Intoduction to Quantum Mechanics

Answers for questions for study session 1.

Spring Quarter, 1999-2000

April 4, 2000

2. One can improve the result of a variational calculation of the ground state energy by using more parameters because when one minimizes the exceptation value then the introduction of a new parameter can only lower the minimum.

3.  One can use the variational calculation for states other than the ground state if the system has a symmetry (like reflection or rotation). In that case the eigenstates can be classified according to the eigenvalues of the symmetry operator (even or odd, or being an eigenstate of L² ).  If one restricts the variation to states having a given eigenvalue under the symmetry operator then one gets an upper bound for the lowest energy state having the same eigenvalue.

4.  Variational calculations are more successful if one has an idea about the structure of the state.  The physical idea behind using a Helium atom wavefunction  built from hydrogenlike wavefunctions ~ e^-Zr/a, with Z as a variational parameter is that in the two electron system one of the electrons partially screens the nucleus for the other electron.  Thus, the electrons only see a partially screened charge, Z, with 1<Z<2.

5.  The requirements for the wavefunction of the H₂⁺ ion when the distance of the two H atoms goes to infinity is that it becomes a superposition of two states, each of them is the product of a proton state and a Hydrogen atom.  When the distance goes to 0 then it should become the wavefunction for the Helium atom.

6.  Molecular wavefunctions are built from sums of products of single atom wavefucntions.  When such a wavefuction is normalized or the expectation value of an operator is calculated then integrals appear in which the product of two wavefunctions (one is complex conjugated)  of an electron appears  in which the electron is centered around different atoms. Such integrals are called overlap integrals as their value depends critically on the overlap of the wavefunctions of the two atoms.

7. The Heitler-London valence bond orbital is a two electron wavefunction that is the linear combination of two terms, in both of which are products of atomic wavefunctions, for the two electrons centered around different atoms.

8.  Increasing overlap of the electron wavefunctions always increases the energy because the expectation value of the Coulomb repulsion term of electrons increases.  In molecules, however the electrons have an important effect of creating the bond between ions.  This role is played the most effectively if the electrons spend most of their time between the two atoms.  For molecules the attraction energy created this way is larger than the repulsion energy stemming from the fact that the electrons are closer to each other.

9.  The WKB method is based on the fact that for slowly varying potentials the wavefunction of a paraticle is similar to that of a free particle, having a relatively fast varying phase and slowly varying amplitude.

10. The WKB method can be made into a systematic expansion if we expand around classical physics, by eassuming a form e^iS/h and expand the amplitude in a power series of h.

11. The classical momentum, p(x),  of a particle of potential energy V(x) and energy E is [2m(E-V(x)]^1/2.

12. A quantum mechanical state such that the momentum operator acting on the state gives this p(x), multiplied by the wavefunction (almost like having an `eigenvalue' of p(x)) is constructed as e^iS/h where S is the integral of the function p(x') to an upper limit x.

13. The classical turning point defined by the vanishing of the momentum, it is given by the solution of the equation E=V(x) for x.

14. The problem with the semiclassical method at the classical turning points is that when the differential equations are solved for S then one encounters a sum of two terms -S^'2/h²+iS''/h. One solves this equation using the principle of expansion in h. by neglecting the second term. At a classical turning point, however, S'=p(x) vanishes, while S'' does not and the expansion breaks down.

15. Classical turning points at vertical walls where the momentum does not vanish, but it is just reversed, constitute an exception to that.

16. The wavefunction oscillates in the classically allowed regions and it has an exponentially decreasing or increasing behavior in classically forbidden regions.

17. The tunnelling rate for broad and high barriers decreases exponentially, like e^-2g where g  is an integral over |p(x)| across the barrier.

18. The tunnelling rate is connected to the lifetime of bound systems through a simple classical picture.  Classically the particle is bouncing back an forth inside the potential well. Its energy determines its velocity and one can calculate the time it takes for the particle to go through the well and hit the wall.  As each time that happens one knows the tunneling rate, one can find the tunneling (or decay) rate per second, which is the inverse of the life time.

19. To connect WKB solutions in the classically allowed and forbidden regions across the classical turning point one needs to find an "exact" solution of the Schrodinger equation in the immediate neighborhood of the classical turning point.  This can be done and one finds that the Airy functions provide such solutions.  Then in the next step one can match these solutions with the WKB solutions inside and outside the well.