Thermodynamics, Statistical Physics, and Quantum Mechanics

236 SOLUTIONS 4.104 Electrons and Holes (Boston, Moscow Phys-Tech) a) Let the zero of energybe the bottom of the conduction band ...

THERMODYNAMICS AND STATISTICAL PHYSICS 237 since a hole isdefined as the absence of anelectron. We maythenwrite where is the ene ...

238 SOLUTIONS Therefore, Equating (S.4.104.3) and (S.4.104.11), we can find the chemical potential for an intrinsic semiconducto ...

THERMODYNAMICS AND STATISTICAL PHYSICS 239 We then obtain, from (S.4.105.2), The cross derivatives of (S.4.105.3) are equal so T ...

240 SOLUTIONS c)The heat capacity may be written as the integral Substituting into (S.4.105.8), we have Using the heat capacity ...

THERMODYNAMICS AND STATISTICAL PHYSICS 241 4.106 Critical Field in Superconductor (Stony Brook, Chicago) a) If the external fiel ...

242 SOLUTIONS b) The latent heat if the transition occurs at a constant temperature, is given by If the transition is from super ...

5 Quantum Mechanics One-Dimensional Potentials 5.1 Shallow Square Well I (Columbia) The groundstate energyEmust be lessthan zero ...

244 SOLUTIONS Dividingthese two equations produces theeigenvalueequation The equation given by the rightmost equals sign is an e ...

QUANTUM MECHANICS 245 c) For a positive energy state set where is the wave vector outside thesquare well. Inside the square well ...

246 SOLUTIONS The bound state for has the form We have already imposed the constraint that be continuous at This form satisfies ...

QUANTUM MECHANICS 247 Substituting (S.5.3.2) into (S.5.3.9) and using theresult of (S.5.3.7), we obtain Finally,using(S.5.3.5)yi ...

248 SOLUTIONS We use the first equation to eliminateAin the secondequation. Then each term has a factor of which is canceled: Mu ...

QUANTUM MECHANICS 249 Matching at either gives the pair of equations: Eliminating the constants A and B gives the final equation ...

250 SOLUTIONS 5.6 Transmission Through a Delta Function Potential (Michigan State, MIT, Princeton) On the left the particle has ...

QUANTUM MECHANICS 251 Using (S.5.4.1), we match the difference in the derivatives at with the amplitude of the delta function po ...

252 SOLUTIONS where the amplitude of the transition is given by The amplitude for the particle to remain in its ground state is ...

QUANTUM MECHANICS 253 5.9 One-Dimensional Coulomb Potential (Princeton) a) Since the electron is confined to the right half-spac ...

254 SOLUTIONS If both electrons are in the spin state (spin up), then the spin part of the wave function is symmetric under exch ...

QUANTUM MECHANICS 5.11 Square Well (MIT) 255 a) The most general solution is We evaluate the coefficients by using the initial c ...