Physics Commons^™

Many-Particle Systems, 1, David Peak

Many Particles

Wavefunctions for more than one particle: Different kinds of particles

Introductory examples of quantum mechanical wavefunction calculations involve a single particle moving about in a “magic” potential energy—e.g., a particle trapped inside a square well or an electron in a hydrogen atom. But, potential energy arises from interaction, so these situations must inevitably include more than one particle. Even the simplest atom—hydrogen— consists of two particles: the electron and the proton. So, how should the Schrödinger Equation be generalized to account for multiple particles?

Many-Particle Systems, 5, David Peak

Many Particles

Photons as T → 0 K

Photons are massless bosons. Because they are massless, any number of them can be added or subtracted without changing the photon system energy. (For example, a 1 eV photon can be exchanged for one hundred 0.01 eV photons, without changing system energy.) As a result, the chemical potential for photons is zero.

Many-Particle Systems, 2, David Peak

Many Particles

Multi-electron atoms

The rich diversity of chemical processes and structures is directly related to the diversity of electronic states of multi-electron atoms that, in turn, is dictated by the Pauli Exclusion Principle. To see how the Pauli Exclusion Principle produces atomic diversity, it is useful to begin simply, in particular, by considering the most elementary multi-electron “atom”: the hydrogen anion, H^–.

Many-Particle Systems, 7, David Peak

Many Particles

Electronic energy bands in crystalline solids: The finite well model

Previously, we have considered the “conduction electrons” in a metal as if they were an ideal gas in a 3D infinite well. What allows us to do that? A crystalline solid consists of a periodic array of atoms, packed so close to one another that “flow” (long range relative motion of groups of atoms) is essentially impossible. It is useful to think of a solid as a giant (rigid) molecule. The periodic structure of the atoms has profound consequences for the behavior of the solid’s electrons. While, in detail, the …

Many-Particle Systems, 9, David Peak

Many Particles

Superconductivity phenomenology

Superconductors are materials that exhibit zero (or close to zero) resistance to electrical currents as well as perfect diamagnetism (the Meissner Effect). When a current is started in a superconducting loop, it persists for a very long time without an applied potential difference. The resistivity of a superconductor is measured to be less than 4x10^–25 Ω-m (for comparison, the resistivity of an ordinary good conductor is about 10^–8 Ω-m), and the associated decay time for the current is estimated to be greater than 100,000 years (as opposed to about 1 µs for an ordinary good …

Many-Particle Systems, 4, David Peak

Many Particles

Absolute temperature

When a system is in statistical equilibrium it can usefully be characterized by a few macroscopic variables. Temperature is one of the most important of these. The absolute temperature scale (measured in kelvins, K) has the following properties. (1) T = 0 K is the temperature of a macroscopic system found permanently in its ground state. Such a system has no excitations; it has its lowest possible energy and is completely isolated from the rest of the universe.

Many-Particle Systems, 8, David Peak

Many Particles

The classical picture of how electrons migrate through a resistor driven by an applied potential difference draws an analogy with a kind of pinball machine. In a pinball machine, gravity accelerates the pinball down the table, but the ball’s progress is impeded by collisions with bumpers, as depicted to the right. (θ is the angle the table makes with the horizontal direction.) When averaged over many collisions the ball’s average equation of motion is ma = mg sinθ − mv/τ , where a and v are directed down the table, and τ is the average time between collisions. If the …

Many-Particle Systems, 3, David Peak

Many Particles

Bare essentials of statistical mechanics

Atoms are examples of many-particle systems, but atoms are extraordinarily simpler than macroscopic systems consisting of 10²⁰-10³⁰ atoms. Despite their great size, many properties of macroscopic systems depend intimately on the microscopic behavior of their microscopic constituents. The proper quantum mechanical description of an N -particle system is a wavefunction that depends on 3N coordinates (3 ways of moving, in general, for every particle) and 4N quantum numbers (3 motional quantum numbers and 1 spin quantum number for every particle). (If the “particles” are molecules there might be additional quantum …

Many-Particle Systems, 11, David Peak

Many Particles

Quantum information

In Mn10 we discussed the rudiments of “classical computation.” Classical, conventional computation involves combinations of transistors that convert low- and high-voltage inputs into different low- and high-voltage outputs. These voltages are interpreted as the binary digits 0 and 1, i.e., as bits. How bits are changed into other bits leads to such things as text preparation and storage, numerical calculations and symbolic manipulations, image and sound generation, game playing, intercontinental communication—in short, the modern world of information.

Many-Particle Systems, 10, David Peak

Many Particles

Intrinsic semiconductors

Intrinsic semiconductors have negligible concentrations of impurity atoms. Their electrical conductivity arises primarily from electrons excited into the otherwise empty conduction band from the otherwise filled valence band—usually by absorbing sufficient energy from phonons at finite temperature. Exciting an electron into the conduction band leaves a vacant state in the valence band. An electron at lower energy in the valence band can fill this vacant state. That, in turn, makes available a possible state for yet another valence band electron to fill. In other words, the excitation of the electron provides a mobile charge in the conduction band …