Lyman series (invisible)
The atom exists in the excited state
for a short time before making a
transition to a lower energy state with
emission of a photon. At ordinary
temperatures, almost all hydrogen
atoms exist in n = 1 state. Absorption
therefore gives the Lyman series.
Balmer series (visible)
When sunlight passes through the
atmosphere, hydrogen atoms in
water vapor absorb wavelengths of
the Balmer series giving dark lines in
the absorption spectrum.
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• 
de Broglie matter waves led ultimately to a wave equation
describing particle motion. First, let’s review the physics of waves.
The “displacement” of a wave is
• 
This is a solution to the wave equation
• 
Define the wave number k and the angular frequency ω as:
and
• 
The wave function is now
Ψ(x, t) = A sin (kx - wt)
2
• 
• 
When two or more waves traverse the same region, they act
independently of each other
Combining two waves yields
⎡ Δk
Δω ⎤
Ψ(x,t) = Ψ1 (x,t) + Ψ2 (x,t) = 2Acos⎢ x −
t cos(k av x − ω av t)
⎣ 2
2 ⎥⎦
The combined wave oscillates within an envelope that denotes the
maximum displacement of the combined waves.
€•  By combining many waves with different amplitudes and
frequencies, a pulse, or wave packet, can be formed which moves
with a group velocity
• 
ugr = Δω / Δk.
3
1
1 2π
Δx = λenv =
⇒ ΔkΔx = 2π
2
2 Δk /2
1
1 2π
Δt = Tenv =
⇒ ΔωΔt = 2π
2
2 Δω /2
€
4
• 
• 
• 
• 
Claus Jönsson of Tübingen,
succeeded in 1961 in showing
double-slit interference effects for
electrons by constructing very
narrow slits and using relatively
large distances between the slits
and the observation screen
This experiment demonstrated that
precisely the same behavior occurs
for both light (waves) and electrons
(particles)
To determine which slit the electron went through: We set up a light
shining on the double slit and use a powerful microscope to look at the
region. After the electron passes through one of the slits, light bounces
off the electron; we observe the reflected light, so we know which slit
the electron came through.
Use a subscript “ph” to denote variables for light (photon). Therefore the
momentum of the photon is
pph =
• 
The momentum of the electrons will be on the order of
pel =
• 
h > h
d
λph
h ~
λel
.
h
d
The difficulty is that the momentum of the photons used to determine
which slit the electron went through is sufficiently great to strongly
modify the momentum of the electron itself, thus changing the direction
of the electron! The attempt to identify which slit the electron is passing
through will in itself destroy the interference pattern.
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• 
It is impossible to measure simultaneously, with no uncertainty, the
precise values of k and x for the same particle. The wave number k
may be rewritten as
k=
• 
2π 2π
2π p
=
=p
=
ћ
h
λ h/p
For the case of a Gaussian wave packet we have
ΔkΔx =
• 
Δp
ћ
1
2
Thus for a single particle we have Heisenberg’s uncertainty principle
Δpx Δx ≥
• 
Δx =
ћ
2
The Gaussian packet turns out to give the minimum product of
uncertainties (any other shape gives a larger product)
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