Particle in a 3 Dimensional Box just extending our model from 1D to 3D

Transcription

1 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-1 Particle in a 3 Dimensional Box just extending our model from 1D to 3D A 3D model is a step closer to reality than a 1D model. Let s increase the dimensionality of our particle in a box model by including directions x, y and z. Schrödinger Equation Ĥψ = Eψ Earlier, we derived the Schrödinger Equation for a 1D box, (x direction only). h 2 8π 2 m d 2 dx 2 ψ = Eψ 3D box (x,y,z directions) with the same rules (V = 0 in the box; can t get out) h 2 8π 2 m # 2 x y + & % 2 (ψ = Eψ $ 2 z 2 ' must take 2 nd partial derivative w.r.t. x, y and z otherwise no diff.!

2 Laplacian more new mathematical symbols CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-2 The Laplace operator 2, or Laplacian is used as shorthand notation for: 2 2 x y z 2 So # 2 x y & % $ z 2 ( ψ = 2 ψ ' Note: The symbol (nabla) denotes the gradient (or gradient vector field) of a scalar function. It is like the 1 st derivative of a curve (slope) except that it is in 2D or 3D space (gradient) think of a topological map mountains & valleys. The Laplacian 2 is then the divergence ( ) of the gradient ( f). It should be zero anywhere the gradient isn t changing.

3 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-3 Particle in 3D Square Box: 3D Standing Waves! resemblance to orbitals! Note, a function of x, y, and z can be rewritten as a product of functions: ψ(x, y, z) = ψ x (x) ψ y (y) ψ z (z) Solving for the wavefunction for a SQUARE box (L x = L y = L z =L): $ ψ = asin n xπx' $ & ) sin n yπy' $ & ) sin n zπz' & ) % L ( % L ( % L ( n x = 1, 2, 3, n y = 1, 2, 3, n z = 1, 2, 3, HOMEWORK: What would the ψ 1,1,2 wavefunction look like? How about the ψ 2,1,2 wavefunction?

4 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-4 Energies for 3D Square Box Model E = (n x 2 + n y 2 + n z 2 ) h 2 8 ml 2 Degeneracies Recall: States are degenerate if they have the same energy. Energies of particle in 3D box are. E 1 (1) = 3h 2 /8 ml 2 for ψ 1,1,1 E 2 (3) = 6h 2 /8 ml 2 for ψ 2,1,1 or ψ 1,2,1 or ψ 1,1,2 E 3 (3) = 9h 2 /8 ml 2 for ψ 2,2,1 or ψ 1,2,2 or ψ 2,1,2 E 4 (3) = 11h 2 /8 ml 2 for ψ 3,1,1 or ψ 1,3,1 or ψ 1,1,3 etc. Number in brackets is how many ways this energy can be arrived at. i.e., the number of degenerate states at that energy

5 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-5 Particle in a 2D Box electrons on surfaces STM images nanoscience! The 2D box model actually has a real life application. Atoms on surfaces can be probed using Scanning Tunneling Microscopy (STM)...we can see the electron waves! Quantum Corrals e.g. Iron on Copper (111) surface

6 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-6 How Does STM Work? The STM tip is brought very close to the sample surface. As it scans over the surface, it passes through regions of high and low electron density. A voltage is applied across the tip and sample and thus a current flows. higher e- density = higher current STM reveals the structures of surfaces atom by atom. The instrument s versatility extends to investigators in the field of physics, chemistry and biology. DNA

7 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-7 The Hydrogen Atom - an electron in the potential field of a nucleus (proton). Using quantum mechanics (i.e., wave mechanics) we can now solve the electronic structure of an H atom (2-body problem) properly. Schrödinger Equation Ĥψ = Eψ We must solve the Schrödinger Equation for an H atom. Unlike the particle in a box model, the potential energy is not zero (V 0). The electron exists in the potential field of the nucleus, so the potential energy is electrostatic and a function of distance from the nucleus r. Potential Energy is given by e2 r electrostatic attraction (+1 nucleus and -1 electron) The total energy E is just the sum of kinetic and potential contributions. We ve actually already figured out the kinetic contribution T using the particle in a 3D box model! So now we just have to put the pieces together

8 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-8 Kinetic and Potential Energy Contributions H atom From the particle in a 3D box model, here is the solution for the kinetic energy: h 2 8π 2 m 2 ψ = E kinetic ψ and the potential energy is just a simple electrostatic attraction: e 2 r ψ = E potentialψ E TOTAL = E KIN + E POT So h 2 8π 2 m 2 ψ + (e 2 /r)ψ = Eψ Schrödinger Equation for H Atom

9 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-9 Polar Spherical Coordinates easier than Cartesian coordinates Solutions are best represented by polar spherical coordinates instead of x, y, z Cartesian coordinates, we use r, θ (theta), φ (phi)

10 H atom Wavefunctions in Polar Spherical Coordinates CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-10 For an atom, describing the electron s position in space is much easier using the distance from the nucleus r, and two angles θ & φ (which are kind of like latitude and longitude on a globe). Instead of using ψ(x,y,z), the wavefunction is written as the product of the radial R and angular Y components:- Ψ(r,Θ,Φ) = R(r)Y (Θ,Φ) Actually working through the math to solve for the wavefunctions is beyond the scope of this introduction to Q.M., so let s just look at the answers (the solutions) The atomic orbitals are the mathematical solution to the H atom problem! Remember, we describe atomic orbitals using three spatial quantum numbers: n, l and m l

11 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-11 Solutions (Atomic Orbitals) The spatial wavefunctions are specified by THREE spatial quantum numbers: n, l, m 1 So, here are two examples of the many (infinite number of) solutions 1s: ψ 1s = 2e r radial 1 2 π angular 2p x : ψ 2 px = 1 re r /2 2 6 radial 3(sinθ cosφ) 2 π angular The point is that the atomic orbitals are real mathematical solutions to the Schrödinger equation for the H atom.

12 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-12 Quantum Numbers, n The solution to the Schrödinger equation imposes certain restrictions on the values that the quantum numbers can have. The principle quantum number n = 1, 2, 3, We saw n in the particle in the box model. It determined which normal mode of vibration we were looking at (fundamental or one of the overtones). The energies for the allowed states of the hydrogen atom are given by: $ E n = 2πm ee 4 ' & % h 2 ) 1 ( n 2 E n = k n 2 where n =1, 2, 3,... k = 13.6 ev (1 st IP of H)

13 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-13

14 CHEM 2060 Lecture 20: Particle in a 3D Box; H atom L20-14 Quantum Numbers, l l determines angular momentum determines # of planar nodes Runs in integral steps l = 0 n-1. Can be thought of as the shape quantum number l = 0 s orbital l = 1 p orbital l = 2 d orbital l = 3 f orbital (n l -1) determines # of radial nodes