Symmetries in Quantum Mechanics

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De afgelopen examens

Maandag 14 januari 2013

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Woensdag 1 februari 2012

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Maandag 23 januari 2012

1. 1. Explain the relation between symmetry and conservation laws in quantum mechanics.
   2. If turning on a perturbation has the effect of splitting the energy levels, then what can you generally infer about the system before the perturbation was turned on?
2. Let $|njm>$ be the energy eigenstates of a particle in some spherically symmetrical potential, with j,m the usual angular momentum quantum numbers. A perturbation $H\to H+e(t)W$ is applied, which will cause transitions between these states. Give as many selection rules as you can for the first order transition matrix element $<njm|W|njm>$, when (i) $W=e^{-r^2}$, (ii) $W=(x^2-y^2)e^{-r^2}$ , (iii) $W=L_x$.
3. A system consists of 100 spin 2 particles. Construct a state with total spin quantum numbers (J,M)=(200,199).
4. Just by counting degeneracies, find the total angular momentum of the ground state of eight non-interacting identical spin 1/2 particles in a 3D harmonic oscillator potential. Take into account the Pauli exclusion principle.
5. A system with rotational and time reversal invariance whose energy levels are nondegenerate, except for the degeneracies implied by rotational invariance cannot have a permanent electric dipole moment in any energy eigenstate. Show this by combining time reversal invariance with the Wigner-Eckart theorem (to relate dipole and Angular momentum matrix elements).

Vrijdag 20 januari 2012

1. The wigner eckart theorem: what's it good for? (in a few lines)
2. Under what circumstances is it possible to simultaneously measure with arbitrary precision energy and (i) position (ii) momentum (iii) angular momentum and (iv) velocity, of particle in a static electric field? And in a magnetic field?
3. A particle is in a state with wave function $\psi (x,y,z)=(x+y)f(r)$ where $r=\sqrt{x^2+y^2+z^2}$.
   1. argue using simple symmetry reasoning that $<\chi |\psi >=0$ for any $|\chi >$ that is even under parity or odd under the exchange of x and y coordinates.
   2. what are the possible outcomes for measurements of $L^2$ and $L_z$, and with what probabilities?
   3. let |nlm> be hydrogen wave function. For which values of l,m is it guaranteed that $<nlm|z|\psi >=0$? Same for $<nlm|x^2-y^2|\psi >$ and $<nlm|L+|\psi >$.
4. A system consists out of 100 spin 1 particles. Can you explicitly construct a state with total spin quantum number |J=100,m=99>?
5. 1. two atoms are slowly brought towards each other. what will happen to their energy spectra?
   2. three helium 3 atoms walk into a bar. all energy levels of this trio are double degenerate. what can be done to undo this doubling?

Maandag 16 januari 2012

1. symmetries in quantum mechanics: what are they good for?
2. Say we have an orthonormal set of three states {psi_x, psi_y, psi_x} (say of some atom), on which rotations act in the standard vector representation, i.e. they transform among each other in the same way as a vector (x,y,z) in R³
   1. How do the operators corresponding to angular momentum Lx, Ly and Lz act on these states?
   2. What are the possible values of Lz?
   3. if we prepare a beam of these atoms, prepared to be in states restricted to be linear combinations of the above three states, and we first pass this beam through a filter that allows through only atoms with maximally positive spin in the z-direction, next through a filter that allows only atoms with maximally positive spin in the x-direction, and last through a filter that allows only atoms with maximally negative spin in the z-direction, then what fraction of the original beam will survive?
   4. Can this result change if the beam passes through some homogeneous magnetic field between two subsequent detectors?
3. 1. A spin 1 particle at rest decays into a spin 1/2 particle B and a spin 1/2 particle C. What are the possible values of the z-components of spin B and C assuming the final (center of mass) orbital angular momentum is measured to be zero?
   2. What are the other possible values of the final orbital angular momentum that could be measured, and what are the corresponding values of the spins of B and C?
   3. How do the possibilities get further reduced if you know the intrinsic parities of A, B and C are all even?
   4. Restricting again to the zero orbital angular momentum sector, and assuming all initial spin states are equally likely, what are the probabilities for the z-components of the spin of B and C?
   5. How would you solve this last problem for a spin 3 particle A decaying into a spin 1 particle B and a spin 2 particle C? (optionally: solve it)