Understand Quantum Physics — Understand E.2 Quantum physics - IB Physics

SECTION 1

Historical Context & Motivation

By the late 1800s, classical physics — the physics of Newton and Maxwell — seemed nearly complete. Scientists could predict the motion of planets, describe electromagnetic waves, and explain heat engines. Yet several stubborn experiments refused to fit the classical framework. The spectrum of light emitted by hot objects, the way light ejects electrons from metals, and the discrete colors produced by glowing gases all pointed toward something fundamentally new. These puzzles ignited a revolution that became quantum physics, a theory describing nature at the smallest scales.

1900 • Planck's Quantum Hypothesis

Max Planck proposed that energy is emitted in discrete packets called quanta, solving the ultraviolet catastrophe problem for blackbody radiation.

1905 • Einstein & the Photoelectric Effect

Albert Einstein explained the photoelectric effect by treating light as a stream of energy packets called photons, each carrying energy E = hf.

1913 • Bohr's Atomic Model

Niels Bohr introduced quantized energy levels for electrons in atoms, explaining the line spectra of hydrogen.

1924 • de Broglie's Matter Waves

Louis de Broglie proposed that all matter has wave-like properties, with a wavelength inversely proportional to momentum.

1926 • Schrödinger's Wave Equation

Erwin Schrödinger formulated a wave equation that describes the probability of finding a particle in a given region of space.

The central question driving all of these breakthroughs was deceptively simple: How do energy and matter actually behave at very small scales? The answer — that particles can act like waves and waves can act like particles — shattered centuries of intuition and gave rise to the quantum world we explore in IB Physics E.2.

SECTION 2

Core Principles of Quantum Physics

Quantum physics rests on a handful of ideas that differ dramatically from everyday experience. In the classical world, a ball is always a particle and an ocean ripple is always a wave. In the quantum world, that neat separation breaks down. Below are the foundational concepts you need to master for IB Physics E.2.

Quantization of Energy

Energy is not continuous — it comes in discrete amounts. A photon carries energy E = hf, where h is Planck's constant and f is frequency.

The Photoelectric Effect

Light striking a metal surface can eject electrons, but only if each photon exceeds a minimum energy called the work function (φ). Increasing intensity alone does not help if the frequency is too low.

Wave–Particle Duality

Light can behave as a wave (diffraction, interference) or as a particle (photoelectric effect). Matter particles such as electrons also exhibit wave behaviour with a de Broglie wavelength.

Atomic Energy Levels

Electrons in atoms occupy specific energy levels. When an electron transitions between levels, a photon is emitted or absorbed with energy equal to the difference between those levels.

The Heisenberg Uncertainty Principle

It is impossible to simultaneously know a particle's exact position and momentum. The more precisely one is measured, the less precisely the other can be known: Δx · Δp ≥ h / (4π).

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KEY TAKEAWAY

Think of energy quantization like a staircase versus a ramp. In classical physics, energy is like a ramp — you can stand at any height. In quantum physics, energy is like a staircase — you can only stand on specific steps. Photons are the 'jumps' between steps, and each jump has a precise energy determined by E = hf.

SECTION 3

Visualizing the Photoelectric Effect

The photoelectric effect is one of the most important experiments in quantum physics. When light shines on a metal surface, electrons can be ejected — but only if the frequency of light is high enough. The diagram below shows the key features of this phenomenon and the graph that convinced physicists that light is quantized.

Left: Graph of maximum kinetic energy vs. frequency showing the threshold frequency f₀ below which no electrons are emitted. The slope of the line equals Planck's constant h. Right: Schematic of the photoelectric apparatus showing photons striking a metal plate and ejecting electrons.

Notice that the graph is a straight line with a positive slope starting at the threshold frequency f₀. Below this frequency, no electrons are emitted regardless of how bright the light is. This was the key observation that disproved the classical wave theory of light: if light were purely a wave, even dim light of any frequency should eventually provide enough energy to free electrons. Instead, quantum theory tells us each photon must individually carry enough energy (hf ≥ φ) to liberate a single electron.

SECTION 4

Mathematical Framework

Quantum physics at the IB level relies on a small set of powerful equations. Each one connects a measurable quantity to Planck's constant h = 6.63 × 10⁻³⁴ J·s, the fundamental constant of quantum mechanics. Let's walk through the key relationships.

PHOTON ENERGY

E = hf = hc / λ

Where E = photon energy (J), h = Planck's constant (6.63 × 10⁻³⁴ J·s), f = frequency (Hz), c = speed of light (3.00 × 10⁸ m/s), λ = wavelength (m).

PHOTOELECTRIC EQUATION

E_k(max) = hf − φ

Where $E_k$ (max) = maximum kinetic energy of ejected electrons (J), φ = work function of the metal (J) = the minimum energy needed to free an electron.

DE BROGLIE WAVELENGTH

λ = h / p = h / (mv)

Where λ = de Broglie wavelength (m), p = momentum (kg·m/s), m = mass (kg), v = velocity (m/s). This equation shows that every moving object has a wavelength, but for macroscopic objects it is far too small to detect.

ENERGY LEVEL TRANSITIONS

hf = E₂ − E₁

A photon emitted or absorbed during an electron transition has energy exactly equal to the difference between two energy levels E₂ and E₁. This explains why atomic spectra consist of discrete lines rather than a continuous rainbow.

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IB DATA BOOKLET TIP

All four equations above are provided in the IB data booklet. Your job on the exam is to recognize which equation to apply, substitute values with correct units, and interpret the result. Practice converting between eV and joules: 1 eV = 1.60 × 10⁻¹⁹ J.

SECTION 5

Atomic Energy Levels & Emission Spectra

One of the most striking pieces of evidence for quantum physics is the line emission spectrum of hydrogen. When hydrogen gas is excited by an electric current, it glows with a characteristic pinkish-purple color. Passing this light through a prism or diffraction grating reveals not a continuous rainbow but a set of distinct colored lines. Each line corresponds to electrons dropping from one quantized energy level to another, releasing a photon of a very specific wavelength.

The energy level diagram for hydrogen shows quantized levels from n = 1 (ground state at −13.6 eV) to n = ∞ (ionization at 0 eV). Colored arrows represent photon emission during transitions. The Balmer series (transitions ending at n = 2) produces visible light lines, while transitions to n = 1 (Lyman series) produce ultraviolet and transitions to n = 3 (Paschen series) produce infrared.

Notice how the energy levels get closer together as n increases. This means the energy differences between high levels are small, producing low-energy (long-wavelength, infrared) photons. Transitions down to the ground state (n = 1) involve the largest energy jumps and produce high-energy ultraviolet photons. The visible Balmer series — the only transitions you can see with your eyes — all end at n = 2.

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ABSORPTION SPECTRA

The reverse process also works. When white light passes through a cool gas, atoms absorb photons at exactly the same frequencies they would emit. This creates dark lines in the otherwise continuous spectrum — an absorption spectrum. Emission and absorption spectra are like photographic negatives of each other.

SECTION 6

Worked Example: Photoelectric Effect Calculation

Let's solve a typical IB-style problem involving the photoelectric effect. This example brings together several of the equations and concepts from earlier sections.

Finding the Maximum Kinetic Energy of Photoelectrons

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Step 1 — Read the Problem

Ultraviolet light of wavelength 200 nm strikes a metal surface with a work function of 4.50 eV. Determine (a) the energy of each incident photon in eV, and (b) the maximum kinetic energy of the emitted photoelectrons in eV.
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Step 2 — Identify Given Values

Wavelength: λ = 200 nm = 200 × 10⁻⁹ m = 2.00 × 10⁻⁷ m. Work function: φ = 4.50 eV. Planck's constant: h = 6.63 × 10⁻³⁴ J·s. Speed of light: c = 3.00 × 10⁸ m/s. Conversion: 1 eV = 1.60 × 10⁻¹⁹ J.
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Step 3 — Calculate Photon Energy

Using E = hc / λ: E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / (2.00 × 10⁻⁷) = (1.989 × 10⁻²⁵) / (2.00 × 10⁻⁷) = 9.945 × 10⁻¹⁹ J. Converting to eV: E = 9.945 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ = 6.22 eV.

Photon energy E = 6.22 eV
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Step 4 — Apply the Photoelectric Equation

Using $E_k$ (max) = hf − φ, which can also be written as $E_k$ (max) = E_photon − φ: $E_k$ (max) = 6.22 − 4.50 = 1.72 eV.

Maximum kinetic energy $E_k$ (max) = 1.72 eV
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Step 5 — Interpret the Result

Since the photon energy (6.22 eV) exceeds the work function (4.50 eV), electrons are indeed emitted. The surplus energy, 1.72 eV, goes into the kinetic energy of the fastest ejected electrons. Note that "maximum" kinetic energy refers to electrons from the surface; electrons from deeper in the metal lose some energy escaping and emerge slower.

SECTION 7

Classical Physics vs. Quantum Physics

To truly appreciate quantum physics, it helps to compare its predictions side by side with those of classical physics. The table below highlights the key differences that the IB syllabus expects you to understand.

Feature	Classical Prediction	Quantum Reality
Photoelectric threshold	Any frequency of light should eject electrons if the intensity is high enough.	Only photons with f ≥ f₀ can eject electrons; intensity only changes the number, not the energy.
Time delay for emission	Dim light should require time for energy to accumulate at the surface.	Emission is instantaneous when hf ≥ φ, because a single photon delivers all its energy at once.
Effect of intensity	Higher intensity should increase electron kinetic energy.	Higher intensity increases the number of emitted electrons (current) but not their maximum kinetic energy.
Nature of light	Light is a continuous electromagnetic wave.	Light has both wave and particle properties (wave–particle duality).
Atomic spectra	Atoms should emit a continuous range of wavelengths.	Atoms emit only specific wavelengths corresponding to energy-level differences.

Classical vs. Quantum Predictions

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KEY TAKEAWAY

Classical physics is like buying groceries by weight — you can get any amount. Quantum physics is like buying eggs — they only come in whole numbers. You cannot buy half a photon, and a photon of the wrong frequency cannot do the job no matter how many you throw at the problem. This 'all-or-nothing' nature of photon interactions is what makes quantum physics fundamentally different from everyday experience.

SECTION 8

Connections to Advanced Quantum Theory

The concepts in E.2 form the foundation for much deeper physics. While you don't need to master these advanced topics for the IB exam, understanding where they fit helps you see the bigger picture and prepares you if you continue studying physics at university.

IB E.2 Concept	Advanced Extension
E = hf (photon energy)	Quantum electrodynamics (QED) — photons as mediators of the electromagnetic force, Feynman diagrams.
λ = h/p (de Broglie wavelength)	Schrödinger wave equation — probability amplitudes, wave functions, electron orbitals as standing waves.
Discrete energy levels	Quantum numbers (n, l, $m_l$ , $m_s$ ) — electron configurations, periodic table structure, spectroscopy.
Heisenberg uncertainty principle	Quantum tunnelling — particles crossing barriers they classically shouldn't, tunnel diodes, nuclear fusion in stars.
Wave–particle duality	Quantum superposition and entanglement — the basis of quantum computing and quantum cryptography.

From IB E.2 to University-Level Quantum Physics

The de Broglie wavelength, for example, leads directly into the Schrödinger equation — the central equation of quantum mechanics. Instead of thinking of an electron as a tiny ball orbiting the nucleus, the wave function describes a cloud of probability. The electron is most likely to be found where the cloud is densest. This picture not only explains chemistry's electron orbitals but also underpins technologies like electron microscopes, which exploit the tiny de Broglie wavelengths of fast electrons to image structures smaller than visible light can resolve.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why increasing the intensity of light below the threshold frequency does not produce any photoelectrons, no matter how bright the light becomes.

Show answer

Intensity increases the number of photons per second but does not change the energy of each individual photon. Since each photon's energy is E = hf, a photon with f < f₀ has energy less than the work function φ. No single photon carries enough energy to liberate an electron, and in quantum physics, a single electron interacts with a single photon — it cannot accumulate energy from multiple photons. Therefore, no electrons are ejected regardless of intensity.

Question 2: BASIC CALCULATION

Calculate the energy, in electron-volts, of a photon of green light with wavelength 520 nm. Use h = 6.63 × 10⁻³⁴ J·s and c = 3.00 × 10⁸ m/s.

Show answer

Using E = hc / λ: E = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / (520 × 10⁻⁹) = 1.989 × 10⁻²⁵ / 5.20 × 10⁻⁷ = 3.83 × 10⁻¹⁹ J. Converting to eV: E = 3.83 × 10⁻¹⁹ / 1.60 × 10⁻¹⁹ = 2.39 eV. So each green photon carries approximately 2.39 eV of energy.

Question 3: INTERMEDIATE

An electron in a hydrogen atom transitions from the n = 4 level (−0.85 eV) to the n = 2 level (−3.40 eV). Calculate the wavelength of the emitted photon and determine which region of the electromagnetic spectrum it belongs to.

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The energy of the emitted photon equals the difference between levels: ΔE = (−0.85) − (−3.40) = 2.55 eV. Converting to joules: ΔE = 2.55 × 1.60 × 10⁻¹⁹ = 4.08 × 10⁻¹⁹ J. Using E = hc / λ rearranged to λ = hc / E: λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸) / 4.08 × 10⁻¹⁹ = 4.87 × 10⁻⁷ m = 487 nm. This falls in the visible blue-cyan region of the electromagnetic spectrum, and it corresponds to one of the Balmer series lines of hydrogen.

Question 4: APPLIED

An electron microscope accelerates electrons through a potential difference of 50.0 kV. Calculate the de Broglie wavelength of these electrons. The electron mass is 9.11 × 10⁻³¹ kg. (Hint: first find the kinetic energy, then the momentum, then the wavelength.)

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The kinetic energy gained by an electron accelerated through 50.0 kV is

E_k

= eV = 1.60 × 10⁻¹⁹ × 50,000 = 8.00 × 10⁻¹⁵ J. Using

E_k

= p² / (2m), we solve for p: p = √(2mE_k) = √(2 × 9.11 × 10⁻³¹ × 8.00 × 10⁻¹⁵) = √(1.458 × 10⁻⁴⁴) = 1.21 × 10⁻²² kg·m/s. The de Broglie wavelength is λ = h / p = 6.63 × 10⁻³⁴ / 1.21 × 10⁻²² = 5.48 × 10⁻¹² m ≈ 0.00548 nm. This is much smaller than visible light wavelengths (≈ 400–700 nm), explaining why electron microscopes achieve far greater resolution than optical microscopes.

Question 5: CRITICAL THINKING

A student claims: 'Since de Broglie tells us every object has a wavelength, a baseball thrown at 40 m/s must show wave behaviour just like an electron.' Evaluate this claim by calculating the de Broglie wavelength of a 0.145 kg baseball moving at 40 m/s, comparing it to the size of an atom (≈ 10⁻¹⁰ m), and explaining why we do not observe wave behaviour for everyday objects.

Show answer

Using λ = h / (mv): λ = 6.63 × 10⁻³⁴ / (0.145 × 40) = 6.63 × 10⁻³⁴ / 5.80 = 1.14 × 10⁻³⁴ m. This wavelength is roughly 10⁻²⁴ times smaller than an atom and approximately 10⁻¹⁹ times smaller than a proton. For wave effects like diffraction to be observable, the wavelength must be comparable to the size of the obstacles or slits the object encounters. Since no physical slit is anywhere near 10⁻³⁴ m, the wave nature of the baseball is utterly undetectable. The student is technically correct that the baseball has a wavelength, but practically wrong to say it 'shows wave behaviour.' Wave–particle duality is real for all objects, but only measurable for very small, low-momentum particles like electrons and neutrons.

SUMMARY

Quantum Physics — Key Concepts Review

Summary

Quantum physics reveals that energy is quantized — it comes in discrete packets called photons, each carrying energy E = hf. The photoelectric effect shows that photons must exceed a metal's work function φ to eject electrons, with the maximum kinetic energy given by $E_k$ (max) = hf − φ. Atoms have discrete energy levels, and transitions between them produce line spectra with photon energies equal to hf = E₂ − E₁.

Wave–particle duality extends to matter: all particles have a de Broglie wavelength λ = h / p, though this wavelength is only measurable for very small, low-momentum particles. The Heisenberg uncertainty principle sets a fundamental limit on how precisely we can simultaneously know a particle's position and momentum. Together, these ideas replace the deterministic world of classical physics with a probabilistic description of nature at the atomic scale — and they form the foundation for modern technologies from lasers to semiconductors.