Thermodynamics, Statistical Physics, and Quantum Mechanics

At time we start in state Later this state becomes The amplitude for pointing in the negative is found by taking the matrix elem ...

We assume that We determine the eigenvalue frequency by inserting the above form for into (S.5.30.8), which gives a quadratic eq ...

The probability of spin up is and that of spin down is b) In the usual NMR experiment, one chooses the field so that in which ca ...

respect to the variational parameter Denote by the value at this minimum: This result for is higher than the exact eigenvalue. 5 ...

5.33 Linear Potential II (MIT, Tennessee) The wave function must vanish in either limit that Two acceptable variational trial fu ...

The optimal value of called is obtained by finding the minimum value of Note that this result is also the first asymmetric state ...

The minimum energy is obtained by setting to zero the derivative of. with respect to This gives the optimal value and the minimu ...

Note theform of thekineticenergy integralK,which again is obtained from by an integration by parts. Again set the derivative of ...

where we set and Evaluate the integral So, we have Finally, To minimize this function, find corresponding to Now, 284 SOLUTIONS ...

Therefore, We should have the inequality (see Problem5.33) where is the true ground state energy. b) To find the exact ground st ...

5.37 Helium Atom (Tennessee) In the ground state of the two-electron system, both orbitals are in 1s states. So the spin state m ...

One can exchange the order of integration in one of the integrals and demonstrate that it is identical to the other. We evaluate ...

The action of the displaced momentum operator on the new eigenstates is so the Hamiltonian gives and the eigenvalues are simply ...

5.40 Circle with Field (Colorado, Michigan State) The perturbation is if weassume the field is in the The same result is obtaine ...

5.41 Rotator in Field (Stony Brook) The eigenfunctions and eigenvalues are b) The electric field interacts with the dipole momen ...

where is a constant of integration. We chose to make the potential continuous at b) For a single electronbound to a point nucleu ...

The result from first-order perturbation theory is obtained by taking the integral of the perturbation with the ground state wav ...

5.44 Relativistic Oscillator (MIT, Moscow Phys-Tech, Stony Brook (a)) a) The classical Hamiltonian is given by whereas the relat ...

Introducing the standard creation and annihilation operators (see Problems 5.15 and 5.16): we find that Using these results, we ...

Second solution: Instead of using operator algebra, we can find a wave function in the momentum representation, where The Hamilt ...