CHEM 1310 A/B Fall 2006 Chapter 16: Quantum Mechanics and the Hydrogen Atom

Quantum Mechanics and the Hydrogen Atom

Overview

Waves and Light
Paradoxes in Classical Physics
Planck, Einstein, and Bohr
Waves, Particles, and the Schrödinger equation
The Hydrogen Atom

Questions Addressed

What is quantum mechanics?
When do we need it?
What does it do?
How does it apply to the H atom?

Quantum Mechanics (QM) Defined

QM is the set of rules obeyed by small systems (molecules, atoms, and subatomic particles).
It's one of the two greatest achievements of 20th-century physics.
QM is the basis for new research into smaller electronic devices (e.g., quantum dots).
It is required to understand chemistry.

The Two-Slit Experiment

Firing small particles at a barrier with two tiny slits:
- Classical expectation: Particles go through one slit or the other, resulting in two distinct bands.
- Actual Result: An interference pattern is observed, indicating wave-like properties.
Even stranger, the wave-like interference pattern happens even when sending one electron through at a time.
If we watch to see which slit a particle goes through, the interference pattern disappears, and we see the expected pattern!
Richard Feynman: "I think it is safe to say that no one understands quantum mechanics."

Historical Background of QM

Near the end of the 19th century, physicists thought they knew everything.
Several key experiments showed something really unknown was going on.
QM developed to explain these unusual experiments in the early 1900s (~1900-1930s).
Developed around same time as theory of relativity.

Electromagnetic Spectrum

Illustrates the range of electromagnetic radiation, from radio waves to gamma rays.
Visible light is a small portion of the spectrum.
Wavelength and frequency are inversely proportional.

The Ultraviolet Catastrophe

Classical physics predicted that blackbodies should emit infinite energy at high frequencies (short wavelengths), which was not observed.
This discrepancy was known as the "ultraviolet catastrophe."

Planck's Solution

In 1900, Planck postulated that the blackbody is made of tiny oscillators with energies proportional to the frequency of oscillation.
$E = nhν$ , where:
- $E$ is the energy.
- $n$ is an integer.
- $h$ is Planck’s constant ( $6.626 \times 10^{-34}$ J s).
- $ν$ is the frequency.
Energy is quantized; only certain values are allowed.
Using this hypothesis, blackbody radiation curves can be predicted accurately.

The Photoelectric Effect

Light can cause electrons to be ejected from a metal surface.
Problem: KE of electrons does not depend on intensity, but does depend on frequency $ν$ .

Einstein's Explanation

Borrowed Planck’s “quantum” idea --- maybe light might have quantized energy levels, too!
Light comes in “packets” of energy $E = hν$ , called “photons.”
Explains the photoelectric effect --- higher $ν$ , more energy in each light packet (photon), kicks out electron with more KE.

Photoelectric Effect Explained

Minimum energy to remove an electron is $hν_0$ , the “work function” of the metal.

Atomic/Molecular Spectra

Emission Spectrum: Excited sample emits light and pass through prism, resulting in individual lines.
Absorption Spectrum: White light source shines through an absorbing sample and then through a prism, it results in dark lines.

H Atom Spectrum

The lines follow a particular pattern.
Lines fit the “Rydberg formula” $ν = (1/n^2 – 1/m^2)(3.29 \times 10^{15} s^{-1})$ where n and m are integers. Amazing!

Bohr's Explanation of H Atom Spectrum

Bohr(1913) borrowed ideas of quantization from Planck and Einstein and explained the H atom spectrum.
Bohr argued that angular momentum was quantized, leads to quantization of H atom energy levels.
Bohr frequency condition: $\Delta E = hν$
Equations match the Rydberg formula to an accuracy not seen previously in all of science.

Bohr’s Solution

Quantization of angular momentum.
Leads to quantization of radii (“Bohr orbits”).
Leads to quantization of energies.
Assume the “Bohr frequency condition”.
Yields the same “Rydberg formula” for allowed energy levels!!!
Constants: $a0 = 1$ bohr ( $0.529$ Å), $Ry = 1$ Rydberg = $2.17987 \times 10^{-18}$ J

H Atom Spectrum Explained

Energy-level diagram for the hydrogen atom, showing how transitions from higher states into some particular state lead to the observed spectral series for hydrogen.

“New Quantum Theory”

The “quantization” idea was groundbreaking, but it did not have a firm foundation.
De Broglie (1924) realized that if light can act as a wave and a particle, then maybe particles like electrons can also act like waves!
“Wave/particle duality” also works for matter!
Can relate momentum (particle property) to wavelength (wave property) via the de Broglie relation $\lambda = h / p$ ( $p = mv$ )

The Schrödinger Equation

1925: Schrödinger developed new mechanics for “matter waves” shown by de Broglie. Quantum mechanics!

The Schrödinger Equation (Components)

Nuclear kinetic energy
Electron kinetic energy
Nuclear/electron attraction
Nuclear/nuclear repulsion
Electron/electron repulsion

The Schrödinger Equation (Wave Function)

$\Psi$ is the wave function. It gives the amplitude of the matter wave at any position in space (for more than 1 electron; need the coordinates $xi = {xi, yi, zi}$ for each particle i)
$\Psi(x1, x2, …, x_n)$ for n particles.
Focus on wave function for a single particle (like an electron) for now…

Classical Standing Waves

String tied to the wall at both ends ( $x=0$ and $x=L$ ).
Have to fit a half-integer number of wavelengths $\lambda$ in the length L.
Number the standing waves $n=1$ , $n=2$ , …
Max amplitude for standing wave n is $un(x) = An \sin(nπx/L)$
# of nodes increases with n; energy also increases with n (more nodes -> higher energy).
Just like $un(x)$ gives amplitude of vibration at a given point x, $\Psin(x)$ gives the “amplitude” of the matter wave.

Interpretation of Ψ

Most commonly accepted interpretation due to Max Born.
Assume only one particle for now.
$\Psi^*(x, y, z) \Psi(x, y, z) \Delta x \Delta y \Delta z$ is the probability that the particle will be found in a box of size $\Delta x \Delta y \Delta z$ centered at point x,y,z.
Seems crazy--we never actually know where the particle is, only the probability of finding it there. Even worse – these “probability waves” can interfere constructively/destructively!

Wave Picture Justifies Bohr’s Assumption for H Atom!

To avoid destructive interference, an electron in a Bohr orbit must have its wavefunction match itself after going around once.
$2 πr = n \lambda$
But also $\lambda = h / mv$
… and so $2 πr = nh / mv$ , or $mvr = nh / 2 π$ , as Bohr assumed!

Schrödinger Equation for H Atom

Can solve and obtain the same energy equation as Bohr found. But now we also get the wave function $\Psi_{nlm}(x, y, z)$ , depending on three integers n, l, and m
- n = “principal quantum number” (the same n in energies $E_n$ ), starts counting from 1
- l = “angular quantum number” l = 0, 1, …, n-1
- m = “magnetic quantum number” m = -l, -l+1, …, 0, 1, …, l
Actually there’s also a 4th quantum number, ms, giving the spin ( $\frac{1}{2}$ for “up” spin α, - $\frac{1}{2}$ for “down” spin β)

Wave functions for H atom

Energy depends only on n for H atom, not on l or m
Shape of wave function depends on n, l, and m
A function of one particle is called an “orbital”
- l=0 is an s orbital
- l=1 is a p orbital (m=-1, 0, 1 => px, py, pz)
- l=2 is a d orbital (m=-2, -1, 0, 1, 2 => dxy, dxz, dyz, dx2-y2, dz2)
- l=3 is an f orbital (7 of these)… etc…
All these functions are 3D functions; hard to plot…

Summary of H atom orbitals

Energy depends only on n.
For a given l, increasing n increases the average distance of electrons from the nucleus (& the size of the orbital). 3s larger than 2s.
$\Psi_{nlm}$ has l angular nodes and n-l-1 radial nodes (total of n-1 nodes).
Only for s orbitals does $\Psi_{nlm}$ remain nonzero as r→0. Only s orbitals “penetrate to the nucleus”.
Note: orbitals are only rigorous for H atom or other 1-electron atoms! For multiple electrons, need molecular orbital theory (even for atoms). Solve multi-electron Schrödinger equation.

Heisenberg Uncertainty Principle

“Bohr orbit” idea violates the uncertainty principle!
Certain pairs of variables (e.g., x and px; E and t; r and L) can’t be known exactly at the same time
E.g., $(\Delta x)(\Delta p_x) ≥ \frac{h}{4π}$ , where $\Delta x$ denotes an uncertainty in x, etc. Clearly both uncertainties can’t be zero if RHS is nonzero…
Deep result, NOT a mere technical problem with measurement.