Properties of Nuclei
Z protons and N neutrons held together with a short-ranged force gives binding energy
m p = 938.3 MeV / c 2 M nucleus = Zm p with A= Z+ N mn = 939.6 MeV / c 2
+ Nmn E bind AmN mN = u = 9315 MeV / c 2 .
P and n made from quarks. Most
General Structure of Wave Mechanics (Ch. 5)
Sections 5-1 to 5-3 review items covered previously use Hermitian operators to represent observables (H,p,x) eigenvalues of Hermitian operators are real and give the expectation values eigenvectors for differen
Harmonic Oscillators
F=-kx or V=cx2. Arises often as first approximation for the minimum of a potential well Solve directly through calculus (analytical) Solve using group-theory like methods from relationship between x and p (algebraic)
V
Classically F=
Many Particle Systems
can write down the Schrodinger Equation for a many particle system
( x1 , x2 K xn ) H ( x1 , x2 K xn ) = ih t
with xi being the coordinate of particle i (r if 3D) the Hamiltonian has kinetic and potential energy
1 2 1 2 H = h 2 (
Operator methods in Quantum Mechanics
Section 6-1 outlines some formalism dont get lost; much you understand define ket and bra vectors and dot product
( x ) x | * ( x) | x | = * ( x ) ( x )dx
add in operators to this formalism. Let A be an operator
A
Time Dependent Perturbation Theory
Many possible potentials. Consider one where V(x,t)=V(x)+v(x,t) V(x) has solutions to the S.E. and so known eigenvalues and eigenfunctions
n eigenfunct ions of V ( x )
n ( x , t ) = e
iE n t / h
n ( x)
let perturbat
Perturbation Theory
Only a few QM systems that can be solved exactly: Hydrogen Atom(simplified), harmonic oscillator, infinite+finite well solve using perturbation theory which starts from a known solution and makes successive approximations start with t
The Real Hydrogen Atom
Solve SE and in first order get (independent of L):
13.6eV En = n2
can use perturbation theory to determine: magnetic effects (spin-orbit and hyperfine e-A) relativistic corrections Also have Lamb shift due to electron selfintera
Special Theory of Relativity
Up to ~1895, used simple Galilean Transformations x = x - vt t = t But observed that the speed of light, c, is always measured to travel at the same speed even if seen from different, moving frames c = 3 x 108 m/s is finite a
Spin and Magnetic Moments
(skip sect. 10-3)
• Orbital and intrinsic (spin) angular momentum
produce magnetic moments
• coupling between moments shift atomic energies
· Look first at orbital (think of current in a loop)
µ = I • A = current × area
=
• πr 2
Finite Square Well Potential
For V=finite outside the well. Solutions to S.E. inside the well the same. Have different outside. The boundary conditions (wavefunction and its derivative continuous) give quantization for E<V 0 longer wavelength, lower Ener
Solving Schrodinger Equation If V(x,t)=v(x) than can separate variables
h 2m
2
2 ( x ,t ) x 2
+ V ( x) = ih t
d dt
assume ( x, t ) = ( x) (t )
h 2m
2
2
d 2 dx
2
+ V ( x) (t ) ( x) = ih
[
h 2 d 2 2 mdx
+ V
] =
1
ihd dt
=G
G is separation constant valid an
Developing Wave Equations
Need wave equation and wave function for particles. Schrodinger, Klein-Gordon, Dirac not derived. Instead forms were guessed at, then solved, and found where applicable So Dirac equation applicable for spin 1/2 relativistic part
Superconductivity
Resistance goes to 0 below a critical temperature Tc element Ag Cu Ga Al Sn Pb Nb Tc resistivity (T=300) -.16 mOhms/m -.17 mOhms/m 1.1 K 1.7 mO/m 1.2 .28 Res. 3.7 1.2 7.2 2.2 9.2 1.3
T
many compounds (Nb-Ti, Cu-O-Y mixtures) have Tc up
Pre-quantum mechanics Modern Physics
Historical problems were resolved by modern treatments which lead to the development of quantum mechanics need special relativity EM radiation is transmitted by massless photons which have energy and momentum. Mathema
Multi-Electron Atoms
Start with Helium: He+ - same as H but with Z=2 He - 2 electrons. No exact solution of S.E. but can use H wave functions and energy levels as starting point. Well use some aspects of perturbation theory but skip Ritz variational tech
Nuclear Decays
Unstable nuclei can change N,Z.A to a nuclei at a lower energy (mass)
: N
Z A
Z 2
Nn
A 4
+ He
2
4
: Z N A Z 1N nA + e + /
If there is a mass difference such that energy is released, pretty much all decays occur but with very different l
all fundamental with no underlying structure Leptons+quarks spin while photon, W, Z, gluons spin 1 No QM theory for gravity Higher generations have larger mass
P461 - particles I 1
When/where discovered
Nobel Prize?
Mostly Europe 1895-1920 Roentgen (sor
Mixing in Weak Decays
Charged Weak Current (exchange of Ws) causes one member of a weak doublet to change into the other e + e +
e
W
e
Taus and muons therefore decay into the lightest member of the doublet (their neutrinos) electrons are stable as the
Solids - types
MOLECULAR. Set of single atoms or molecules bound to adjacent due to weak electric force between neutral objects (van der Waals). Strength depends on electric dipole moment No free electrons poor conductors easily deformed, low freezing te
Quantum Statistics
Determine probability for object/particle in a group of similar particles to have a given energy derive via: a. look at all possible states b. assign each allowed state equal probability c. conserve energy d. particles indistinguishabl
Boson and Fermion Gases
If free/quasifree gases mass > 0 non-relativistic P(E) = D(E) x n(E) - do Bosons first let N(E) = total number of particles. A fixed number (E&R use script N for this)
N = n( E ) D ( E )dE =
0
0
D( E ) dE E / kT ee 1
D(E)=densit
3D Schrodinger Equation
• Simply substitute momentum operator
• do particle in box and H atom
• added dimensions give more quantum numbers.
Can have degeneracies (more than 1 state with
same energy). Added complexity.
r
r
p ⇔ −ih∇ or
−h 2
2m
∂2
∂x 2
⇒∇ ⇒
Physics 395, Laboratory 12
Operational Amplifiers
Overview
The purpose of these experiments is to measure the basic parameters of an operational amplifier,
including gain, bias, and noise rejection, and to use operational feedback to form an amplifier.
Ba
Physics 375, Laboratory 13
Analog Math
Overview
The purpose of these experiments is to use op-amps to perform mathematical operations on input
voltages. Operations include sum, difference, logarithm, differential and integral.
Background
An operational am
Physics 475, Laboratory 15
Active Rectifiers
Overview
The purpose of these experiments is to use op-amps in circuits with diodes to improve the ability
to rectify signals and select the peak signal generated.
Background
An operational amplifier (op-amp) i
Physics 475, Laboratory 16
Power Supplies
Overview
The purpose of these experiments is to use integrated circuits to generate power sources of fixed
DC voltage from unregulated sources.
Components
The TL071 op-amp is an integrated circuit based on JFET te
Physics 475, Laboratory 18
Clocks
Overview
The purpose of these experiments is to study the properties of clocks and square-wave oscillators,
both as discrete-component circuits and in integrated circuits (555).
Components
The CA3140 op-amp is an integrat
Orbital Angular Momentum
In classical mechanics, conservation of angular momentum L is sometimes treated by an effective (repulsive) potential 2
L 2 mr
2
Soon we will solve the 3D Schr. Eqn. The R equation will have an angular momentum term which arises
MOLECULES
BONDS Ionic: closed shell (+) or open shell (-) Covalent: both open shells neutral (share e) Other (skip): van der Waals (HeHe)Hydrogen bonds (in DNA, proteins, etc) ENERGY LEVELS electronic vibrational rotational
P461 - Molecules 1
Ionic Bonds