Physics comprehensive

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Gas molecules have an average translational kinetic energy of ³/₂ k T (three-halves times Boltzmann’s constant times the absolute temperature). The pressure arises due to the many molecular collisions per second with the walls so it increases for higher speeds v (v increases if temperature increases) and for higher density. The molecules have a cross sectional area for interaction  and so sweep out a volume  v t in time t. The mean time between collisions is found as the time for a molecule to sweep out the average volume per molecule.

 v t_{collision free} = ^V/_N so t_free = ^V/_Nv

The mean free path _free is the distance traveled to sweep out ^V/_N.  _free = ^V/_N.

Ex. 17) Cubical tanks X and Y have the same volumes and share a common wall. There is 1 gram of helium in tank X and 2 grams of helium in tank Y, and both samples are held at the same temperature. Which of the following is the same for both samples? (A) the number of molecular collisions per second on the common wall (B) the average speed of the molecules (C) the pressure exerted by the helium (D) the density of the helium (E) the mean free path of the molecules

isothermal: process at constant temperature

isobaric: process at constant pressure

iso-entropic: process at constant entropy

adiabatic: process in which there is no heat transfer; insulated from the environment

isochoric, isometric, iso-volumetric -------- process at constant volume

number of moles: n

ideal gas constant R = 8.31 (^J/ _mol-K)

number of molecules: N

Boltzmann’s constant: k = 1.38 x 10^{-23 J}/_K
LaChatlier’s Principle: A stable system acts to counter any change in its parameters. If a the volume of a sample of gas is reduced, its pressure increases to fight additional decreases in volume.
(Ex. 18) If one mole of an ideal gas doubles its volume as it undergoes an isothermal expansion, it pressure is

(A) quadrupled

(B) doubled

(C) unchanged

(D) halved

(E) quartered

The process is isothermal (same temperature). P₁ V₁ = n R T = P₂ V₂

D

Use ratios whenever possible.

Entropy Change: Ideal Heat Engine Efficiency:

Use absolute temperatures. 0 ⁰C = 273 K
Ex 19.) A power plant takes in steam at 527 ⁰C to power turbines and then exhausts the steam at 127 ⁰C. In any given time, it consumes 100 megawatts of heat energy from the steam. The maximum output power of the plane is

(A) 10 MW (B) 20 MW (C) 50 MW (D) 75 MW (E) 100 MW

Using 0 ⁰C = 273 K, T_high = 800 K and T_low = 400 K. The ideal thermal efficiency for a reversible engine operating between these temperatures is 50% so the maximum possible mechanical/electrical power out is 50 MW. C

The information that can be known about a system is encoded in a wavefunction (x,t) that satisfies the Schrödinger equation. The probability to find the particle between x and x + dx is (x,t)(x,t) dx (Born interpretation). Additional information can be extracted from the wavefunction using the operator for the dynamical quantity of interest. The operator for x is x. The operator for p_x is - i ^_x. Dynamic quantities in Hamiltonian mechanics are functions of position, momentum and time. The quantum operator for the quantity is formed by keeping the coordinates and the time while replacing each momentum p_q with –i^/_q. p_x  - i ^_x; p_- i ^_; …..

Shrödinger’s equation identifies two operators for energy, the Hamiltonian and i ^_t. In other words, the Hamiltonian represents the total energy (kinetic + potential), and it is the time-development operator for wavefunction (multiplied by ^–i/_).

The values E_n are the energy eigenvalues, the total energies, a set of real values that are characteristic of the problem and the associated temporal dependence is: T_n(t) = .

A general solution has the form, .

Operators operate on wavefunctions to return other (or perhaps the same) wavefunctions.

Operators operate on their eigenfunctions to return a scalar multiplying the same wavefunction.

Eigenvalue equation: p_x u(x) = –i ^_x u(x) = (momentum eigenvalue) u(x) = p u(x)

–i ^_x u(x) = p u(x)  u(x) = A

 A e ^{i (kx - t)} is a momentum eigenfunction with eigenvalue

Operators for physical observables are Hermitian (or self-adjoint) an, as such, they have real eigenvalues and expectation values.

All the functions oscillate in the classically allowed region and decay in the forbidden regions. The decay is more rapid for larger energy deficits. Each higher state has an additional node.

Notice that the spatial rate of oscillation increases as the kinetic energy increases (or the ‘local wavelength decreases).

The n = 36 QHO wavefunction is plotted for t = 0. Note the more rapid oscillation around x = 0 where the kinetic energy is greatest.

Energy eigenstates are stationary  the associated probability density * is time-independent.

A hydrogen atom in an n = 4 state can make a spontaneous transition to an n =2 state emitting a photon of energy 2.55 eV.

- ^{13.6 eV}/₄2 – (- ^{13.6 eV}/₂2) = 2.25 eV   = 486 nm a fantastic shade of blue or blue-green!
Commutation: If the operators for two quantities do not commute, then those quantities have a minimum uncertainty product. [x,p_x] = i   x p_x  ½ .

General Uncertainty Relation:

If operators commute, they can have simultaneous eigenvalues. For the hydrogen problem, H, L² and L_z (plus the operator for electron spin) commute. The hydrogen atom states are labeled by nm representing the eigenvalues of energy ( - ^{13.6 eV}/_n2), orbital angular momentum squared ([+1]²) and z component of orbital angular momentum (m) (plus spin – up or down).
Generalized Uncertainty Principle. Experimental measurements are performed on systems in order to extract average values and the corresponding uncertainties. In quantum systems, constraints can arise between two uncertainties due to the nature of the corresponding operators. Given two operators A and B, and knowledge of the commutator [A,B] = iC, then there exists a hard lower limit on the product of uncertainties A B >= ½ ||. Example: [x,p_x] = i   x p_x  ½ .

If two operators commute, then there is no lower-level uncertainty constraint on knowledge of the simultaneous values of operators (the outcomes of measurements), and the expectation values of these operators serve as a very useful label to identify and characterize a system. For the bare Coulomb potential (simple H-atom problem), the operators H, L², and L_z commute, so the values of the three can be known exactly at the same time. Therefore we identify individual states by the eigenvalues (energy, a.m. magnitude, a.m. z-comp) or their corresponding labels n,,m (quantum numbers).

returns the energy, - 3.40 eV, the energy for the n = 2 states, then the atom is left in a state of the form by the measurement: immediately after the measurement. (All the n = 2 parts remain; the parts for other n’s are removed.) The act of making a precise measurement of a physical observable collapses the state to a state that is consistent with the measured value. (Removes the parts that are not consistent; preserves remainder with relative phases.)

After the collapse, the state will time develop as directed by the Hamiltonian, and the state may not be consistent with the measured eigenvalue after some time has passed. If the operator for the eigenvalue commutes with Hamiltonian, the expectation value for the quantity will be time independent.

time development of expectation values

for operators Q without explicit time dependence

Time independent operators that commute with the hamiltonian have time independent expectation values.

Example with non-zero commutator:

The time rate of change is momentum is the ‘force’. Ehrenfest’s Theorem: Quantum mechanical expectation values obey the corresponding classical equations of motion.

According to L. Pauling, “the power of an atom in a molecule to attract electrons to itself.” Electronegativity is a measure of the tendency of an atom to attract a bonding pair of electrons. Electro-negativity increases from bottom to top of a column.

→ Atomic radius decreases → Ionization energy increases → Electronegativity increases →

The inner electrons screen the nuclear charge seen by the outer electrons. The apparent unscreened charge increases as you move across the table filling an electron shell. It is a maximum for group VIII and a minimum for the single electron outside a closed shell in group I. The effective unscreened charge increases as you move down the table to the higher Z atoms. Group I has one electron outside a closed shell and are the easiest to ionize (easiest at top after H; lowest screened charge). In group VII, the effective charge gets higher as you move down the table  most electron hungry (highest electron affinity). Electron affinity increases as you move down and across from I to VII. VIII is the closed shell nobles; they wish neither to give nor receive electrons. Outer valence shell has two S plus 6 P states.