Lecture 19: The Hydrogen Atom

Lecture 19: The Hydrogen Atom • Reading: Zumdahl 12.7-12.9 • Outline – The wavefunction for the H atom • Know what wave functions look like for a part...
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Lecture 19: The Hydrogen Atom • Reading: Zumdahl 12.7-12.9 • Outline – The wavefunction for the H atom • Know what wave functions look like for a particle trapped in a “box”; now we need to know what they look like for an electron attracted to a nucleus; and the energy of each wave function. – Quantum numbers and nomenclature – Orbital (i.e. wavefunction) shapes and energies • Problems (Chapter 12, 5th Ed.) – 48, 49, 50, 52, 54, 55, 56, 57, 60

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H-atom wavefunctions • Recap: The Hamiltonian is a sum of kinetic (KE, or T) and potential (PE, or V) energy.

Hˆ = Tˆ + Vˆ

The ‘bar’ means average over the position of the electron.

E = T + V = 12 V

er

P+

V (Potential E.)

• The hydrogen atom potential energy is given by: r

0

−e ˆ V = V (r) = r 2

2

The Coulombic PE (V) can be generalized − Ze 2 V (r ) = ( 4πε o ) r F e=− NA

e′2 =

e-

e2

( 4πε o )

2

p T= where p = mv 2m

r

Z + P

• Z = atomic number (= 1 for hydrogen) • r is the distance between the electron and the nucleus • Only one electron allowed (for now). 3

H-atom Coordinates Frame • The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates. r = interparticle distance (0 ≤ r ≤ ∞)

e-

p+

Major (azimuthal) angle θ = angle from z to“xy plane” (0 ≤ θ ≤ π) Minor angle φ = rotation in “xy plane” (0 ≤ φ ≤ 2π) 4

H-atom Allowed Energies When we solve the Schrodinger equation using the Coulomb potential, we find that the bound-state energy levels are quantized or discrete: 4 2 ⎛ ⎞ ⎛ Z me Z ⎞ En = − 2 ⎜ 2 2 ⎟ = − ⎜ 2 ⎟ ⋅ 2.178 x10−18 J n ⎝ 8ε 0 h ⎠ ⎝n ⎠ 2

• n (an integer counter) is the principal quantum number, and ranges from 1 to infinity. n=1 is the lowest energy (level) or ground state for an electron bound to a hydrogen-like nucleus. •This is the same formula Bohr gave us. •Compare and contrast these energy levels with those of the particle in a box. 5

Solve the Wave Equation for the Electron bound to the Nucleus • Set up the Schrödinger equation (SE) for the wave function in terms of x,y and z coordinates, then rewrite in polar coordinates (because V depends only on r). • Solve the SE the same way Schrödinger did: Look the answer up in a math book (Courant and Hilbert, in his case). • The solution gives a set of wave functions, and the energy of each wave function. • The wave functions (and energies) are distinct and countable (although in principle there are an infinite number of wavefunctions). • The wavefunctions are now called orbitals as they describe the probability of the electron in the vicinity of the nucleus. They are not orbits but regions of space wherein the electron orbits, hence orbitals. 6

Form of WaveFunctions (for Orbitals) • Like the particle in a box the wave function depends on the coordinate and a quantum number (like x and n). • There are three coordinates so the wave function is a product of a part that – Depends only on r and has n (and l) with it – Depends only on theta and has l (and m) with it – Depends only on phi (and has m with it)

• The total wave function has the form: Ψ n ,l ,m ( r , θ , φ ) = Rn ,l ( r ) Θl ,m (θ ) Φ m (φ ) x = r sin θ cos φ Relation between Cartesian and polar coordinates.

y = r sin θ sin φ z = r cos θ

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Orbitals • Orbitals are a description of where the electron resides (like a house) • Quantum numbers are like the address of the house. • The orbital does exist even without the electron (so an empty orbital is called a virtual orbital).

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Energy levels The energy expression for the QM result is the same as Bohrs, because the Virial Relation (which is also true for planets going around the sun) is also true for Quantum Mechanics and embodies the balance between potential and kinetic energy. p2 Ze 2 KE = T = >0 PE = V = −