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## Quantum Mechanical Model Systems

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**Quantum Mechanical Model Systems**Erwin P. Enriquez, Ph. D. Ateneo de Manila University CH 47**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**SOLUTION:Allowed energy levels**v = 0,1, 2, 3, … Quantum Harmonic Oscillator (H.O.) Schrödinger Equation Potential energy**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)**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**v=1**v=2 v=3 v=4 Quantum Harmonic Oscillator Nomalization constant Hermite Polynomialsgenerated through recursion formula Example: What is SOLUTION**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)**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.**Reduced mass-CM coordinate system**Separable differential equation**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.**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**Anharmonicity**Vibrational motion does not follow the parabolic potential especially at high energies. CORRECTION: ce is the anharmonicity constant**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:**The Rigid Rotor**Classical treatment Shrödinger equation Energy Wavefunctions: Spherical Harmonics Properties**The Rigid Rotor**• 2D (on a plane) circular motion with fixed radius. • 3D: Rotational motion with fixed radius (spherical) The Rigid Rotor**Classical treatment**Linear velocity Linear frequency Motion defined in terms of • Angular velocity • Moment of inertia • Angular momentum • Kinetic energy The Rigid Rotor**Quantum mechanical treatment**Shrödinger equation In spherical coordinate system Laplacian operator in Spherical Coordinate System The Rigid Rotor**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**Plots of spherical harmonics and the corresponding square**functions From WolframMathWorld (just Google ‘Spherical harmonics’**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**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**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**Angular momentum operators**NOTE: SAME AS FOR RIGID ROTOR CASE**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**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.**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**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**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**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:**Bohr radius, ao**Radial probability density (or radial distribution function) Node for 2s orbital Nodes for 3s orbital See also Figure 10.5 Silbey**Electron cloud picture**2s 1s 3s 2p**Shapes of y (orbitals)**NOTE: This is not yet the y2.**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**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**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**More complicated systems…**MANY-ELETRON ATOMS**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**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.**He-atom approximation**• As a first approximation, neglect the e-e repulsion part.**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.**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:**Permutation operator**• Permutation operator • Permutation operator squared • Eigenvalues • f is symmetric function • f is antisymmetric**Including spin states**SINGLE ELECTRON SYSTEM TWO- ELECTRON SYSTEM (e.g., He atom: SEATWORK 1: Which of the functions above are antisymmetric, symmetric?