Thermodynamics, Statistical Physics, and Quantum Mechanics

as found in thefirst solution. b) Thefirst-orderenergyshift from would be zero (no diagonal elements in the matrix). Theleadingc ...

5.45 Spin Interaction (Princeton) In first-order perturbation theory the change in energy is since and the matrix element of is ...

b) In the spin–orbit interaction we take the derivative and find The matrix element is a constant, which simplifies the calculat ...

The last factor in (S.5.47.3) is the spin-wave function for the singlet in terms of up and down spin states. Since the spin stat ...

Since the two states have no matrix elementswith the other two states, we can omit them from the remaining steps in the calculat ...

QUANTUM MECHANICS 301 One interaction potential is One can use parity and other group theory arguments to show that the only non ...

302 SOLUTIONS 5.50 Hydrogen in Capacitor (Maryland, Michigan State) For time-dependent perturbations a general wave function is ...

QUANTUM MECHANICS 303 a) For the state the probability is zero. It vanishes because the matrix element of is zero: because of pa ...

304 SOLUTIONS a) If the initial state is at then the amplitude of the state for is given by The last equation is the probability ...

QUANTUM MECHANICS 305 5.52 Decay of Tritium (Michigan State) We use the sudden approximation to calculate the probability that t ...

(^306) SOLUTIONS Imposing the condition yields We know that in this approximation 5.54 Truncated Harmonic Oscillator (Tennessee) ...

QUANTUM MECHANICS 307 The integral on the left equals The easiest way to see this result is to use the change of variables and t ...

308 SOLUTIONS The last integral equals Writing we find We have to determine E. Equation (S.5.55.5) is a quadratic equation for t ...

QUANTUM MECHANICS 309 5.57 Charge and Plane (Stony Brook) In the WKB approximation or, betweenturningpoints, Substituting(S.5.57 ...

310 SOLUTIONS c) Using the boundary conditions at we obtain It implies that the odd states, for which are not affected by whilee ...

QUANTUM MECHANICS 311 The factor is thephase changewhen the particle goesthrough the turning point where For the present problem ...

312 SOLUTIONS The last integral is found in standard tables. To evaluate the phase shift, we need to evaluate this expression in ...

QUANTUM MECHANICS 313 a) The wave functions on theleft and right of the origin are where and are the amplitudesof the reflected ...

314 SOLUTIONS where and the theta function is 1 if and 0 if For the solutions are in the form of or Instead, write it as where t ...

QUANTUM MECHANICS 315 5.64 3D Delta Function (Princeton) For a particle of wave vector Schrödinger’s equation for the radial par ...