St Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:

Transcription

1 St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S. Friedman, ISBN: X Another nice treatment may be found in Principles of Quantum Mechanics as applied to chemistry and chemical physics, Donald D. Fitts, ISBN: Lecture notes from Quantum Mechanics: Principles and Applications, Dr D. E. Manolopoulos and Prof. J. M. Brown, Michaelmas Term Second Year. A nice introduction to the spectroscopic aspects of this work is Atomic Spectra, T. P. Softley, ISBN: He gives a clear introduction to the coupling of angular momenta and selection rules in a qualitative sense. There are many, many other textbooks on Quantum Mechanics and Angular Momentum out there, but you should be aware that this is a vast topic and that many texts may be at an excessively high level for Part IA. Topics Please cover the following topics in your reading, making notes as appropriate. Rotational motion and the analogy with translational motion. Quantum mechanical angular momentum: definition of angular momentum operators, their commutation relations and eigenfunctions. Illustrative examples which you should know: particle on a ring, particle on a sphere, eigenfunctions of L 2 (orbitals). Electron spin: evidence for the existence of electron spin (Stern-Gerlach experiment, fine structure in the hydrogen atom spectrum), spin functions for a single electron (both α, β and the Pauli matrices), spin functions for two electrons, singlet and triplet states. The Pauli principle, antisymmetric wavefunctions. Methods of approximation: the variational principle. (The Quanutm Mechanics supplementary subject lectures will also cover time-dependent and time-independent perturbation theory, which would be useful to know by your third year.) Atomic spectra (you should have covered much of this last year, if not please see Softley): Atomic orbitals; electron configurations; selection rules; spectra of the hydrogen atom, hydrogenic atoms, the helium atom and alkali metal atoms; SCF procedure; penetration and shielding; quantum defects. You should also know about: singlet and triplet states, LS coupling, treatment of spin-orbit coupling, the Zeeman effect (magnetic fields) and the Stark effect (electric fields). Comparison of Russell-Saunders and jj coupling regimes. Term symbols. Hund s rules.

2 Recap of material covered in first year maths You should have covered the following material during your first year maths course, I list it here for reference only, please do not spend long on this. A good source for any revision needed is Classical Mechanics by T. W. B. Kibble and F. H. Berkshire, Hooke Library shelf mark Ma7. Definition of vector moment of a force about a point τ = r F. ( 3.3) Definition of vector angular momentum of a body J = r p. ( 3.3) Conservation of angular momentum (under the action of a central force). ( 3.4) Plane polar, spherical polar and cylindrical polar coordinates: definitions. ( 3.5) Definition of angular velocity vector ω = r v. ( 5.1) Definition of angular kinetic energy. ( 9.2 and attached PDF) Definition of moment of inertia about an axis. ( ) Inertial tensor. Definition of torque vector. (Attached PDF) Power of and work done by a torque. (Attached PDF) Rotation is not a vector quantity. (Attached PDF) Analogy between translational and rotational motion. Translational motion Rotational motion Displacement dr Angular displacement dφ Velocity v = dr/dt Angular velocity ω = dφ/dt Acceleration a = dv/dt Angular acceleration α = dω/dt Mass m Moment of inertia I = ρ ω r 2 dv Force F = ma Torque τ = r F = Iα Work W = F dr Work W = τ dφ Power P = dw/dt = F v Power P = dw/dt = τ ω Kinetic energy K = mv 2 /2 = p 2 /2m Kinetic energy K = Iω 2 /2 Momentum p = mv Angular momentum L = mr v Questions Please submit a complete set of answers to following problems. Answers should be sent to me via the R pigeon hole in the PTCL by 2pm Wednesday. Questions 1. The Hamiltonian for a particle of mass m, moving freely in two dimensions is: ( Ĥ = h2 2 2m x ) y 2

3 (i) Use the chain rule: ( ) x ( ) y y x = = r x y r r y r x φ + x y φ φ + y φ φ φ x r r to transform this Hamiltonian into polar coordinates r = (x 2 +y 2 ) 1/2 and φ = arctan(y/x) and hence show that the Hamiltonian for a particle of mass m confined to a ring of radius r can be written as: Ĥ = h2 2I 2 φ 2 where I = mr 2 is the moment of inertia of the particle about the origin. (ii) Evaluate the commutator [H,L z ] where L z = i h φ is the operator for the z component of the angular momentum and comment on the physical significance of your result. (iii) Verify that the functions: ψ m (φ) = (2π) 1/2 e +imφ, m = 0,±1,±2,... are simultaneous eigenfunctions of Ĥ and L z subject to the periodic boundary condition ψ(φ + 2π) = ψ(φ) and determine the corresponding energy and angular momentum eigenvalues. Is sin(mφ) also a simultaneous eigenfunction of Ĥ and L z? N.B. The m in this part is a quantum number, not a mass as in part (i). 2. The angular momentum L of a particle about the origin is given classically by the vector product: L = r p where r is the position of the particle and p is its linear momentum. The quantum mechanical operators for the components L x, L y and L z of the angular momentum are therefore given in terms of the position operators x, y and z and the linear momentum operators p x, p y and p z of the particles as: L x = yp z zp y L y = zp x xp z L z = xp y yp x Note the cyclic symmetry, which you should exploit as much as possible in the following: (i) Use the chain rule to show that this expression for L z is equivalent to the simpler expression L z = i h / φ when p x and p y are written out explicitly as p x = i h / x and p y = i h / y. (ii) Use these definitions of p x and p y to show that [x, p x ] = i h, [x, p y ] = 0 and [p x, p y ] = 0, and hence go on to show that: [L x,l y ] = i hl z [L y,l z ] = i hl x [L z,l x ] = i hl y

4 (iii) Verify the commutator identity [A,B 2 ] = [A,B]B + B[A,B] for two operators A and B and use this identity to show that: [ Lz,Lx 2 ] = +i h(lx L y + L y L x ) [ Lz,Ly 2 ] = i h(lx L y + L y L x ) (iv) Hence show that the squared angular momentum operator L 2 = L 2 x + L 2 y + L 2 z commutes with each component L x, L y and L z of the angular momentum. What is the significance of this result in light of your answer to part (ii)? (v) In spherical polar coordinates and L 2 = h 2 ( 1 sinθ θ sinθ θ + 1 sin 2 θ L Z = i h φ 2 ) φ 2 show that the function Y?,? (θ,φ) = sinθe +iφ is a simultaneous eigenfunction of both of these operators and identify the appropriate values of the quantum numbers l and m by comparing with the eigenvalue equations and L 2 Y l,m = h 2 l(l + 1)Y l,m L z Y l,m = hmy l,m Y?,? (θ,φ) is the angular part of what kind of hydrogenic orbital? 3. Consider the coupling of two angular momenta J 1 and J 2 (which may be either orbital or spin angular momenta) to give a resultant J = J 1 + J 2. (This coupling might be brought about, for example, by the spin-orbit interaction considered in the following question, where J 1 = L, J 2 = S, and the interaction which couples these two sources of angular momenta together to give a resultant J = L+S is the spin-orbital coupling Hamiltonian H so = ζ L S.) The operator for the square of the total angular momentum is: J 2 = (J 1 + J 2 ) 2 = J J J 1.J 2 = J J (J 1x J 2x + J 1y J 2y + J 1z J 2z ) and the operator for the projection of the total angular momentum in the z axis is: J z = J 1z + J 2z where the operators J ix, J iy and J iz satisfy the usual commutator relations: [J 2 i,j ix ] = [J 2 i,j iy ] = [J 2 i,j iz ] = 0 [J ix,j iy ] = i hj iz, [J iy,j iz ] = i hj ix, [J iz,j ix ] = i hj iy, for i = 1 and 2. Note also that each component of J 1 commutes with each component of J 2 because the two operators operate on different degrees of freedom.

5 (i) Use the basic angular momentum commutation relations to verify that: and [J 2,J 2 1] = [J 2,J 2 2] = 0 [J z,j 2 1] = [J z,j 2 2] = 0 and hence that it is possible to specify all four of the observables corresponding to J 2 1, J 2 2, J2 and J z simultaneously and precisely (ii) Go on to show that [J 2,J 1z ] 0 and [J 2,J 2z ] 0 and hence that the projections of the individual angular momenta J 1 and J 2 on the z axis cannot be specified at the same time as the square of the total angular momentum J. 4. Several different electronic states can arise from an atomic electron configuration (nl) k where k is less than the number required for a closed shell. Electrostatic interactions between open shell electrons cause moderately large energy splittings between these states, which are labelled by orbital and spin angular momentum quantum numbers L and S and which are referred to as terms. Within each term, smaller splittings occur due to spin-orbit coupling (a magnetic interaction between the spin magnetic moment and the magnetic field generated by the motion of charged particles about the electron in question). These splittings are referred to as fine structure and result in individual levels labelled by the total electronic angular momentum quantum number J. Assuming that L, S and J are all good quantum numbers, which is a reasonable approximation for light atoms, each atomic energy level E LSJ is given by: E LSJ = E LS + LSJ H so LSJ where E LSJ is the term energy, H so = ζ L S is the spin-orbit interaction Hamiltonian and ζ is the spin-orbit coupling constant of the atom. (i) Use the fact that the total electronic angular momentum operator J is given by the vector sum J = L + S to express H so in terms of the operators L 2, S 2 and J 2, and hence obtain an expression for E LSJ in terms of the quantum numbers L, S and J. (ii) An analysis of the electronic spectrum of a certain atom shows that the ground term is split into three levels with relative energies of 0, 14 and 42 cm 1. Use your expression from part (a) to assign J values to the three levels and obtain a value for ζ. What are the possible values for L and S for this term? 5. A quartic oscillator has a potential proportional to the forth power of the displacement from equilibrium so that the Hamiltonian is: Ĥ = h2 d 2 2µ dx kx4 Find the best variational value of the parameter α in the trial function: ψ(x) = e 1 2 αx2 and determine the corresponding energy. How might this approximate wavefunction be improved? 6. (i) What is meant by the Pauli principle and what is its effect on the symmetry of allowed molecular states?

6 (ii) Using the orbital approximation, write down the possible wavefunctions for the ground state and first excited state (1s2s) of the He atom, showing which are singlets and which are triplets. (iii) Treating the atomic Hamiltonian as H = h 1 + h r 12 where h 1 and h 2 are the single electron Hamiltonians and r 1 12 represents the electronelectron Coulombic replulsion (in atomic units), show that the triplet functions are usually lower in energy than the corresponding singlet states. Comment on the validity of the simple orbital approximation in this case. 7. (i) Give a brief account of the Stern-Gerlach apparatus. (ii) What happens when a beam of Ag atoms is passed through this apparatus and why? (iii) We now consider the case of two Stern-Gerlach apparatus. We pass a beam of silver atoms through the first apparatus (arranged with the field gradient parallel to z, and the beam moving along y), and then block the lower of the two emerging beams. The top beam, however, is fed into another Stern-Gerlach apparatus (also parallel to z). Describe the resulting pattern after this second magnet. (iv) We now insert a third magnet between the two described above. This Stern-Gerlach magnet is oriented parallel to the x-axis, and we again only allow one beam to progress from this magnet to the third. Explain what happens in this arrangement.