Physical Sciences and Mathematics Commons^™

Physics 3710 – Problem Set #3, David Peak

Problems

Physics 3710 – Problem Set #3 Relativistic kinematics, I

Physics 3710 – Problem Set #5, David Peak

Problems

Physics 3710 – Problem Set #5 Relativistic dynamics, I

Problems 1-5 refer to: One mass, m₁ = 1 (in some units), collides head-on with a second mass, m₂ = 2 , and sticks to it, forming a composite body of mass M . There are no external forces. Observer O records m₁ as initially moving with dimensionless velocity, u1 = +0.9 in the x - direction, while m₂ is recorded to be at rest. Do not make unwarranted assumptions about M , please; that’s the point of this set of problems.

Physics 3710 – Problem Set #7, David Peak

Problems

Physics 3710 – Problem Set #7 Newtonian gravity

Physics 3710 – Problem Set #8, David Peak

Problems

Physics 3710 – Problem Set #8 Relativistic gravity, I

Special Relativity, 1, David Peak

Special Relativity

Newton and Maxwell

We have seen that simple magnetic forces are incompatible with Newtonian mechanics. There are far more profound incompatibilities between electromagnetism and Newton. In particular, Maxwell’s equations predict that an accelerating charge produces a time changing magnetic field, which, in turn, produces a time changing electric field, which, in turn, produces a time changing magnetic field, which, in turn, produces … . This self-sustaining production of magnetic and electric fields propagates away from the source charge at a finite speed given by 1 µ0ε 0 (in empty space), the numerical value of which is approximately 3x108 m/s. But, …

Special Relativity, 6, David Peak

Special Relativity

Energy-momentum vector

Background, 1, David Peak

Background

This course deals with the structure of matter at its extreme length scales: cosmological on the large end (on the order of 1026 m), sub-nuclear on the small (less than 10-19 m). It also deals with the now firmly established realization that the organization of matter on these two phenomenally different scales is actually intimately connected. This course is about science in its most alive and vibrant state: what we think we know about the big and small of the universe changes virtually daily. Satellite observatories and ground-based particle accelerators make what was formerly “common knowledge” obsolete at a rapid …

General Relativity, 2, David Peak

General Relativity

Newton’s law of gravitostatics is incompatible with special relativity. To see this, suppose at time t in frame O m₁ is at x₁(t) and m₂ is at x₂(t). Newton’s gravitational force law says F_{1on 2}(t) = Gm₁m₂ / [x₂(t) − x₁(t)]² and relativistic dynamics says dp₂ /dt = F_{1on 2}. Transforming to another frame Oʹ moving relative to O leads to dp′₂/dt′ = F′_{1on 2}. But what is F′_{1on 2} ? If x₁(t) and …

General Relativity, 9, David Peak

General Relativity

Cosmic development

As discussed in GR 8, the cosmic scale factor a in the FLWR s-t obeys the Friedmann equation

Structure Of Matter, 3, David Peak

Structure of Matter

The particle zoo

Prior to the 1930s the fundamental structure of matter was believed to be extremely simple: there were electrons (each with mass about 0.5 MeV), e− , photons (no mass), γ , and protons (mass about 938 MeV), p⁺ . Starting in 1932 the world began to get a lot more complicated. First came Dirac’s positron ( e⁺ , with same mass as the electron), postulated in 1928 but mostly ignored until Anderson’s accidental discovery (see SM 1). Soon after, the neutron ( n ) was identified (mass about 940 MeV). In beta decay, the neutron …

Structure Of Matter, 6, David Peak

Structure of Matter

Quantum Flavor Dynamics (QFD), I

That each generation of the quark and lepton periodic tables (i.e., electron, muon, and tauon–see SM 1, p.1) has two members (a neutrino and a charged particle) that can be flipped into one another by emission or absorption of W bosons is reminiscent of how the angular momentum spin-1/2 component of a charged particle can be flipped between “up” and “down” orientations along some direction ( z ) in space by emission or absorption of photons. This analogy is made more graphic by labeling flavor rows by a new kind of “spin” (completely unrelated to …

Structure Of Matter, 7, David Peak

Structure of Matter

More about the matter with mass

To repeat, the basic premise of QED is that the physical world doesn’t care about the phase of the electron wavefunction; calculations and observations strongly support that idea. The basic premise of QCD is that the physical world doesn’t care about the color of the quark wavefunction; calculations and observations strongly support that idea. The basic premise of QFD is that the physical world doesn’t care about the flavor (isospin) of the quark or lepton wavefunctions. But that’s not true! For example, a “free” d quark can certainly emit a (virtual) W particle and …

Physics 3710 – Problem Set #2, David Peak

Problems

Physics 3710 – Problem Set #2 Newtonian relativity

Problems 1-4 refer to: Sound travels at about 330 m/s in still air. Observer O is at rest with respect to still air, observer O′ travels with constant velocity +50 m/s in the common x, x′ direction. Event A is the emission of a sound pulse from a stationary source at the origin of O; it occurs at x_A = 0 at t_A = 0. Event B is the reflection of the pulse at x_B = +100 m. Event C is the detection of the reflected pulse at x …

Physics 3710 – Problem Set #10, David Peak

Problems

Physics 3710 – Problem Set #10 Relativistic gravity, III

Problems 1-3 refer to: The maximum measured z value for a galaxy is 11.1. As on page 1, GR7, z = λ_d − λ_e / λ_e.

Physics 3710 – Problem Set #12, David Peak

Problems

Problem Set #12 Quarks and gluons

In the following solid lines represent quarks or antiquarks and dotted lines represent gluons. Time increases upward.

Physics 3710 – Problem Set #13, David Peak

Problems

Physics 3710 – Problem Set #13 Some weak interaction stuff

Questions 1-4 refer to the diagram at the right. In it, a particle p₁ absorbs a particle X and transforms into a particle p₂. Time increases vertically.

Special Relativity, 4, David Peak

Special Relativity

More kinematic consequences of the Lorentz transformations

Light cones: A “light cone” is a set of world lines corresponding to light rays emanating from and/or entering into an event.

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.