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Unformatted text preview: 22. Lecture 22
22.1 Photons
As already discussed, the correct explanation of the photoelectric eﬀect is that electromagnetic waves are made of small quanta called photons. Each photon has energy and
momentum given in terms of the frequency and wavelength by:
Eph. = ω = hf, Pph. = Eph.
c = h
λ (22.1) That photons have momentum can be seen by shining light on a reﬂective surface and
observing that light exerts a pressure over the surface, namely transfers momentum to
it. The value of , the Planck constant is given
c = 197M eV f m (22.2) where c is the speed of light and as usual 1M eV = 106 eV , 1f m = 10−15 m. Notice that
has units of Energy × time as expected from the formula Eph. = ω . It turns out
that usual electromagnetic waves are constituted by huge number of photons and for
that reason the particle nature of light is not apparent. However, when interacting with
electrons, each photon interacts with an individual electron and the quantum nature
of light becomes manifest.
22.2 Hydrogen spectrum and Bohr atomic theory
Now we consider the interaction of light with the simplest atom, namely hydrogen,
which is made out of a proton and an electron orbiting around it. When a photon
hits the electron it can break it free from the atom in the hydrogen version of the
photoelectric eﬀect. However, if the energy is not enough the electron will jump to
another orbit and stay bounded to the proton. Similarly, once excited it can decay to
a lower orbit and emit a photon. According to classical mechanics the energy of the
emitted photon can take any value, the spectrum of energies is continuous. However,
very surprisingly at the time, it was observed that when heated up, hydrogen emits
photon of very well deﬁned frequencies (or wavelengths) as shown in ﬁg.120. The
same happens with absorption, the same frequencies which are emitted are also the
ones that are absorbed by the atom. In fact, experimentally it is seen that the energy
of the emitted or absorbed photons ﬁt the very simple formula:
1
1
Eph. = 13.6 eV
−
(22.3)
n2 n2
1
2 – 142 – where n1,2 are two positive integers such that n2 > n1 . As Bohr pointed out, the only
explanation is that the electron cannot be in any arbitrary orbit but only in orbits with
energy:
13.6eV
Eel. = −
, n = 1, 2 , 3 , . . .
(22.4)
n2
The negative sign means that the electron has less energy than a free electron, namely
it is bounded to the proton. When an electron jumps from one orbit to another it can
only emit or absorb photons of energies equal to the diﬀerence in energy between two of
these orbits thus explaining eq.(22.3). Although this explains the hydrogen spectrum it
is quite extraordinary. The ﬁrst surprising point is that the lowest energy is attained for
n = 1 called the ground state. Classically the electron would radiate all its energy and
get stuck to the proton in a state of inﬁnite negative energy. In quantum mechanics this
is not what happens, there is a minimum energy that is attained. An explanation for
this fact is given in the next subsection in terms of Heisenberg’s uncertainty principle.
Moreover, above the minimal energy only very speciﬁc energies are allowed as we will
explain in terms of the de Broglie proposal that particles behave also as waves. To
summarize, we are in a realm where Newtonian mechanics does not apply any more
and we have to discuss which principles allow us to understand the physical reality at
the atomic scale.
22.3 Uncertainty principle
One of the basic principles of quantum mechanics is the uncertainty principle proposed
by Heisenberg. In classical mechanics the state of a system is determined by giving the
position and momentum (or velocity) of each particle. In a quantum mechanical state,
however, the position and momentum of a particle are not simultaneously well deﬁned.
If the particle is localized at a point then the momentum is completely undetermined
and vice versa if the momentum is well deﬁned, the particle is completely unlocalized.
For that reason in quantum mechanics we talk about the probability distribution of
position and momentum. If the particle is localized in a region of size ∆x and the
momentum is in the range (p, p + ∆p) then the uncertainty principle establishes that
∆ x∆ p ≥
2 (22.5) We can make ∆x small at the expense of making ∆p large and vice versa.
Consider now the case of the hydrogen atom. The Coulomb potential is attractive
and tries to localize the electron as close to the proton as possible. However, if we
localize the electron very close to the proton, the momentum distribution is very spread.
p2
Since the kinetic energy is given by K = 2m that means that the average value of the – 143 – Figure 120: When light is emitted by hydrogen only certain wavelengths are present as
seen in this spectrum where hydrogen light is split using a grating or other similar device. kinetic energy will be large. This is the way in which the Heisenberg principle operates.
The potential energy tends to localize particles at the minimum of the potential. On
the other hand the kinetic energy prefers that the particle is spread. For example in
metals the conduction electrons are spread all over the metal and conduct electricity
whereas there are other electrons which are localized around the atoms and do not
contribute to the current. The more we want to localize a particle the stronger the
potential we need. For that reason to study the physics at very small scales large
energies per particle are required.
Going back to the hydrogen atom we can make a quantitative prediction if we write
the total energy as
1
e2
p2
e2
¯
E = mv 2 −
=
−
(22.6)
2
4π0 r
2m
r
where we used that the momentum is p = mv and also deﬁned
e2 =
¯ e2
= 1.44 M eV f m
4π0 (22.7) The last value is computed using the electron charge e = −1.6 × 10−19 C . It has the
correct units of energy × length. If we localize the electron in a region of size r, the – 144 – Figure 121: Bohr proposed that only certain discrete orbits are possible based on a quantization principle. momentum will be spread ∆p ∼
r and therefore we estimate:
p2 ∼ The total energy therefore is
2
e2
¯
E
−
= e2
¯
2
2mr
r 2
r2 2
1
−
2 r2
2me
¯
r where we deﬁned
r0 = 2
2me 2
¯ (22.8)
e2
¯
=
r0 2
r0 r0
−
r2
r (22.9) (22.10) The radius r0 has the value
r0 = 2
( c ) 2
1972 M eV 2 f m2
=
=
= 2.7 × 104 f m = 2.7 × 10−11 m (22.11)
2
2 e2
2me
¯
2mc ¯
1M eV 1.44 M eV f m – 145 – where we used that mc2 = 0.5M eV , the energy equivalent of the electron mass. We
see now more quantitatively what happens. If r is very small the kinetic energy grows,
1
1
if r is large then the potential energy grows. We plot the function f (x) = x2 − x where
x = rr0 in ﬁg.122 and see that it attains a minimum at x = rr0 = 2. If we replace r 2r0
in the previous computation of the energy we obtain
e2 1 1
¯
e2
¯
E
−
=−
= −13.6eV
(22.12)
r0 4 2
4 r0
in very good agreement with the experimental result in eq.(22.4) for n = 1, the ground
state. To be completely honest the full agreement is a coincidence. The uncertainty
principle only allows us to get an estimate and this should work well. Namely we expect
the energy to be of order of tens of electron volts and it is. That it comes exactly equal
is as we said a coincidence and does not always work that way.
In any case, summarizing, the uncertainty principle tells us that we need energy to
localize a particle therefore there is a compromise radius where the particle is localized
so that the Coulomb energy is low but not too localized that the kinetic energy will
grow. Given the mass of the electron and the strength of the interaction (given by the
charge) the radius is of order 10−10 m and the energy of order 10 eV .
This principle allows us to understand why there is a minimum energy but we still
do not have a principle that tells us that the higher energy states are quantized. We
now have to introduce the idea of wave mechanics. – 146 – Figure 122: The function f (x) =
with f (2) = −1/4. 1
x2 − 1
x is plotted and seen to have a minimum at x = 2 – 147 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue UniversityWest Lafayette.
 Fall '11
 NA
 Energy, Momentum, Photon

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