The Hydrogen Atom

1. z e- q r + x f y The Hydrogen Atom Coordinates Systems e- r spherical polar coordinates re Z R + rp X Y center of mass or barycenter z=r cos(q) y=rsin(q)sin(f) x=rsin(q)cos(f) reduced mass

2. moment of inertia: I=mr2 angular momentum: kinetic energy: T=L2/(2I) Rotation If mp>>me: m≈me r ≈ re f + e- tangential component: radial component: angular velocity

3. angular momentum: vectorial representation quantum mechanics: me Analogies: Translation Rotation m I v w p=mv L=Iw

4. z e- q Coordinates Transformation r f + x y cartesian coordinates spherical polar coordinates

5. coulombic potential: U=-1/r Z 2-D r=|√x2+ y2| X Y 1-D r=|x| 3-D r=|√x2+ y2+z2|

6. Space: f(x) TIme: f(t) Equivalent to two ordinary (not partial) differential equations: Space: X(x) Time: T(t)

7. Schrödinger equation: kinetic, rotational, coulombic free particle

8. Schrödinger equation: radial, angular wavefunctions

9. Wavefunctions (solutions)

10. Quantum numbers: Energy: Principal: n: 1,2,3,…….. Angular: l: 0,1,2,…(n-1) Magnetic: m: +l,(l-1)…0….-l z component of the angular momentum: Magnitude of the angular momentum:

11. Effective potential The effective potential energy of an electron in the hydrogen atom. When the electron has zero orbital angular momentum, the effective potential energy is the Coulombic potential energy. When the electron has nonzero orbital angular momentum, the centrifugal effect gives rise to a positive contribution which is very large close to the nucleus. We can expect the l = 0 and l  0 wavefunctions to be very different near the nucleus. eigenvalues

12. Radial wavefunctions

13. Atomic orbitals What is an atomic orbital? Orbitals and orbits When the a planet moves around the sun, you can plot a definite path for it which is called an orbit. A simple view of the atom looks similar and you may have pictured the electrons as orbiting around the nucleus. The truth is different, and electrons in fact inhabit regions of space known as orbitals. Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them. The impossibility of drawing orbits for electrons To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do this for electrons. The Heisenberg Uncertainty Principle says - loosely - that you can't know with certainty both where an electron is and where it's going next. (What it actually says is that it is impossible to define with absolute precision, at the same time, both the position and the momentum of an electron.) That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and find a way around it.

14. It is not possible to determine the exact location of an electron in an atom. However, the probability of finding an electron at a given position can be calculated. The higher the probability of finding an electron at a given position, the larger the electron density at that position. An atomic orbital

16. Some interesting Websites…. http://winter.group.shef.ac.uk/orbitron/ http://www.orbitals.com/orb/ http://www.falstad.com/qmatom/ http://bouman.chem.georgetown.edu/atomorbs/aovis.html http://electron6.phys.utk.edu/qm2/modules/m1-3/hydrogen.htm http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/HydrogenAtom.htm