Quantum Mechanical Model Systems

1. Quantum Mechanical Model Systems Erwin P. Enriquez, Ph. D. Ateneo de Manila University CH 47

2. Based on mode of motion • Translational motion: Particle in a Box • Infinite potential energy barrier: 1D, 2D, 3D • Finite Potential energy barrier • Free particle • Harmonic Oscillator • Rotational motion

3. Harmonic Oscillator

4. Classical Harmonic Oscillator

5. SOLUTION:Allowed energy levels v = 0,1, 2, 3, … Quantum Harmonic Oscillator (H.O.) Schrödinger Equation Potential energy

6. Solving the H.O. differential equation Power series method Trial solution: Substituting in H. O. differential equation: Rearranging and changing summation indices: Mathematically, this is true for all values of x iff the sum of the coefficients of xn is equal to zero. Thus, rearranging: 2-TERM RECURSION RELATION FOR COEFFICIENTS: Two arbitrary constants co (even) and c1(odd)

7. General solution • Becomes infinite for very large x as x ∞. • This is resolved by ‘breaking off’ the power series after a finite number of ters, e.g., whenn = v Thus, our recursion relation becomes: When n > v, coefficient is zero (truncated series, zero higher terms) v = 0,1, 2, … also, QUANTIZED E levels

8. v=1 v=2 v=3 v=4 Quantum Harmonic Oscillator Nomalization constant Hermite Polynomialsgenerated through recursion formula Example: What is  SOLUTION

9. General properties of H.O. solutions • Equally spaced E levels • Ground state = Eo = ½ hn (zero-point energy) • The particle ‘tunnels’ through classically forbidden regions • The distribution of the particle approaches the classically predicted average distribution as v becomes large (Bohr correspondence)

10. Molecular vibration • Often modeled using simple harmonic oscillator • For a diatomic molecule: In Cartesian system, the differential equation is non-separable. This can be solved by transforming the coordinate system to the Center-of-Mass coordinate and reduced mass coordinates.

11. Reduced mass-CM coordinate system Separable differential equation

12. Separation of variables (DE) Particle of reduced mass 'motion' (just like Harmonic oscillator case) Center-of-mass motion, just like Translational motion case The motion of the diatomic molecule was ‘separated’ into translational motion of center of mass, and Vibrational motion of a hypothetical reduced mass particle.

13. H. O. model for vibration of molecule • E depends on reduced mass, m • Note: particle of reduced mass is only a hypothetical particle describing the vibration of the entire molecule

14. Anharmonicity Vibrational motion does not follow the parabolic potential especially at high energies. CORRECTION: ce is the anharmonicity constant

15. Selection rules in spectroscopy • For excitation of vibrational motions, not all changes in state are ‘allowed’. • It should follow so-called SELECTION RULES • For vibration, change of state must corrspond to Dv= ± 1. • These are the ‘allowed transitions’. • Therefore, for harmonic oscillator:

16. The Rigid Rotor Classical treatment Shrödinger equation Energy Wavefunctions: Spherical Harmonics Properties

17. The Rigid Rotor • 2D (on a plane) circular motion with fixed radius. • 3D: Rotational motion with fixed radius (spherical) The Rigid Rotor

18. Classical treatment Linear velocity Linear frequency Motion defined in terms of • Angular velocity • Moment of inertia • Angular momentum • Kinetic energy The Rigid Rotor

19. Quantum mechanical treatment Shrödinger equation In spherical coordinate system Laplacian operator in Spherical Coordinate System The Rigid Rotor

20. Substituting into Schrodinger equation: Since R is fixed and by separation of variable: SOLUTION: SPHERICAL HARMONICS (Table 9.2: Silbey) l = azimuthal quantum number Degeneracy = 2l+1 The Rigid Rotor

21. Plots of spherical harmonics and the corresponding square functions From WolframMathWorld (just Google ‘Spherical harmonics’

22. Notes: • E is zero (lowest energy) because, there is maximum uncertainty for first state given by • We do not know where exactly is the particle (anywhere on the surface of the ‘sphere’) The Rigid Rotor

23. For a two-particle rigid rotor • The two coordinate system can be Center of Mass and Reduced Mass • since radius is fixed, the distance between the two particles R is also fixed • The kinetic energy for rotational motion is: • The result is the same: Spherical Harmonics as wavefunctions (but using reduced mass) The Rigid Rotor

24. Angular momentum and the Hydrogen Atom

25. Angular Momentum • This is a physical observable (for rotational motion) • A vector (just like linear momentum) • Recall: right-hand rule • L2 =L∙ L=scalar The Rigid Rotor

26. Angular momentum operators NOTE: SAME AS FOR RIGID ROTOR CASE

27. Angular momentum eigenfunctions Are the spherical harmonics: l =0,1,2,… m=0, ±1,…, ±l The z-component is also solved (Lx and Ly are Uncertain) REMINDER: SKETCH ON THE BOARD. FIGURE 9.9 and 9.10 SILBEY

28. RECALL: HCl rotational energies (l is called J) Angular momentum and rotational kinetic energy RECALL The spherical harmonics are eigenfunctions of both Hamiltonian and Angular Momemtum Square operators.

29. Hydrogen Atom

30. H-atom: A two-body problem: electron and nucleus describes translational motion of entire atom (center of mass motion) To be solved to get the wavefunction for the electron Note that the reduced mass is approx. mass of e-. Thus

31. Shrödinger equation for electron in H-atom SPHERICAL HARMONICS RADIAL FUNCTIONS (depends on quantum numbers n and l Quantized energy (as predicted by Bohr as well), Ryd = Rydberg constant. E depends only on n Degenerary = 2n2 E1s = -13.6 eV

33. Hydrogen atom wavefunctions • Are called atomic orbitals • Technically atomic orbital is a wavefunction = y • Given short-cut names nl: • When l = 0, s orbital l = 1 p l = 2, d

34. Plotting the H-atom wavefunction • Probability density = • Radial probability density (r part only)= gives probability density of finding electron at given distances from the nucleus Probability = • The spherical harmonics squared gives ‘orientational dependence’ of the probability density for the electron:

35. Bohr radius, ao Radial probability density (or radial distribution function) Node for 2s orbital Nodes for 3s orbital See also Figure 10.5 Silbey

36. Electron cloud picture 2s 1s 3s 2p

37. Shapes of y (orbitals) NOTE: This is not yet the y2.

38. Shapes of y (orbitals)

39. Properties for Hydrogen-like atom • H, He+, Li2+ • Energy depends only on n • Degeneracy: 2n2 degenerate state (including spin) • The energies of states of different l values are split in a magnetic field (Zeeman effect) due to differences in orbital angular momentum • The atom acts like a small magnet: Magnetic dipole moment Magnetogyric ratio of the electron Orbital angular momentum

40. Electron spin • Spin is purely a relativistic quantum phenomenon (no classical counterpart) • Shown by Dirac in 1928 as a relativistic effect and observed by Goudsmit and Uhlenbeck in 1920 to explain the splitting (fine structure) of spectroscopic lines • There is a intrinsic SPIN ANGULAR MOMENTUM, S for the electron which also generates a spin magnetic moment: • The spin state is given by s=1/2 for the electron, and with two possible spin orientations given by ms = +1/2 or -1/2 (spin up or spin down) • Spin has no classical observable counterpart, thus, the operators are postulated, and follows closely that of the angular momentum (Table 10.2 of Silbey) ge = 2.002322, electron g factor

41. Pauli Exclusion Principle • The wavefunction of any system of electrons must be antisymmetric with respect to the interchange of any two electrons • The complete wavefunction including spin must be antisymmetric: • In other words, each hydrogen-like state can be multiplied by a spin state of up or down • Thus, “No two electrons can occupy the same state = otherwise, each must have different spins” or “no two electrons in an atom can have the same 4 quantum numbers n, l, ml, ms.” Spatial part Spin part

42. More complicated systems… MANY-ELETRON ATOMS

43. He atom - r12 r1 • Three-body problem (non-reducible) • Not solved exactly! • Use VARIATIONAL THEOREM to find approximate solutions +2 - r2 Kinetic energy of e-s Electrostatic repulsion between e’s and attraction of each to nucleus

44. Variational Theorem (or principle) • One of the approximation methods in quantum mechanics • States that the expectation value for energy generated for any function is greater than or equal to the ground state energy • Any function f is a “Trial Function” that can be used, and can be parameterized (f = f(a)) wherein a can be adjusted so that the lowest E is obtained.

45. He-atom approximation • As a first approximation, neglect the e-e repulsion part.

46. Applying variational principle • Calculating the ‘expectation value from this trial function’ yields: • 2E1s=8(-13.6) eV=-108.8 eV • Subtracting repulsive energy of two electrons by evaluating: • Total energy is -74.8 eV versus experimental -79.0 eV.

47. Parameterization of the trial function • The trial wavefunction may be ‘improved’ by parameterization • For the He atom, the ‘effective nuclear charge’ Z is introduced in the trial wavefunction and its value is adjusted to get the lowest variational energy:

48. Going back to Pauli exclusion… implications…

49. Permutation operator • Permutation operator • Permutation operator squared • Eigenvalues • f is symmetric function • f is antisymmetric

50. Including spin states SINGLE ELECTRON SYSTEM TWO- ELECTRON SYSTEM (e.g., He atom: SEATWORK 1: Which of the functions above are antisymmetric, symmetric?