing the average number of photons arriving per unit time and area. It does not increase each photon's energy hf. If the beam carries total energy E_beam with photons of the same frequency, the approximate number will be E_beam/(hf).

This does not eliminate the wave description: interference and diffraction are still observed. Quantum physics predicts both the statistical pattern of many events and the localized detection of each event. Classical images of a “spread-out material wave” and a “little ball” are limited analogies. The quantum formalism does not require a particle to be literally a small sphere oscillating like a string.

The photoelectric effect: interpreting a threshold

Illuminating a suitable surface can emit electrons. In the simple one-photon-per-electron photoelectric description, some of the received energy is needed to free the electron from the material. This minimum energy is called the work function, φ, and depends on the surface. Energy conservation leads to K_max = hf − φ, when hf is sufficient to allow emission.

The threshold frequency is f₀ = φ/h. Below it, the calculation hf − φ would be negative: that does not mean the electron escaped with negative kinetic energy. It means emission is not allowed in this model. Above the threshold, increasing frequency increases maximum kinetic energy; increasing intensity at fixed frequency can increase the number of emitted electrons, subject to surface and device conditions.

In an experiment with a retarding potential, the magnitude U_stopping needed to prevent the most energetic electrons from arriving satisfies eU_stopping = K_max. Not all emitted electrons have maximum energy, so this relationship describes one end of the distribution. The introductory explanation assumes suitable radiation and operating conditions; multiphoton phenomena and other details require more complete models.

Matter with wavelength and probabilistic predictions

Diffraction is not exclusive to light. Electrons and other material particles can also produce interference patterns. The de Broglie relationship associates wavelength with momentum: λ = h/p. For a particle of mass m moving sufficiently slowly compared with c, we can use p ≈ mv; the momentum relationship needs a relativistic correction at high speeds.

A macroscopic particle with large momentum in SI units has an extremely small wavelength. This helps explain why we do not normally see a ball diffract when passing through a doorway. An electron, however, can have a wavelength comparable to the separation between atoms in a crystal, allowing observable effects. Comparing scales matters as much as the formula.

The quantum state allows calculation of probabilities for measurement results. This is not merely a lack of experimental care: experiments prepared in the same way produce distributions predicted by the theory. The uncertainty principle for position and momentum can be written ΔxΔp ≥ h/(4π), where Δx and Δp represent appropriate statistical spreads. It does not claim that “nothing can be known”, nor is it merely an instrument malfunction; it limits simultaneous preparation of these distributions.

Energy levels and spectral lines

An atom contains a nucleus and electrons. Electrical attraction matters to atomic structure, but imagining electrons travelling in classical planetary orbits does not adequately describe all phenomena. In the quantum treatment, allowed electronic states have characteristic energies and spatial distributions; an orbital is not a rigid track along which an electron is observed circulating.

When moving from a higher-energy state E_i to a lower one E_f, the atom can emit a photon with hf = E_i − E_f. In absorption, an energy difference can be supplied by radiation. Because bound-state energies do not fill an arbitrary continuum, lines associated with certain differences are observed. Selection rules and other details also exist: matching energies is an important condition, not a complete description of all transition probabilities.

For hydrogen levels in an introductory model, E_n ≈ −13,6 eV/n², with n a positive integer and zero reference at the ionization threshold. The negative sign indicates a bound state. The state n = 1 is lower than n = 2. The formula predicts a basic structure of a one-electron atom well, but must not be used unchanged for all atoms. The modern quantum description also explains aspects that the Bohr model does not describe.

The nucleus and types of radioactive transformation

The nucleus contains protons and, in the usual isotopes except the simplest hydrogen, neutrons. The proton number Z identifies the element. Mass number A counts protons and neutrons; it is not mass in kilograms. Isotopes of the same element have the same Z and different neutron numbers. Electron transitions and nuclear transformations belong to different scales and mechanisms.

Some nuclei are unstable and transform spontaneously. In alpha emission, the nucleus emits a particle containing two protons and two neutrons: the residual nucleus has Z reduced by 2 and A by 4. In beta-minus emission, a neutron becomes a proton with emission of an electron and an antineutrino: Z increases by 1 and A does not change. The beta electron is not simply an electron that was already orbiting the nucleus and was pulled away. In gamma emission, the nucleus enters a lower-energy state by emitting a photon, without changing Z or A.

In reactions, the appropriate quantities of the complete system are conserved, including energy, momentum and charge. Ionization means removing or adding electrons to an atom; it is not the same as altering its nucleus. Ionizing radiation can produce ionization, but activity, energy per event and the effect in a material are different quantities. This lesson covers models without guiding experiments with radioactive sources.

A statistical law is not a clock for each nucleus

For a large population of nuclei of a given radionuclide in the exponential-decay model, the expected number not yet transformed is N(t) = N₀e^−λt. The decay constant λ is positive and has units of inverse time. The negative sign describes the decrease of the original population. Transformed nuclei have not ceased to exist as matter: they have produced decay products.

The half-life T_1/2 satisfies N(T_1/2) = N₀/2. The exponential relationship gives λ = ln 2/T_1/2 and N(t) = N₀(1/2)^t/T₁/₂. After two half-lives, one quarter remains on average, not zero. The law does not choose which nucleus transforms first. For small populations, relative fluctuations matter.

Activity A = λN measures the expected number of transformations per second when λ is in s⁻¹; the unit is becquerel, Bq. Do not confuse activity with energy released per second, which also depends on each event's energy and the products. In decay chains, products may still be unstable; a single exponential may not describe all measured activity. Always specify which population and process were modeled.

Reference frames, the speed of light and rest energy

Special relativity starts from equivalence of physical laws in inertial frames and invariance of the speed of light in vacuum, c, for those observers. An inertial frame is one in which a body with no net force has uniform straight-line motion. The theory describes space and time measurements between observers; it does not claim all statements are relative or that any value can be chosen.

Comparing clocks and lengths involves defining events and simultaneity in each frame. One consequence is the factor γ = 1/√(1 − v²/c²). For a particle with rest mass m, its total free-particle energy is E = γmc², rest energy is E₀ = mc² and kinetic energy is K = (γ − 1)mc². When v is much smaller than c, the approximation recovers K ≈ mv²/2.

As v approaches c, γ grows without limit; a massive particle cannot be accelerated to c by a finite energy input. A photon has zero rest mass and energy E = pc: we must not substitute m = 0 into γmc² with v = c, since γ is not finite there. Formulas need their domains, just as in algebra.

What changes in a reaction that releases energy

A system's rest mass is linked to its energy in the frame where total momentum is zero. If a collection loses energy to the outside, the mass of the remaining system can decrease. In nuclear physics, a difference between total reactant and product rest masses can correspond to kinetic energy and radiation: reaction Q = (initial m − final m)c², correctly including all relevant particles.

It is not correct simply to say that “mass disappears” and forget the energy that left. The balance for the complete system still conserves energy and momentum. Nor can all of an object's mass be arbitrarily transformed into useful energy in a device: the fact that mc² is large does not describe an available mechanism or an efficiency.

These ideas connect topics already studied. A photon carries energy and momentum; a nucleus can emit a photon and recoil; a transition must respect the system's balance. Understanding the principles prepares later study of quantum and relativistic equations. Here we seek conceptual foundations and introductory calculations, without presenting a complete theory in a few pages.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

A new theory makes everything that came before useless.

Classical models remain useful in regimes where they are good approximations; extended theories explain their limits.

More intensity gives more energy to each photon.

Individual energy depends on frequency. Intensity can change the average number of incident photons.

An orbital is an imprecisely drawn planetary orbit.

A quantum electronic state describes energies and probability distributions; it is not simply an unknown classical trajectory.

After two half-lives the original sample disappears.

The expected remaining value is one quarter. The law describes a radionuclide's population and allows statistical fluctuations.

Activity directly measures all radiation energy or every radiation effect.

Becquerels measure transformations per second. Energy per event, products, material and geometry are additional information.

E = mc² means kinetic energy is always mc².

mc² is rest energy. For a free massive particle, total E = γmc² and K = (γ − 1)mc².