Chapter 3: The experimental basis of quantum physicsa prologue to quantum mechanics

3.1Discovery of the x ray and the electron

In the 1890s, physicists studied cathode rays (음극선), generated from one of the metal plates in an evacuated tube across which a large electric potential had been established. (The same cathode-ray tube, or CRT, was inside every old television and computer monitor.) They surmised that the rays had something to do with atoms.

Wilhelm Röntgen (1845–1923) found that cathode rays striking various materials produce a new kind of ray, which he called x rays. They are unaffected by magnetic fields and far more penetrating than cathode rays: Röntgen obtained an image of the bones in a hand when x rays were allowed to pass through it, and medical applications followed immediately. He received the first Nobel Prize in Physics (1901). (In Korean and Japanese, x rays are still often called "Röntgen", 렌트겐.)

J. J. Thomson (1856–1940) had studied cathode rays for several years. He deflected them in electric and magnetic fields and showed that they are negatively charged particles, electrons (Nobel Prize 1906). With the magnetic field off, an electron of charge \(q\) and mass \(m\) entering plates of length \(\ell\) with horizontal speed \(v_0\) feels a vertical force

The deflection angle \(\theta\) is measurable, so \(q/m\) can be determined if \(v_0\) is known. To find \(v_0\), turn on \(\vec B\) and adjust it until there is no deflection: \(\vec F = q\vec E + q\vec v\times\vec B = 0\), so \(|\vec E| = |v_x||\vec B|\) and \(v_x = E/B = v_0\). Then

Example 3.1. With plates 5.0 cm long, \(E = 1.2\times10^4\) V/m, a deflection of 30° without the magnetic field, and no deflection with \(B = 8.8\times10^{-4}\) T: \(v_0 = E/B = 1.4\times10^7\) m/s and \(q/m = 1.8\times10^{11}\) C/kg. Thomson's own value was about 35% below the accepted \(1.76\times10^{11}\) C/kg. He realized it was much larger than expected and a factor of 1000 larger than any \(q/m\) previously measured (for hydrogen), so either \(m\) is small or \(e\) is large, and the carriers of electricity are far more penetrating than atoms or molecules.

3.2Determination of electron charge

Robert A. Millikan (1868–1953) measured the charge itself. Charged oil drops from an atomizer drift between horizontal plates with a uniform field \(E = V/d\), where \(V\) is the voltage and \(d\) the plate separation. Adjusting the voltage suspends a drop at rest, so \(F_E = qE = mg\) and

The drop's radius \(r\) (and with the oil density \(\rho\), its mass) is determined by turning off the field and measuring the terminal velocity \(v_t\) of the falling drop. Neglecting buoyancy, \(mg = bv_t\) with the frictional force \(\vec F_f = -b\vec v\) and Stokes's law \(b = 6\pi\eta r\) (\(\eta\) the viscosity of air), so \(\tfrac43\pi r^3\rho g = 6\pi\eta rv_t\) and \(r = 3\sqrt{\eta v_t/2g\rho}\) (Problems 6 and 7). Friction in the atomizer sometimes charges the drops. From thousands of measurements Millikan showed that charge is quantized in units of a basic electronic charge

\(q\) is always an integer multiple of \(e\).

3.3Line spectra

By the early 1800s it was known that each chemical element produces its own unique wavelengths (colors) when burned in a flame or excited in an electrical discharge. In a spectrometer, light from a discharge tube passes through a slit and a diffraction grating. Light from neighboring slits a distance \(d\) apart differs in path length by \(d\sin\theta\), and the diffraction maxima occur where

Hydrogen's visible lines form the Balmer series, which Balmer described in 1885 by

with the Rydberg constant for hydrogen \(R_H = 1.096776\times10^7\ \text{m}^{-1}\). More generally,

with \(n = 1\) (Lyman, ultraviolet), \(n=2\) (Balmer, visible and ultraviolet), and \(n = 3, 4, 5\) (Paschen, Brackett, Pfund, all infrared).

Helium was discovered in the Sun before it was found on Earth: a yellow line seen in the solar spectrum during the eclipse of 1868, near but distinct from the sodium lines, belonged to no known element. Lockyer named it after the Greek helios, and Ramsay found it on Earth in 1895.

3.4Quantization

Are there "building blocks" of matter or energy? Millikan's experiment showed the quantization of electric charge in units of \(e\). (Modern theory adds quarks with charges \(\pm e/3\) and \(\pm2e/3\), never directly observed in isolation.) Atomic weights are not continuous but take discrete values, close to integral multiples of a unit mass. And line spectra are a prime example of quantization: Niels Bohr used some simple assumptions based on the new quantum theory to model the atom and successfully predict these wavelengths (Chapter 4).

3.5Blackbody radiation

When matter is heated it emits radiation in the form of electromagnetic waves, with different "colors" at different temperatures. All bodies simultaneously emit and absorb radiation. When a body's temperature is constant in time, it is in thermal equilibrium with its surroundings, absorbing thermal energy at the same rate as it emits it. An ideal absorber and emitter is a blackbody, well approximated by a small hole into a cavity: light entering reflects around inside until it is absorbed. Let \(\mathcal I(\lambda,T)\) be the intensity, the power radiated per unit area per unit wavelength at temperature \(T\); the total power per unit area is \(R(T) = \int_0^\infty\mathcal I(\lambda,T)\,d\lambda\). Two important observations:

1. The maximum of the distribution shifts to shorter wavelengths as the temperature increases: Wien's displacement law,

   \[ \lambda_{\max}T = 2.898\times10^{-3}\ \text{m}\cdot\text{K}\qquad(\lambda_{\max}\propto T^{-1}). \]

2. The total power radiated increases with temperature: the Stefan–Boltzmann law of Josef Stefan,

   \[ \begin{gathered} R(T) = \int_0^\infty\mathcal I(\lambda,T)\,d\lambda = \epsilon\sigma T^4,\\ \sigma = 5.6704\times10^{-8}\ \frac{\text{W}}{\text{m}^2\,\text{K}^4}, \end{gathered} \]

   with emissivity \(\epsilon = 1\) for an idealized blackbody.

Example 3.5: the Sun. Its spectrum peaks near 500 nm, so \(T = 2.898\times10^{-3}\ \text{m}\cdot\text{K}/500\ \text{nm}\approx5800\) K, and \(R = \sigma T^4 = 6.42\times10^7\ \text{W/m}^2\). Over the Sun's surface, \(4\pi(6.96\times10^8\ \text{m})^2\), that is \(3.91\times10^{26}\) W. The Earth intercepts the fraction \(\pi r_E^2/4\pi R_{ES}^2 = 4.57\times10^{-10}\), or \(1.79\times10^{17}\) W, about \(1400\ \text{W/m}^2\) of exposed area: close to the measured value above the atmosphere, so the Sun does act roughly as a blackbody.

The dilemma. Classical electromagnetic theory and thermodynamics, through the equipartition theorem (Chapter 9), assign \(k_BT\) to each possible wave configuration in the cavity. Lord Rayleigh and James Jeans obtained

which approaches the data at long wavelengths but, as \(\lambda\to0\), grows as \(\lambda^{-4}\to\infty\): the ultraviolet catastrophe. Experimentally, \(\mathcal I(\lambda,T)\to0\) as \(\lambda\to0\).

Max Planck assumed that the radiation in the cavity is emitted and absorbed by some sort of "oscillators" in the walls, whose energies are integral multiples of \(hf\), with \(f\) the frequency of the oscillating wave and \(h\) Planck's constant. Using Boltzmann's statistical methods, he obtained

where the last factor is the Planck distribution, a special case of the Bose–Einstein distribution (Chapter 9). This required two important modifications of classical theory:

1. The oscillators can only have certain discrete energies, \(E_n = nhf\), with \(n\) an integer and Planck's constant \(h = 6.6261\times10^{-34}\) J·s.
2. The oscillators can absorb or emit energy only in discrete multiples of the fundamental quantum \(\Delta E = hf\).

Planck found these results disturbing and spent years trying to recover the agreement with experiment while letting \(h\to0\). Every attempt failed, and his quantum result became a cornerstone of modern science. From Planck's law one can derive Wien's law (Example 3.6: \(\lambda_{\max}T = hc/4.966k_B\)), the Stefan–Boltzmann law (Example 3.7: \(\sigma = 2\pi^5k_B^4/15h^3c^2\)), and the Rayleigh–Jeans law at long wavelengths (Example 3.8), and it avoids the ultraviolet catastrophe (Example 3.9): at short wavelengths \(hf\gg k_BT\), few oscillators can acquire such large quanta, and the radiated energy stays finite.

3.6The photoelectric effect (광전효과)

The photoelectric effect is the most compelling and simplest evidence for the quantization of radiation energy. "Free" (valence) electrons in metals move easily from atom to atom, and they can escape the material by thermionic emission (heat), secondary emission (bombardment by fast particles), field emission (a strong external field), or the photoelectric effect: incident light gives electrons enough extra kinetic energy to escape. The ejected electrons are photoelectrons, and the minimum extra energy needed to escape is the work function \(\phi\) of the material. In the experiment, light shines on an emitter (cathode) in a vacuum tube, and a power supply makes the collector (anode) either accelerating (\(V>0\)) or retarding (\(V\lt 0\)) while an ammeter measures the photocurrent \(I\). The pertinent experimental results:

1. The kinetic energies of the photoelectrons are independent of the light intensity. A stopping potential \(-V_0\) is sufficient to stop all photoelectrons, no matter what the intensity, and a maximum photocurrent is reached as the voltage increases from negative to positive values.
2. The maximum kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light: a larger \(|V_0|\) is required for larger \(f\).
3. The smaller the work function \(\phi\), the lower the threshold frequency that can eject photoelectrons. No photoelectrons are produced below the threshold, no matter how intense the light.
4. When photoelectrons are produced, their number is proportional to the intensity of the light.
5. The photoelectrons are emitted almost instantly (within \(3\times10^{-9}\) s) after illumination, independent of the intensity.

Classical interpretation. Classical theory says a light wave's energy grows with its intensity, so more intense light should give the electrons more kinetic energy; it cannot explain results 1 and 2, the threshold frequency in result 3, or the dependence on frequency. It does predict result 4. And for very weak light it predicts a long delay before any electron accumulates enough energy: for sodium (\(\phi = 2.36\) eV) in light of \(10^{-8}\ \text{W/m}^2\), about 14.7 years to produce a 1 eV photoelectron (Example 3.10). In reality the electron is produced immediately.

Einstein's theory (1905) proposed that the electromagnetic radiation field itself is quantized: the energy of a light ray spreading out from a point source "is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units." These quanta are now called photons, each with energy

consistent with Planck's quantum of energy (with \(\lambda f = c\), the factor in Planck's law is \(1/(e^{hf/k_BT}-1)\)). A photon delivers its entire energy to a single electron, which needs \(\phi\) to overcome its binding in the material:

(At these energies of a few eV the nonrelativistic kinetic energy is safe.) The retarding potential is the opposing potential needed to stop the most energetic electrons. In the quantum interpretation, \(V_0\) is fixed by \(f\) alone, while the photocurrent is proportional to the number of photons, i.e. the intensity. Plotting \(eV_0\) against \(f\) gives straight lines of slope \(h\) and intercept \(-\phi\), as Millikan confirmed in 1916. Einstein (1921) and Millikan (1923) received Nobel Prizes for this work. In summary, electromagnetic radiation consists of photons, each of energy \(E = hf = hc/\lambda\), and the total energy is the sum of the photons' energies.

If the collector and emitter are made of different metals, a contact potential difference \(\phi_E-\phi_C\) appears between them, and the work function in \(eV_0 = hf-\phi\) is in fact that of the collector, not the emitter as most textbooks imply. Photoelectric devices include photomultiplier tubes, light sensors in door openers and intrusion alarms, and solar panels.

3.7X-ray production

The inverse of the photoelectric effect: can an electron (or any charged particle) give up its energy and create a photon? An electron may give up part or all of its kinetic energy and still be the same electron. Interacting with the strong electric field of an atomic nucleus, it is accelerated and radiates electromagnetic energy. Classically it should radiate continuously; in the quantum picture it emits a series of photons of varying energies, \(E_f = E_i - hf\) for each. This is bremsstrahlung (German for "braking radiation"). In an x-ray tube, electrons from a hot filament are accelerated through a high voltage \(V_0\) into a target. The x rays include photons of many wavelengths, plus sharp peaks produced by atomic excitations (Section 4.6). The minimum wavelength corresponds to the maximum frequency, when an electron gives up all its kinetic energy to create one photon, the inverse photoelectric effect:

the Duane–Hunt rule: \(\lambda_{\min}\) depends only on \(V_0\), and is the same for all targets. Heavier elements, with stronger nuclear fields, are more effective at making the electrons radiate; the intensity grows roughly as the square of the target's atomic number and of the voltage, which is why x-ray tubes use high voltages and tungsten anodes (tungsten also withstands high temperatures and currents). Measuring \(\lambda_{\min}\) is a way to monitor the electron energy in devices such as electron microscopes (Conceptual Example 3.14). Example 3.15: for a tungsten anode (\(\phi = 4.63\) eV) at 35 kV, the few-eV initial energies and work functions are negligible against 35,000 eV, and \(\lambda_{\min} = 1.240\times10^{-6}/35.0\times10^3 = 3.54\times10^{-11}\) m.

3.8The Compton effect

Classically, a light wave striking an electron makes it oscillate at the light's frequency, and the electron re-radiates at the same frequency (Thomson scattering). In the early 1920s Arthur Compton found that x rays scattered from matter contain a component, the "modified" wave, with a longer wavelength than the original "unmodified" wave. This can be explained only by Einstein's photon concept. Treat the scattering as a collision of a photon, with energy \(E = hf\) and momentum \(p = E/c = h/\lambda\), and an electron essentially at rest, with rest energy \(mc^2\). Energy and momentum conservation give

Squaring and adding the momentum equations, \(p_e^2 = (h/\lambda)^2 + (h/\lambda')^2 - 2(h/\lambda)(h/\lambda')\cos\theta\). Using \(E_e^2 = (mc^2)^2 + p_e^2c^2\) with \(E_e = h(f-f') + mc^2\),

so \(\frac{h}{mc^2}(1-\cos\theta) = \frac{f-f'}{ff'} = \frac1c(\lambda'-\lambda)\):

The photon scatters elastically from an essentially free electron, since its energy is much larger than the electron's binding energy. If instead it scatters from a tightly bound inner electron, the electron is not dislodged and the photon scatters from the entire atom: the mass in \(h/mc\) is then thousands of times larger than \(m_e\), and \(\Delta\lambda\) is correspondingly negligible. That explains the unmodified wavelength that Compton also observed. The success of the theory convincingly demonstrated the correctness of the quantum concept and the particle nature of the photon (Nobel Prize 1927).

3.9Pair production and annihilation

High-energy gamma rays passing through matter can create an electron and its antiparticle, the positron predicted by P. A. M. Dirac: \(\gamma\to e^+ + e^-\). This cannot happen in free space, because energy and momentum cannot both be conserved (Example 3.17): energy conservation requires \(hf = E_+ + E_-\), and with \(E_\pm = \sqrt{p_\pm^2c^2 + m^2c^4} > p_\pm c\) this gives \(hf > p_+c + p_-c\), while momentum conservation allows at most \(hf = p_+c\cos\theta_+ + p_-c\cos\theta_- \le p_+c + p_-c\). Near a nucleus, which can absorb momentum, the reaction is possible:

When positrons and electrons are close together, even briefly, they annihilate each other and produce photons, \(e^+ + e^-\to\gamma+\gamma\). Two photons are needed to conserve both energy and momentum: for an electron–positron pair at rest (positronium), \(2m_ec^2\approx hf_1 + hf_2\) and \(0 = hf_1/c - hf_2/c\), so \(f_1 = f_2\) and

consistent with experiments. This pair of photons in opposite directions is the basis of positron emission tomography (PET): a positron-emitting chemical containing a nucleus such as \(^{15}\)O, \(^{11}\)C, \(^{13}\)N, or \(^{18}\)F is injected into the body, and detecting the two 0.511 MeV photons in coincidence locates where the chemical has concentrated (Conceptual Example 3.18).

The positron is the antiparticle of the electron, with the opposite charge but the same mass. The antiproton was discovered in 1955 by E. G. Segrè and O. Chamberlain (Nobel Prize 1959), and we now believe every particle has an antiparticle. For some, like the photon and the neutral pion, particle and antiparticle are the same; for most, such as the neutron and proton, they are distinct. Matter and antimatter cannot coexist for long in our world, because their ultimate fate is annihilation.

This completes Chapter 3. Chapter 4 uses these ideas to build a model of the atom.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 3.
• A. Einstein, "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt," Ann. Phys. 17, 132 (1905); English translation by A. B. Arons and M. B. Peppard, Am. J. Phys. 33, 367 (1965).
• R. A. Millikan, "A direct photoelectric determination of Planck's h," Phys. Rev. 7, 355 (1916).
• A. H. Compton, "A quantum theory of the scattering of x-rays by light elements," Phys. Rev. 21, 483 (1923).
• H. A. Kramers, "On the theory of x-ray absorption and of the continuous x-ray spectrum," Phil. Mag. 46, 836 (1923).