Ohio State University Marion Campus

by Gordon Aubrecht

©copyright 2004 all rights reserved

These notes sketch the topics that will be the focus of each class. You may use them to refresh your mind, and to help you structure your “lecture” notes.

The difference between boundary conditions and initial conditions. Boundary conditions tell what sorts of functions we have. Initial conditions tell which of the possible functions picked out by the boundary conditions are present.

Standing waves are set up only for certain values of the parameters.

The wave equation:

For a rectangular solid, the three-dimensional wavefunction for the standing waves present is given below, as is the wave’s corresponding angular frequency, which the wave equation allows us to determine:

Of course, one- and two-dimensional waves can also be written even more easily.

The energy for the wave will depend on the absolute value squared of the wavefunction, which is the extension of the dependence on the square of the amplitude.

The Heisenberg Uncertainty Principle:

The de Broglie relation, p = h/l, allows us to find the particle property [momentum] corresponding to this wave. We may write

where k is the wavenumber.

Similarly, we can find the kinetic energy. If the particle is massive, the kinetic energy is given by p^{2}/2m; if it is massless, by pc.

Consider electrons having a certain kinetic energy in a region of space in which a certain potential energy U exists. The potential U defines the confines of the region in which the electron can exist. For U = 0, the electron is free. For U = infinity, the wavefunction must vanish. We can think of the electron wave as existing only in the regions of space where the potential energy U is not infinite. The condition U = infinity acts like the fixed end of a string, pinning the wavefunction down to a value of zero.

Schrödinger studied ways to use an equation that could be solved to allow us to determine the wave behavior. He looked at how one would solve in the one-dimensional case, and guessed accordingly, obtaining a result identical mathematically to Heisenberg’s.

The Schrödinger equation describes the behavior of a quantum-mechanical system. In three dimensions it is given as

Born interpreted the wavefunction y(r) as a probability amplitude. By this expression, we mean that the square of the absolute value of the wavefunction represents a probability per unit volume of finding the particle at that point assuming we preformed many experiments beginning from identically-prepared quantum systems.

Neither wave nor particle description gives a complete description of physical reality. The reality is more than either, and encompasses both.

The Copenhagen interpretation is discussed.

There must be something that acts for rotational motion as force acts for translational motion. That is, a particle’s rotational motion will not change unless that corresponding thing acts. Force by itself is not enough. Both force and displacement of the point of application of the force from the center of rotation are important.

Further, the orientation of the force and displacement vectors are important.

We define torque as the quantity that acts analogously to force for rotational motion.

**t** = **r** x **F**

Similarity to translational equilibrium. There we say it is in equilibrium if:

Sum of the forces is zero.

For rotational equilibrium,

Sum of the torques is zero.

Another way of writing the results for circular motion (Physics 131) is

d**r**/dt = **w **x** r**.

d**v**/dt = d^{2}**r**/dt^{2} = **w **x** v** = -**w **^{2}**r** + **a **x** r**.

By analogy with force and momentum, we seek a quantity that has a derivative equal to the torque.

**L** = **r** x **p**

d**L**/dt = d/dt **r** x **p** = **r** x **F**

Examples of the use of angular momentum are given.

Kepler’s third law is just a different way to describe conservation of angular momentum.

We may see that the total angular momentum is given as the sum of the **orbital angular momentum** and the **spin angular momentum.** For example, for Earth, the orbital angular momentum consists in its angular momentum due to its revolution around the Sun; the spin angular momentum comes from Earth’s rotation about its own axis.

Everything we know from rectilinear motion has an analog for rotational motion.

The total energy of a body that is both translating and rotating is expressible as the sum of the translational kinetic energy of the center of mass and the rotational kninetic energy about the center of mass. This is only possible to have two terms like this when the CENTER OF MASS is used. For example, one could express the total kinetic energy of the system just in terms of rotation with one term

Quantum mechanics involves angular momentum. Note that h-bar has units of angular momentum. The difference in quantum mechanics is that angular momentum, as well as the projection of the angular momentum along some axis, is quantized.

This is obviously not easy to arrange.

Addition of quantized angular momentum is discussed and the way to proceed shown.

Bohr visited Rutherford’s lab shortly after the Geiger-Marsden experiment, and thought up a totally new way to describe the atom’s electrons. He assumed they were quantzed in terms of angular momentum. The electron’s angular momentum was characterized as n times h-bar. Using the known expression for the electric force in the hydrogen atom and this relation, one could find a result for r_{n} and E_{n}.

These states with a fixed value of n were described by Bohr as “stationary states;” they did not obey the usual laws of electrodynamics [described by Maxwell]. This new behavior was a big puzzle for a while, until it was realized that the Bohr model was wrong.

But Bohr was right in so many ways--he was able to calculate the Rydberg constant, seeing where Balmer’s formulas arose. He calculated the energy levels correctly. He found the Bohr radius correctly [of course, it wasn’t known as the Bohr radius until after his calculation].

But it was still wrong. The model only worked for “hydrogenic” atoms, for example. It didn’t give relative line intensities.

The Schrödinger equation gave the correct answer, where the potential energy is -ke^{2}/r. The conditions on the quantum numbers n, *l,* and m arose naturally from solving the equations by separation of variables for the variables r, q, and f.

The quantum number m arose from the f equation and turned out to be the “magnetic quantum number,” the projection of the angular momentum onto an axis (given in multiples of h-bar).

The quantum number *l* is the orbital angular momentum quantum number (given in multiples of h-bar). The differential equation that gave it depended on both *l* and m and was the well-known Legendre equation in q. It gave rise to a term proportional to

(angular momentum) x (angular momentum + 1) = *l*(*l* + 1)

that appears in the differential equation for r. This term contributes what acts like another potential energy term to the Schrödinger equation in r.

This “angular momentum barrier” term was what led to the upturn in the potentials considered in Ch. 23 as r got close to 0. The equation, the differential equation for Laguerre polynomials, leads to the same energy equation as Bohr found in terms of the principal quantum number n. So energy is quantized and the ground state of hydrogen has an energy of -13.6 eV.

The conditions on the quantum number are *l* < n, and m is between -*l* and +*l*.

In a magnetic field, the degeneracy in energy states of different m is lifted. There is a shift in energy.

We note that the electric current due to an electron moving around a fixed positive charge gives rise to a magnetic moment. This magnetic moment tends to line up in an external magnetic field. The magnetic moment can be expressed in terms of the angular momentum (**m** = e**L**/2m_{e}). Thus the energy shift depends on the projection of the quantized angular momentum on the direction of the external magnetic field.

The energy shift is given in terms of the Bohr magneton m_{B} [5.79 x 10^{-5} eV/T] as

DE = m m_{B} B.

The allowed transitions all involve |Dl| = 1 and |Dm| = 1, because photons carry one unit of angular momentum.

The Stern-Gerlach experiment showed that the Bohr model was insufficient to explain the result. Pauli proposed that only two electrons in a quantum mechanical system can be in the same state. This occurs because of the structure of the periodic table.

Shortly thereafter, Uhlenbeck and Goudsmit realized that electrons could be described by a “spin” quantum number. Because there must be two states only, this led to defining the spin quantum number of the electron as + 1/2 and - 1/2 times h-bar. This distinguishes the two electrons and so this idea led to the Pauli Exclusion Principle as we know it today: **No two electrons in a quantum mechanical system can be in the same state.**

The periodic table is now explicable because it arises from the Pauli Exclusion Principle and the existence of quantized angular momenta. Angular momentum states with zero angular momentum are known as s-states; those with angular momentum 1, p-states; angular momentum 2, d-states, etc.

The first period comes from filling the first s-state, 1s (this is only two electrons, because either spin is possible, but we are adding it to 0). We classify H as 1s^{1} and He as 1s^{2}.

The second period comes from the filling of the 2s and 2 p states. There are 2 possible s-states and 6 p-states (2 each from the Pauli Exclusion Principle for m = +1, m = 0, m = -1). We have Li as 1s^{2}2s^{1} and B as 1s^{2}2s^{2}. Then we fill the p shell until Ne, which is 1s^{2}2s^{2}2p^{6}.

The 3d state fills after the 4s and before the 4p, etc.

The ionization energies and other properties of the periodic table are discussed.

Thermal equilibrium and thermometers are introduced. Thermometers are based on thermal expansion. Temperatures are measured in terms of °C or K.

Heat capacity is the ratio of thermal energy transferred between systems at different temperatures and the difference in temperature between them.

Specific heat capacity (specific heat, for short) is heat capacity per kilogram or heat capacity per per mole [this latter is known as the molar specific heat, while the former is referred to simply as specific heat].

When systems at different temperatures are mixed, the energy lost by the system that was originally warmer is gained by the system that was originally colder. Eventually, the entire system is at a single temperature.

The phase changes also lead to contributions to the equilibria of thermal systems. The latent heat is the energy per kilohram needed to change the phase or obtained from the phase change.

The equation of state for an ideal gas is pV = Nk_{B}T = nRT, where T is the absolute temperature and the Boltzmann constant is 1.38 x 10^{23} J/K and the universal gas constant is R = N_{A}k_{B} = 8.314 J/(mol K). If we know macroscopic variables p, V, and T, we may find N or, equivalently, n = N/N_{A}.

The work done by a gas in expanding is dW = p dV.

The First Law of Thermodynamics is

dU = dQ + dW,

where dU is the thermal energy of the system, the sum of the kinetic energies of all the atoms or molecules, dQ is the heat ADDED TO the gas, and dW is the work done ON the gas (the negative of the work done BY the gas). The thermal energy U depends only on temperature (and, of course, number, N), U(N,T).

For a monatomic gas, the molar specific heat at constant volume is c_{v} = 3/2 R and the molar specific heat at constant pressure is c_{p} = c_{v} + R = 5/2 R. For a diatomic gas, c_{v} = 5/2 R and c_{p} = 7/2 R, etc.

We found the work done at constant volume, constant pressure, no heat input (adiabatic), constant temperature, etc.

There is no transfer of heat unless the temperature is different. There are three modes of heat transfer: conduction, convection, and radiation. Conduction and convection involve matter, radiation need not. In conduction, atoms get thermal energy transferred and increase their average kinetic energies. In convection, a material at a higher temperature carries its thermal energy along with it.

Conduction is described by

dQ/dt = kA dT/dx,

where k is the thermal conductivity, A is the cross-sectional area, and dT/dx is the temperature gradient. If there is a temperature difference DT across a material of length L, we may write

dQ/dt = (k/L) A DT = A DT/R,

where R is the R-value of the insulating material.

In convection, dQ/dt ~ DT.

In radiation, we can find dQ/dt = A s T^{4}, where s is the Stefan-Boltzmann constant, 5.67 x 10^{-8} W/(m^{2} K^{4}).

The kinetic therory and equipartition of energy (each degree of freedom gets 1/2 k_{B}T) were considered.

There are many microstates for each macrostate of a system. The Boltzmann factor basically counts the ratio. The Maxwell-Boltzmann distribution refers to systems whose quantum characteristics are unimportant (that is, classical systems).

The Maxwell-Boltzmann distribution function is

e^{-E/kBT}.

The Maxwell-Boltzmann distribution gives the probability that various states will be occupied. The ratio of populations of particles occupying the different energy states E_{1} and E_{2} is expected to be related exponentially, according to the Maxwell-Boltzmann distribution as

n_{2}(T)/n_{1}(T)) = e^{-(E2-E1)/kBT}.

The Maxwell-Boltzmann distribution function is

The way specific heats were calculated, from the law of Dulong and Petit through Einstein to Debye, was considered.

Planck’s breakthrough was considered.

The ionic bond, the covalent bond, and the weaker bonds (hydrogen, van der Waals, and metallic) were discussed. Ionic and covalent bonds are typically around 5 eV; the others lower, around 1 eV.

In the ionic bond, the reason the system is bound is that the state has a lower energy as a bound system than as individual constituents. It costs the ionization energy to ionize sodium, but we get some energy back because the chlorine favoprs a complete shell of electrons (which makes it an ion), and pays energy to obtain it; this is the electron affinity.

The ionization energy is greater than the electron affinity, so overall we would find the system unbound, were it not for the fact that the system of two ions has electric potential energy. This means it gives up an amount of energy ke^{2}/r, which must be greater than the price it cost to ionize the two constituents. Overall, it gives up ke^{2}/r + electron affinity - ionization energy to be bound.

In the covalent bond, electrons are shared quantum mechanically. Each atom gets some higher probability of a closed electron shell, which lowers the energy.

Rotational and vibrational energy states were considered. They are quantized as well.

Hydrogen bond: permanent polarization causes binding.

Van der Waals bond: evanescent polarization effects causes binding.

Metallic bond: The electron is shared among all ions in the metal.

Metallic bond: The electron is shared among all ions in the metal. The mechanism is the phenomenon of band formation. When a quantum system has two identical states, it is okay if the system is composed of bosons, but it is NOT ALLOWED to occur if the system is composed of fermions. This is a result of the Pauli Exclusion Principle.

The quantum system responds by making the energies almost (but not quite) degenerate. In this way, the states are not identical and the system is allowed to exist. So the eelectron can exist in the neighborhood of any ion in the system. It is in this way that it is bound to the metal as a whole. Note that the electron should have a greater probability of being in between the ion locations, and therefore the ions are attracted to the electron locations--so this is similar to the situation for covalent bonding of hydrogen discussed in class.

There are energy gaps--forbidden energies. These gaps separate valence bands from other valence bands and from conduction bands.

Bose-Einstein and Fermi-Dirac statistics apply to systems for which the quantum mechanics is an important consideration. Bose-Einstein stsatistics are followed by particles that can all congregate in the lowest state of the system. The Fermi-Dirac statistics refer back to the Pauli Exclusion Principle, which states that no two identical particles can inhabit any state; or, alternatively, that no more than two identical fermions (not lookng at spin) can be in any quantum energy state in the system.

In practice, the Maxwell-Boltzmann distribution function,

e^{-E/kBT}

appears as part of a denominator 1/[Ce^{E/kBT} +/- 1],

where + 1 goes with F-D, -1 goes with B-E, and C is a constant [1 for B-E, e^{-EF/kBT} for F-D].

The Fermi energy is the maximum energy allowed. It arises because of the Pauli Exclusion Principle forcing the energies to get ever higher to make sure no two electrons are in the same state. Thus the Fermi energy depends on how many particles N there are.

The Fermi distribution function has the inetresting property that at absolute zero, it is 1 for E < E_{F} and 0 for E > E_{F}. The energy distribution function for the Fermi-Dirac particles is:

The work function tells how deeply the average electron is bound to a metal.

This is the amount of energy that must be paid to free the electron in the photoelectric effect.

Doping consists in adding electron donors or acceptors to allow more electrons to move from the valence to the conduction band in a semiconductor.

The Bohr model can be used to calculate the energy levels of these electrons in the semiconductor.

Materials that are doped with acceptor atoms are known as p-type semiconductors. The p indicates that the charge carriers are positive. While they are neutral overall, just as the atoms were neutral overall, they permit the carrying of charge by the holes, which appear to be positively charged (since they are due to the absence of an electron). Materials that are doped with donor atoms are known as n-type semiconductors. The n indicates that the charge carriers are negative. Again, n-type semiconductors are neutral, not charged. In order that the underlying lattice not be greatly disturbed, typical doping replaces only about one atom in a million to one atom in a billion of the lattice atoms by the dopant.

The properties of p-n junctions were considered. Diodes and transistors may be made using such junctions.

The uses of diodes is considered.

Quantum mechanical tunneling is revisited in more detail.

Various applications are discussed.

What is the difference between chemical and nuclear binding?

The curve of the binding energy is discussed.

Activity defined.

Activity & dating are discussed

take me to the journal assignments

aubrecht@mps.ohio-state.edu [latest revision, 19 March 2004]