Physics 228
Today:
Bohr Atom
Lasers
Schrödinger Equation
Reviw: Bohr Model of Hydrogen Atom
Since electrons behave as waves, Niels Bohr reasoned that in the
atom they might form standing waves, or resonances, just like
standing waves on a string.
Bohr proposed that only those electron orbits are allowed where
an integer number of wavelengths fits into a circular orbit:
Bohr Model Takeaway
The Bohr atom is based on E&M plus the assumption that circular
orbits have an integral number of wavelengths. It leads to:
• Quantized radii
• Quantized angular momenta
• Quantized energies
The energy levels are predicted correctly, and the concept of
angular momentum quantization is also correct.
However, the model predicts that the n’th energy level always
has an angular momentum of nħ, which is not confirmed
experimentally.
Thus, the Bohr model allows us to start understanding
quantization, but it is not really right. Full understanding requires
the theory of quantum mechanics, which we start with today.
Light Emission and Absorption
absorption
emission
• The probability of a photon being absorbed must be
proportional to the number of photons in the initial state (in
a given mode). Let’s call this number n+1. The final state then
contains n photons.
• For the time-reversed process (photon emission), the initial
state contains n photons, and the final state contains n+1
photons.
• The rate of the time-reversed process should equal that of
the forward process (“detailed balance”).
• Detailed balance dictates that the probability of photon
emission is thus also proportional to n+1.
Stimulated and Spontaneous Emission
• Photon emission rate: P = const. x (n+1)
• We see that the photon emission rate consists of two terms: One
proportional to the number of photons in the initial state, the
other independent of that number.
• We call the first term “stimulated emission”, as it may be
considered stimulated by the initial state photons.
• The second term is called “spontaneous emission” since it occurs
even if there are no photons present initially.
(here n = 1)
Lasers
• LASER = Light Amplification by Stimulated Emission of
Radiation.
• Stimulalted emission rate is proportional to the number of
photons already present: Number of photons grows
exponentially, like an avalanche:
• To get a laser to work, we need more atoms in a higherenergy state than in a lower-energy state (“population
inversion”).
• This cannot happen in thermal equilibrium.
• How do we produce population inversion?
Population Inversion
• Consider a system with 4 electronic levels.
• We first excite the system (“pump”) by passing current or light
through it.
• Consider one of the excited states having a very long lifetime
compared to the typical excitation lifetime.
• Such a long-lived excited state is called "metastable".
• Now we can have a population inversion of this state (E2 above) with
respect to the lower excited state E1.
iClicker: Two-State System
Can we create population inversion in a two-level system by pumping it
with photons at the transition energy?
a) Yes: pump it hard!
b) Yes: if the lifetime is long enough.
c) No: When the two levels have equal populations, the absorption rate
equals the stimulated emission rate (i.e., they cancel). On top of that
we have spontaneous emission. Thus the system loses energy and can
never be excited past the equal-population point.
d) Dumb question: All systems have more than two levels.
e) I want 10 points deducted from my grade.
A Gas Laser
“Pumping” can be done either electrically or optically.
Wave Equation for Light in 1 Dimension
From Maxwell’s equations, we can derive a wave equation for the
electric field:
A light wave can be represented as E = E0 cos(kx - ωt).
Inserting into wave equation we obtain: -k2 E = (1/c2) (-ω2) E
 k2 = ω2/c2
 c = ω/k.
What is the Wave Equation for Particles
(Matter Waves)?
The wave equation for particles has to be different, since for a nonrelativistic particle we have kinetic energy E = p2/2m (not E = pc as
for light).
We require the de Broglie relations to apply: E = ħω and p = ħk.
Then E = p2/2m ➮ ħω = ħ2k2/2m.
(vs. E = ħω = ħck for light.)
The factor of k2 should look familiar - recall the 2nd derivative with
respect to x (previous slide, wave equation left-hand side) for the
wave equation for light gave: -k2 E(x,t).
The factor of ω is also familiar - that comes from the 1st derivative
of the right hand side of the wave equation.
Wave Equation for Particles
For the particle amplitude let’s use the symbol Ψ.
So we should have an equation that relates the second x-derivative to
the first time derivative of Ψ:
The -ħ2/2m was put on the left to get the kinetic energy, while
the C was put on the right since we are not certain what we will
need there.
Let's assume Ψ(x,t) = A cos(kx-ωt) as before.
Then the L.H.S. gives (ħ2k2/2m) Ψ(x,t) = EΨ(x,t), the r.h.s. of ħω =
ħ2k2/2m.
The R.H.S. gives -ωCA sin(kx-ωt). This has the ω we need for the
l.h.s. of ħω = ħ2k2/2m, but is cannot work with cos(kx-ωt) on one
side and sin(kx-ωt) on the other. ☹ We need something better...
Complex Numbers
• A complex number has the
form z = x + iy, where x is
the real part, and y is the
imaginary part.
• A complex number is
represented by a point in the
complex plane.
• By expanding exp(z), sin(z),
and cos(z) functions into
Taylor series, we can show
that
imaginary
axis
iy
θ
x
real axis
𝑧 = 𝑥 + 𝑖𝑦
𝑧 = 𝑥2 + 𝑦2
𝑧 = 𝑧 cos 𝜃 + 𝑖 sin 𝜃
𝑖𝜃
= 𝑧 𝑒
iClicker
What is the modulus of 𝑒 𝑖𝜑 ?
a.) 𝑒 𝑖𝜑 = 1
b.) 𝑒 𝑖𝜑 depends on the value of 𝜑
c.) 𝑒 𝑖𝜑 = 2
d.) 𝑒 𝑖𝜑 = 0
e.) The modulus is not defined for a complex
argument.
Schrödinger Equation for a Free Particle
Let's now apply the wave equation to the complex wave function:
Ψ(x,t) = A ei(kx-ωt).
(The real part oscillates like a cosine, the imaginary like a sine.)
The second derivative on the LHS gives (-ħ2/2m) (-k2) Ψ(x,t) =
(ħ2k2/2m) Ψ(x,t) = E Ψ(x,t).
The derivative on the RHS gives (-ωi) C Ψ(x,t).
We see that the complex wave function is a solution of the wave
equation (with the proper relationship between momentum and
energy ħ2k2/2m = ħω) if we choose C = iħ:
Meaning of the Wave Function
The wave function Ψ(x,t) has the following
interpretation:
The probability density (probability per unit
volume) of finding the particle at point x at time
t is given by the absolute square of Ψ(x,t):
dP/dV = |Ψ|2