Unit 18: Quantum Physics — Long Questions

The photoelectric effect is the emission of electrons (photoelectrons) from a metal surface when it is exposed to electromagnetic radiation of suitable frequency.

Experimental arrangement An evacuated glass tube contains two electrodes — the anode B, connected to the positive terminal of a battery, and the cathode C, connected to the negative terminal. Monochromatic light of suitable frequency shines on the cathode, ejecting photoelectrons which are attracted to the positive anode, producing a photoelectric current measured by a micro-ammeter. The current stops the instant the light is switched off, confirming it is caused by the incident light.

Stopping potential By reversing the battery so the anode becomes negative, the photoelectrons are repelled. As this retarding potential is made more negative, the photocurrent falls, until at the stopping potential Vₛ the current becomes zero — even the most energetic photoelectrons cannot reach the collector. Hence the maximum kinetic energy of the photoelectrons is (K.E.)max = eVₛ.

Observations (1) Photoelectrons are emitted with a range of energies; the maximum energy depends on the metal and the frequency of the incident light, not on its intensity. (2) There is a minimum (threshold) frequency f₀, below which no electrons are emitted however intense the light, and f₀ varies from metal to metal. (3) Emission is instantaneous, and the intensity of light only affects the number (current) of photoelectrons emitted, not their maximum energy. These results could not be explained by the classical wave theory of light, which predicts that kinetic energy should increase with intensity and that a time delay should occur for weak light — neither of which is observed.

Deduction of the photoelectric equation: Einstein explained these observations using Planck's quantum hypothesis. Light of frequency f consists of photons each carrying energy E = hf. A photon is absorbed by a single electron in the metal. Part of this photon energy is used to overcome the work function φ (the minimum energy needed for the electron to escape the metal surface), and the remainder becomes the kinetic energy of the ejected electron:
hf = φ + ½mv²max
This is Einstein's photoelectric equation. When (K.E.)max = 0, f equals the threshold frequency f₀, so φ = hf₀, and the equation can also be written as hf = hf₀ + ½mv²max. This equation, and the fact that the phenomenon requires light to be absorbed in discrete packets (photons) rather than as a continuous wave, demonstrates the particle nature of light.

In 1924, Louis de Broglie proposed that, just as light (normally thought of as a wave) displays particle-like behaviour (photons, with momentum p = h/λ), material particles (normally thought of as particles) should also display wave-like behaviour. He suggested that a particle of mass m moving with velocity v has an associated wavelength given by the de Broglie relation:
λ = h/p = h/mv

For ordinary macroscopic objects (e.g. a flying bullet), the mass is so large that the de Broglie wavelength is immeasurably small and no wave effects (diffraction, interference) can be observed. For microscopic particles such as electrons, however, the small mass gives a wavelength comparable to atomic spacings (in the X-ray range for typical electron speeds), so wave effects become detectable. This was confirmed experimentally by Davisson and Germer, who showed that a beam of electrons accelerated through a known potential and incident on a nickel crystal is diffracted exactly as X-rays would be, with the observed wavelength matching the de Broglie prediction λ = h/√(2mVe). Diffraction has since also been observed for protons, neutrons, and even whole atoms (hydrogen, helium), confirming that all matter has an associated wave nature.

Wave-particle duality Just as light shows both wave behaviour (interference, diffraction) and particle behaviour (photoelectric effect, Compton effect), matter (electrons, protons, neutrons, atoms, etc.) shows both particle behaviour (e.g. in the measurement of e/m) and wave behaviour (e.g. electron diffraction). Both the wave and particle aspects are genuine and necessary for a complete description, but — as stated in Bohr's principle of complementarity — both can never be observed simultaneously in a single experiment; which aspect is revealed depends on the type of experiment performed. This dual nature of both radiation and matter is known as wave-particle duality.

Isolated atoms (e.g. in a gas or vapour at low pressure) have discrete, quantized electron energy levels, with a ground state (n = 1) of lowest energy and a series of higher excited states (n = 2, 3, 4, ...).

Emission spectrum In a discharge tube, atoms are excited (e.g. by collisions) and their electrons jump to higher energy levels. Since excited states are short-lived, the electrons quickly fall back to lower levels, each transition releasing a photon of energy exactly equal to the difference between the two levels, hf = E_i − E_f. Because only certain discrete energy differences are possible, only certain discrete frequencies (wavelengths) of light are emitted; viewed through a spectrometer, these appear as a series of separate bright lines on a dark background — a line emission spectrum. In atomic hydrogen, these transitions produce the Lyman (ultraviolet), Balmer (visible), and Paschen (infrared) series, described by the Rydberg formula.

Absorption spectrum When white light with a continuous spectrum is passed through a cooler gas, atoms absorb photons of exactly those energies needed to raise electrons to higher levels; these specific wavelengths are removed from the transmitted light, appearing as dark lines in an otherwise continuous spectrum — an absorption spectrum. The dark absorption lines occur at exactly the same wavelengths as the corresponding bright emission lines of the same element.

Identification of elements Since each element has its own unique set of electron energy levels, it produces its own unique pattern (a "fingerprint") of spectral lines, different from every other element. By analysing the characteristic set of wavelengths present in the line emission spectrum (or missing in the absorption spectrum) of light from a mixture of vapours, each constituent element can be identified, even in trace amounts, by matching the observed lines to the known spectral fingerprint of that element.

The transmission electron microscope uses the wave nature of electrons, whose de Broglie wavelength (at the accelerating voltages used) is about a thousand times shorter than the wavelength of visible light, giving it far higher resolution than an optical microscope.

Operation Electrons emitted from a source are accelerated through a potential of several megavolts and focused onto the specimen by an electromagnetic lens called the condenser. The beam passes through the (thin) specimen, and an electromagnetic objective lens forms an image of it. This image is further magnified by an electromagnetic projector lens, and the final image is recorded on a film, called a micrograph. The higher the speed (accelerating voltage) of the electrons, the shorter their de Broglie wavelength and the higher the resolution obtained. (Modern scanning electron microscopes can also produce high-quality three-dimensional images.) Electron microscopes can resolve details down to near-atomic scale, and are used to study cell structure, viruses, and bacteria in microbiology, and to investigate metal fractures, materials, and crystal structures.

Similarities with an optical microscope: Both use a source of radiation, a condenser to focus the beam on the specimen, an objective to form an image, and a further lens system to magnify that image for viewing/recording — i.e. both have an analogous arrangement of a source, condenser, objective and projector/eyepiece stage.

Differences An optical microscope uses visible light and glass lenses, while an electron microscope uses a beam of electrons and electromagnetic (electric/magnetic field) lenses. Because the de Broglie wavelength of accelerated electrons is far shorter than that of visible light, the electron microscope achieves a much higher resolution and magnification than is possible with an optical microscope, which is fundamentally limited by the (comparatively long) wavelength of visible light.

Heisenberg's uncertainty principle (1927) states that the position and momentum of a particle cannot both be measured simultaneously with perfect accuracy. The product of the uncertainty Δx in the position of a particle at some instant, and the uncertainty Δp in the x-component of its momentum at the same instant, is at least of the order of Planck's constant: Δx·Δp ≳ h (more precisely, Δx·Δp ≥ ħ, where ħ = h/2π). There is an analogous relation between the uncertainty in a particle's energy and the time over which that energy is measured: ΔE·Δt ≳ h.

This uncertainty is not a limitation of the measuring instrument — it is a fundamental consequence of the wave-particle duality of matter and radiation. For example, to locate a microscopic particle moving along the x-axis using light of wavelength λ, the position can be determined only to within about one wavelength, Δx ≈ λ. But at most a photon can transfer its entire momentum h/λ to the particle, so the particle's momentum becomes uncertain by Δp ≈ h/λ. Multiplying these gives Δx·Δp ≈ h. Using a shorter wavelength of light reduces the position uncertainty but increases the disturbance to (and hence uncertainty in) the particle's momentum, and vice versa — the two uncertainties cannot both be made arbitrarily small at once.

Significance in particle physics For macroscopic objects in everyday life, this fundamental uncertainty is utterly negligible. But for sub-atomic particles it becomes a dominant fact: a single photon used to observe an electron can drastically change the electron's motion. The uncertainty principle can be used, for instance, to show that electrons cannot exist inside an atomic nucleus — confining an electron to a region as small as a nucleus (~10⁻¹⁵ m) would require a momentum uncertainty corresponding to an energy far in excess of any energy actually observed for electrons bound in atoms. It also sets a fundamental limit on how precisely both members of certain pairs of physical quantities can ever be known together, meaning that, at the sub-atomic scale, physical quantities cannot in principle be predicted or known with complete precision.