Part 4. Quantum mechanics
PHYS 2022, Yi Wang, Department of Physics, HKUST
This part is an introduction of quantum mechanics. We start from
This part is essential.
the key observations that why the world has to be quantized. Af-
terwards we introduce the wave-particle duality, measurements and
uncertainty principle, the Schrödinger equation, square potential, op-
erators and the harmonic oscillator. We end with an overview on the
interpretations of quantum mechanics.
1
Clouds over the classical world
“It was the best of times, it was the worst of times, it was the age of
wisdom, it was the age of foolishness, it was the epoch of belief, it was
the epoch of incredulity, it was the season of Light, it was the season
of Darkness, it was the spring of hope, it was the winter of despair, we
had everything before us, we had nothing before us, we were all going
direct to Heaven, we were all going direct the other way
–
in short,
the period was so far like the present period, that some of its noisiest
authorities insisted on its being received, for good or for evil, in the
superlative degree of comparison only.” – Charles Dickens, “A Tale of
Two Cities”
I wanted to cite the first sentence of the above, but I cannot help to
cite a whole paragraph. This is how in the 1900s the physicists felt
– All the dream, love and faith of their beautiful, harmonic and pre-
dictable classical world had collapsed. Nobody would have imagined
how the real world behaves if the new experimental discoveries were
not thrown onto their faces.
1.1
The UV catastrophe
The equipartition rule
To start to understand how bad things can go, let us first review
Boltamann’s equipartition rule.
For the theory of ideal gas, each
The situation is similar to what if
I send we-chat random red pocket
to the class. Say I prepare $100 and
there are 100 of you. Then each of
you get statistically $1.
degree of freedom (dof) gets on average an energy of
1
2
k
B
T
. That
is to say, in a thermal equilibrium, everybody get statistically equal
share of energy. This rule is expected in general because lower energy
dof tends to get kicked by the higher energy dof and gets statistically
equal share of energy.
For example, for monatomic ideal gas, the internal energy is
3
2
Nk
B
T
,
because each of the
N
particles has 3 directions to move. There are
thus
3
N
dof.