QUANTUM MECHANICS

1. The word "quantum" means packet or bundle. We have already encountered the quantum
of light - called photon - in an earlier lecture. Quanta (plural of quantum) are discrete
steps. Walking up a flight of stairs, you can increase your height one step at a time and not,
for example, by 0.371 steps. In other words your height above ground (and potential
energy can take discrete values only).
2. Quantum Mechanics is the true physics of
the microscopic world. To get an idea of
the sizes in that world, let us start from the
atom which is normally considered to be
a very small object. But, as you can see,
the atomic nucleus is 100,000 times smaller
than the atom. The neutron and proton are
yet another 10 times smaller than the nucleus.
We know that nuclei are made of quarks, but
as yet we do know if the quarks have a size
or if they are just point-like particles.
3. It is impossible to cover quantum mechanics in a few lectures, much less in this single
lecture. But here are some main ideas:
a) Classical (Newtonian) Mechanics is extremely good for dealing with large objects (a
grain of salt is to be considered large). But on the atomic level, it fails.The reason for
failure is the uncertainty principle - the position and momenta of a particle cannot be
determined simultaneously (this is just one example; the uncertainty principle is actually
more general). Quantum Mechanics properly describes the microscopic - as well as
macroscopic - world and has always been found to hold if applied correctly.
b) Atoms or molecules can only exist in certain energy states. These are also called
"allowed levels" or quantum states. Each state is described by certain "quantum
numbers" that give information about that state's energy, momentum, etc.
c) Atoms or molecules emit or absorb energy when they change their energy state. The
amount of energy released or absorbed equals the difference of energies between the
two quantum states.
d) Quantum Mechanics always deals with probabilities. So, for example, in considering
the outcome of two particles colliding with each other, we calculate probabilities to
scatter in a certain direction, etc.
u(ν )
ν
4. What brought about the Quantum Revolution? By the end of the 19th century a number
of serious discrepancies had been found between experimental results and classical theory.
The most serious ones were:
A) The blackbody radiation law
B) The photo-electric effect
C) The stability of the atom, and puzzles of atomic spectra
In the following, we shall briefly consider these discrepancies and the manner in which
quantum mechanics resolved them.
A)Classical physics gives the wrong behaviour for radiation
emitted from a hot body. Although in this lecture it is not
possible to do the classical calculation, it is not difficult to
sh
3
ow that the electromagnetic energy ( ) radiated at
frequency increases as (see graph). So the energy
radiated over all frequencies is infinite. This is clearly
wrong. The correct calcu
u ν
ν ν
lation was done by Max Planck.
Planck's result is shown above, and it leads to the sensible result that u( ) goes to zero at
large . He assumed that radiation of a given frequency could only be emitted and
absorbed in
ν
ν ν
quanta of energy . If the electromagnetic field is thought of as harmonic
oscillators, Planck assumed that the total energy of this large number of oscillators is made
of finite energy elem
ε = hν
ents . With this assumption, he came up with a formula that
fitted well with the data. But he called his theory "an act of desperation" because he did
not understand the deeper reasons.
B
hν
) I have already discussed the photoelectric effect in the previous lecture. Briefly,
Einstein (1905) postulated a quantum of light called photon, which had particle properties
like energy and momentum. The photon is responsible for knocking electrons out of the
metal - but only if it has enough energy.
C) Classical physics cannot explain the fact that atoms are
stable. An accelerating charge always radiates energy if
classical electromagnetism is correct. So why does the
hydrogen atom not collapse? In 1921 Niels Bohr, a great
Danish physicist, made the following hypothesis: if an
electron moves around a nucleus so that its angular
momentum is 􀀽,2􀀽,3􀀽,⋅ ⋅ ⋅ then it will not radiate energy.
In the next lecture we explore the consequences of this
hypothesis. Bohr's hypothesis called for the quantization
a
m
of angular momentum. If the electron is regarded as a De Broglie wave, then there must
be an integer number of waves that go around the centre. Of course, this is not proper
quantum mechanics and cannot be taken too seriously, but it definitely was a major
step in the ultimate development of the subject. Even if one's understanding was imperfect,
it was now possible to understand why atoms had certain energies only, and why the light
radiated by atoms was of discrete frequencies only. In contrast, classical physics predicted
that atoms could radiate at any frequency.
5. One consequence of quantum mechanics is that it explains through the uncertainty principle
the stability of the atom. But before talking of that, let us consider a particle moving
between two walls. Each time it hits a wall, a force pushes it in the opposite direction.
There is no friction, so classically the particle just keeps
moving forever between the two potential walls. In QM
the uncertainty of the particle's position is and so,
from
x a
x p
Δ =
Δ Δ ≥
2 2
2
/ 2, we have . From this we learn
2
that the kinetic energy ( ) . This is telling us
2 8
that as we squeeze the particle into a tighter and tighter
space, the kinetic energy g
pa
p
m ma
Δ ≈
Δ
≈
􀀽 􀀽
􀀽
oes up and up!
6. The story repeats for a harmonic oscillator. So imagine that a mass moves in a potential
of the type shown below, and that its frequency of oscillation is . Classically, the lowest
energy
ω
would be that in which the mass is at rest (no kinetic energy) and it is at the
position where the potential is minimum. But this means that the body has both a
well-defined momentum and position. This is forbidden by the Heisenberg uncertainty
principle. A proper quantum mechanical calculation shows that the minimum energy is
actually 1 . Th
2
􀀽ω is is called the zero-point energy and comes about because it is not
possible for the mass to be a rest. The oscillator's other energy states have energies
1 ,3 ,5 ,7 , We say that
2 2 2 2
􀀽ω 􀀽ω 􀀽ω 􀀽ω ⋅ ⋅ ⋅ the energies are quantized. This is completely
impossible to understand from classical mechanics where we know that we can excite
an oscillator to have any energy we want. Quantized energy levels are indeed what we
observe in so many physical systems: atoms, rotating or vibrating molecules, nuclei, etc.
7. We can understand from both the previous examples why the hydrogen atom does not
collapse even if the electron does not have any orbital angular momentum around the
proton. The uncertainty principle essentially forces the electron to stay away from the
proton - if it tried to get too close, the kinetic energy would rise enormously because
ΔxΔp≥ 􀀽 / 2 says that if Δx becomes small then must become big to compensate.
8. Quantum mechanics predicts what is called "tunneling" of particles through a potential
barrier. Again, we shall use the uncertainty principle but this time
Δp
the energy-time one.
Imagine a particle that moves in a potential of the shape shown below. Suppose it does
not have enough energy to go over the peak and on to the other side. In that case, we
know from our experience that it will just keep oscillating - moving first towards the hill
and then down again, etc. But quantum mechanically, it can "steal" energy for a time
and thi
E
t
Δ
Δ s may be enough to surmount the hill. Of course, the particle must respect
/ 2 so the time is small if it needs a large amount of energy to cross over. Again,
I have given only a rough argu
ΔEΔt ≥ 􀀽
ment here, but in quantum mechanics we can do proper
calculations to find tunneling probabilities.
9. Without tunneling our sun would go cold. As you may know, it is powered by hydrogen
fusion. The protons must somehow overcome electrostatic repulsion to get close enough
so that they can feel the nuclear force and fuse with each other. But the thermal energy
at the core of the sun is not high enough. It is only because the tunneling effect allows
protons to sometimes get close enough that fusion happens.
10. Quantum mechanics is all about probabilities.What is probability? Probability is a
measure of the likelihood that an event will occur. Probability values are assigned on a
scale of zero (the event can never occur) to one (the event definitely occurs). More
precisely, suppose we do an experiment (like rolling dice or flipping a coin) times
where is large. Then if a certain outcome occurs n times, the probability is / .
N
N P= n N
Energy
N
S
↑
↓
11. The simplest system for discussing quantum mechanics is one that has only two states.
Let us call these two states "up" and "down" states, and . They could denote
an electron with s
↑ ↓
pin up/down, or a switch which is up/down, or an atom which can be
only in one of two energy states, etc. Things like were called kets by their inventor,
Paul Dirac.
For definiteness
↑
let us take the electron example. If the state of the electron is known to
be 2 1 , then the probability of finding the electron with spin up is
3 3
P( ) 2 , and with spin down is P( )
3
Ψ = ↑ + ↓
↑ = ↓ = 1 2
2 2
1 2
1. More generally: c denotes
3
an electron with P( ) c and P( ) . This means that if we look at look at
a large number of electrons all of which are in state , then the nu
c
c
N
Ψ = ↑ + ↓
↑ = ↓ =
Ψ
2 2
1 2 1 2
1 2
mber with spin up
is and with spin down is . We sometimes call c a
, and c and .
N c N c c quantum
state c quantum amplitudes
Ψ = ↑ + ↓
12. The Stern-Gerlach experiment illustrated here
shows an electron beam entering a magnetic
field. The electrons can be pointing either up
or down relative to any chosen axis. The field
forces them to choose one of the two states.
That the beam splits into only two parts shows
that the electron has only two states. Other
particle beams might split into 3,4, ⋅ ⋅ ⋅
1 2
1 2 1 2
13. If and are the amplitudes of the two possibilities for a particular event to occur,
then the amplitude for the total event is . Here and are complex numbers
in general.
a a
A=a +a a a
2 2
1 2
1 2
2 * * *
1 2 1 1 2 2 1 2
But the probability for the event to occur is given by . In
daily experience we add probabilities, but in quantum mechanics we add
amplitudes:
P A a a
P P P
P a a aa aa aa
= = +
= +
= + = + + * * *
2 1 1 2 1 2 2 1
* *
1 2 2 1
. The cross
terms are called interference terms. They are familiar to us from the lecture
on light where we add amplitudes first, and then square the sum t
a a P P a a a a
a a a a
+ = + + +
+
2 2
1 2
o find the intensity. Of
course, if we add all possible outcomes then we will get 1. So, for example, in the electron
case P(↑) +P(↓) = c + c = 1. Note that amplitudes can be complex but probabilities
are always real.
14. Let us return to the double slit experiment discussed
earlier. Here the amplitude for an electron wave
coming from one slit interferes with the amplitude
for an electron wave coming from the other slit.
This is what causes a pattern to emerge in which
electron are completely absent in certain places
(destructive interference) and are present in large
numbers where there is constructive interference.
So what we must deal with are matter waves. But
how to treat this mathematically?
15. The above brings us to the concept of a "wave function". In 1926 Schrödinger proposed
a quantity that would describe electron waves (or, more generally, matter waves).
• The wavefunctio
2
n ( , ) of a particle is the amplitude to be at position at time .
The probability of finding the particle at position between and (at time ) is
( , ) . Since the parti
x t x t
x x xdx t
x t dx
Ψ
• +
Ψ
+
2
-
cle has to be somewhere, if we add up all possibilities then
we must get one, i.e. ( , ) 1.
( , ) is determined by solving the "Schrodinger equation" which, unfortunately, I
x t dx
x t
∞
∞
Ψ =
• Ψ
∫
shall not be able to discuss here. This is one of the most important equations of
physics. If it is solved for the atom then it tells you all that is possible to know:
energies, the probability of finding an electron here or there, the momenta with which
they move, etc. Of course, one usually cannot solve this equation in complicated
situations (like a large molecule, for example) and this is what makes the subject both
difficult and interesting.
16. For an electron moving around a nucleus, one can
easily solve the Schrodinger equation an
2
d thus find
the wavefunction ( , ). From this we compute
( , ) , which is large where the electron is more
likely to be found. In this picture, the probability of
finding the el
x t
x t
Ψ
Ψ
ectron inside the first circle is 32%,
between the second and first is 44%, etc
1m
1m