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Lecture 2: Fundamental Principles of Quantum Mechanics
Website with slides and handouts:
http://home.uchicago.edu/ matthewroberts/compton/index.html
Black body radiation
Light comes in individual particles, explaining deviation from Rayleigh-Jeans law at short
wavelengths - waves are particles!
Double-slit experiment
Experiments confirm that electron behave like waves and interfere with themselves - particles
are waves!
Schrödinger Wave Function
ψ(t, x) is a complex-valued function of space and time. (Recall complex numbers have real
and imaginary part, or a magnitude and phase, z = x + iy = reiθ .) The magnitude of the
wave function tells you the probability distribution of where you might find the particle,
ψ ∗ (x)ψ(x) = |ψ(x)|2 R= P (x). Since the sum of all probabilities must be one, the amplitude
of the wave is fixed, |ψ(x)|2 = 1.
Schrödinger’s Equation
2
~
Schrödinger’s equation, i~∂t ψ = Hψ = − 2m
∂x2 ψ + V (x)ψ, tells you how the wave function
2
~
evolves in time. − 2m
∂x2 is the kinetic energy and V is the potential energy, and therefore H
is the total energy in the system. The phase of the wave function oscillates at a frequency
fixed by the energy, ω = E/~.
Expectation Values
You can
R calculate the statistical Raverage for different observations, for instance position
hxi = ψ ∗ xψdx or energy hEi = ψ ∗ Hψdx.
Energy Spectrum
By separation of variables, ψ(t, x) = ψ(x)e−iEt/~ , we can calculate the energy levels of a
system. For instance for a Hydrogen atom, we can find that there are energy levels labelled
by an integer n with energy En = −1 Ry × n12 , 1 Ry = 13.6 eV. This allows
us to predict
the energy of photons emitted when electrons transition, Ephoton = 1 Ry ×
1
n21
−
1
n22
.
Uncertainy Principle
You can’t simultaneously know the exact position and and momentum of a quantum object:
∆x · ∆p ≥ ~. You also can’t measure the energy of a state precisely in a short amount of
time, ∆E · ∆t ≥ ~.
Electron spin and entanglement
Electrons have spin 12 and therefore can be spinning up or down (|+i or |−i). We say a state
made of more than one particle is entangled if the state is not a trivial product state: | + +i
is not entangled, as each particle is spin up regardless of the other. The state | + −i + | − +i
is entangled, as if we measure the first electron to be spin up the second one must then be
spin down, and vice versa. While this is “spooky action at a distance”, you can not send
information faster than the speed of light with it.