Quantum Theory (cont.) 10 Sept. 2008

1. Quantum Theory (cont.)10 Sept. 2008 • Heisenberg Uncertainty Principle • The Schrodinger Wave Equation • Quantum Mechanical Tunneling • The Schrodinger Wave Equation for the Hydrogen Atom • Hydrogen-like Atomic Orbitals • s, p & d orbitals • Electron Spin and the Electron Spin Quantum Number (ms)

2. Classical vs. Quantum Mechanics • Recall that for a classical system made up of particles, you can completely specify the state of the system by giving the position and momentum (or equivalently the velocity) of every particle in the system at any particular time. • If you consider one particle whose position and momentum is known at this initial time, then if you know all the forces acting on that particle you can write down equations which tell you exactly what its position and momentum will be at any future time.

3. Classical vs. Quantum Mechanics In quantum mechanics the situation is a little more complicated . • There are two key differences between these two theories. • First of all, the state of a particle in quantum mechanics is not just given by its position and momentum but by something called a "wave function“, psi. • Secondly, knowing the state of a particle (i.e., its wave function) does not enable you to predict the results of measurements with certainty, but rather gives you a set of probabilities for the possible outcomes of any measurement. • In Quantum Mechanics the systems that you study are still made of particles, and the basic procedure is in some ways similar: • You measure the state of a particle at some initial time, • you specify the forces acting on that particle (or equivalently, the potential energy function describing those forces), • and quantum mechanics gives you a set of equations for predicting the results of measurements taken at any later time.

4. Classical vs. Quantum Mechanics • Psi the wave function is related to the probability of finding the particle in a specific region of space (which now leads us to describe electron orbitals). • So, using quantum mechanics is like a taking a snapshot of the particle’s position at an initial time, t0 and a later snapshot of the particle at a later time tn, but not knowing its position at any given time between t0 and tn. • Bottom line: Quantum mechanics provides a set of probability functions of finding an electron at some given energy level and some distance in space away from a nuclear center.

5. A particle with large uncertainty in position, but well defined wavelength (l) A particle with well-defined position, but large uncertainty in wavelength A a particle with intermediate uncertainty in position and momentum (l) The Heisenberg Uncertainty Principle It is impossible to know simultaneously both the momentum and the position of a particle with certainty. (due to the wavelike properties of matter) where ∆x is the uncertainty of measuring position and ∆p is the uncertainty of measuring the momentum

6. or Schrodinger Wave Equation Complete information about the state of a quantum particle is contained in a function y(x), called the wavefunction. Y2 is related to the probability, P, of finding the particle in a specific location.

7. y as function of x y2 for y as a function of x Probability of the particle in interval a<x>b

8. Quantum (wave) Mechanics Time-independent Schrodinger wave equation with solutions called stationary-state functions (do not change as a function of time). The wave function must satisfy 1. y must be single-valued at all points. 2. The total area under y2(x) must be equal to unity or 3. y must be smooth or dy/dx must be continuous at all points.

9. proportional to d2y/dx2 describes curvature of wavefunction kinetic energy portion potential energy portion Qualitative Aspects of the Wavefunction Ground-state wave function is a compromise to minimize each term.

10. Ground state wavefunction with low potential but high kinetic energy (curvature) Ground state wavefunction with high potential but low kinetic energy (curvature) Exact ground state wavefunction Potential energy of function V(x)

11. For most problems The Schrodinger Wave Equation cannot be solved exactly. One of the few exceptions is the one-dimensional particle in a box with infinite energy barriers, i.e., a particle of mass with potential energy, V, that is confined to a one-dimensional region, a line of length, L

12. kinetic energy portion or Because the potential energy is 0 inside the box the energy of the particle is entirely kinetic. For the region 0 < x < L the Schrodinger Equation becomes….

13. The Particle in a One Dimensional Box Go to http://user.mc.net/~buckeroo/PODB.html

14. Quantum-Mechanical Tunneling Consider a one-dimensional particle in a box with finite energy barriers. For finite potential barriers there is some probability of finding the particle outside the box due to the wavelike properties of matter (QM Tunneling). Quantum Mechanical Tunneling

15. Quantum-Mechanical Tunneling QM Tunneling can be used to explain the spontaneous alpha decay process in which a nucleus spontaneously decays by emitting an alpha particle. Kinetic energy of emitted a: 6 x 10-13 J While the potential energy barrier for escaping the nucleus is about 70 times greater: i.e., 4 x 10-11 J

16. An important practical application of quantum mechanical tunneling is the electron microscope Images of atoms at the Angstrom scale!!

17. Schrodinger Equation for Hydrogen Exact solution in polar spherical coordinates (r, q, f ) results in three quantum numbers that indicate the allowed quantum states. • principal quantum number, n: n = 1, 2, 3, … • angular momentum quantum number, l: l = 0,1, …,n-1 • magnetic quantum number, ml: ml = -l,...-1, 0, +1, …,+l

18. Hydrogen and hydrogen-like atoms orbital energy depends only on n. atomic orbital: wavefunction for a single electron which describes the position (probability) of the electron n>1 multiple orbitals exist corresponding to different combination of n and l. They are • degenerate: have the same energy • collectively called an energy shell

19. subshell: Within an energy shell, a given set of distinct orbitals exist with the same value of l.

20. radial part Radial and Angular Parts of the Wavefunction

22. angular part Radial and Angular Parts of the Wavefunction

24. radial part angular part or Radial and Angular Parts of the Wavefunction A specific orbital is the product of the radial factor and the angular function (think of a 3-dimentional coordinate axis).

26. s Orbitals • Dependent only on r • Spherically symmetric • Defined by the radial function Rn0(r) Radial probability function

27. Boundary surface diagrams: surface containing 90% of the total electron density (defined as y2) in the orbital.

28. Plane in which y , hence electron density is zero, across which the wavefunction changes sign. p Orbitals • Dependent on f, q • Not spherically symmetric (elliptical boundary function) • Defined by the angular parts of the wavefunction • Letter subscript indicates axis along which the orbitals are • oriented • Contain a nodal plane perpendicular to orientation axis

29. Radial part of wavefuntion and radial probability for p Orbitals

30. Orientation of the p Orbitals

31. d Orbitals • Letter subscript contains information about their shape and • orientation • Contain two nodal planes

33. Electron Spin Electron spin quantum number (ms):

34. Experimental Evidence: Stern and Gerlach experiment Experimental evidence demonstrating the existence of electron spin

35. So, what is your intuitive picture of Quantum Mechanics??? Other questions???