Physics Commons^™

Problem Set #8, David Peak

Problems

A bit of stat mech

Problems 1-3 refer to: N identical, noninteracting, and distinguishable spin-1/2 particles (i.e., their separation is much greater than their de Broglie wavelength) are placed in an external magnetic field. Assume the ground state energy of one such particle is 0 and the excited state energy is ε , and the system is in thermal equilibrium at temperature T.

Schrödinger, 1, David Peak

Schrodinger

The Schrödinger equation: the quantum description of one massive, slow-moving particle

To establish a probability wave equation for one photon, it was useful to note that probability density and electromagnetic energy density were proportional. The governing equation for EM radiation fields is the Maxwell wave equation, which is also the governing equation for photon probability wave functions. Converting from EM radiation to photon information is made plausible by identifying energy and momentum operators with time- and space-derivatives, respectively. Thus, the Maxwell wave equation can be interpreted as being equivalent to the energy-momentum relation for photons. Though we don’t have a …

Schrödinger, 5, David Peak

Schrodinger

Transitions

Schrödinger, 4, David Peak

Schrodinger

The sanitized hydrogen atom: separating the variables

Separation of variables in the Schrödinger Equation for the hydrogen problem requires expressing Ψ as a product, Ψ(r,θ,φ,t) = R(r)Θ(θ )Φ(φ)T(t) , substituting into the partial differential equation [(5) in Sc3], and then dividing by Ψ. As in the square well problems, the separation constant for the t part of the separation is the particle’s eigen-energy, E.

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 …

Foundations, 1, David Peak

Foundations

Quantum mechanics is money

Text message and take a picture with your smart phone; watch a movie on your Blu-ray player; get the bar code on your bag of chips scanned; obtain an MRI image of your aching shoulder; take a ride on a maglev train. None of these—and countless other—things would be possible without quantum mechanics! Leon Lederman, Nobel Prize winning physicist, is widely quoted as saying that 1/3 of the world’s economy is due to quantum mechanics. Lederman’s estimate is actually probably too low, but what surely is the case is that computers, lasers, and superconducting magnets (to …

Physics 2710: Introductory Modern Physics, David Peak

Syllabus

No abstract provided.

Foundations, 3, David Peak

Foundations

Our goal is to try to reconcile classical EM with the existence of photons. The electric and magnetic fields associated with EM radiation, propagating in the x -direction, obey the Maxwell wave equation: ∂²E/∂t² = c²(∂²E/∂x²).

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 …

Problem Set #1, David Peak

Problems

A little E&M practice

Problems 1-2 refer to: The electric field in a laser beam is given by E( x,t) = (1000V/m)sin[(πx10⁷rad/m) x + (3πx10¹⁵rad/s)t].

Problem Set #2, David Peak

Problems

A little energy and momentum practice (and units)

Problems 1-2 deal with “rest” energy and relativity.

Problem Set #5, David Peak

Problems

Expectations & 1D finite wells

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.

Problem Set #3, David Peak

Problems

Comparing classical electromagnetic waves with photon probability waves.

Problem 1 refers to: A standing electric field wave (one with lots of photons) in a quantum wire stretching between x = 0 and x = L is described by E(x,t)=E_maxsin(3πx/L)cos(3πct/L). Let L = 900 nm.

Problem Set #4, David Peak

Problems

Some 1D infinite well stuff

Problem Set #6, David Peak

Problems

3D, 1-particle systems

Problem Set #7, David Peak

Problems

Atom stuff

Problem Set #9, David Peak

Problems

Another bit of stat mech

Problems 1-3 refer to: N identical, noninteracting, and distinguishable quantum harmonic oscillators (i.e., their separation is much greater than their de Broglie wavelength) are in thermal equilibrium at temperature T. The energy of each oscillator can be expressed as ε_n = nε , where ε is the level spacing and n = 0, 1, 2, … .

Problem Set #10, David Peak

Problems

Blackbody

Problem Set #12, David Peak

Problems

Solid stuff

Schrödinger, 2, David Peak

Schrodinger

The finite square well

The infinite square well potential energy rigorously restricts the associated wavefunction to an exact region of space: it is infinitely “hard.” Potential energies encountered in more realistic physical scenarios are “softer” in that they permit wavefunctions to spread throughout less well-defined regions. An important toy example of the latter is the finite square well. In this problem, the potential energy function is U(x) = 0, if 0 < x < L, and U₀ otherwise.

Schrödinger, 3, David Peak

Schrodinger

The 3D infinite square well: quantum dots, wells, and wires

In the preceding discussion of the Schrödinger Equation the particle of interest was assumed to be “moving in the x -direction.” Of course, it is not possible for a particle to be moving in one spatial direction only. If that were true, according to the HUP it could be anywhere in the y - and z -directions and therefore be undetectable with finite volume detectors. Now, we consider the more realistic case of motion in all three spatial directions. For this purpose, we start with the 3D infinite square well. …

Foundations, 2, David Peak

Foundations

The double slit experiment in dim light – photons!

Let’s imagine doing the double slit experiment again, but now in very dim light. To do so requires putting the laser, plate, and collector in a sealed, light-tight box. Inserting neutral density filters in the beam between the laser and the double slit plate decreases the intensity of the beam striking the plate. In fact, the experiment can be done at such a low intensity that a human eye will not see any light on the CCD collector; but the CCD can. Under these conditions, the number of pixels that “light …

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 …