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Basic Quantum Theory
Miao Liang
Abstract—Quantum mechanics is the science of the
very small. It explains the behavior of matter and its
interactions with energy on the scale of atoms and
subatomic particles. By contrast, classical physics only
explains matter and energy on a scale familiar to human
experience, including the behavior of astronomical bodies
such as the Moon. Classical physics is still used in much
of modern science and technology. However, towards the
end of the 19th century, scientists discovered phenomena
in both the large (macro) and the small (micro) worlds
that classical physics could not explain.The desire to
resolve inconsistencies between observed phenomena and
classical theory led to two major revolutions in physics
that created a shift in the original scientific paradigm:
the theory of relativity and the development of quantum
mechanics.
I. ST ATE
As in classical mechanics, the state of a particle at
any given time is specified by two variables position an
momentum. If we know these quantities, we can get all
the information of the system we want to measure. For
example, energy is just some function of position and
momentum. We can calculate it if we know the position
and momentum. However, this is impossible in quantum
mechanics. In atom scale, according to the uncertainty
principle, we can’t specify the position and momentum
at the same time. How can can we specify the state of
a system? By the Copenhagen interpretation, a state of a
particle is completely specified by a complex wave function
|ψ〉and it is described to be a probability amplitude, whose
square modulus gives the probability density of finding the
particle at point x at time t.
A mixture of quantum states is also a quantum state.
Quantum states that cannot be writtten as a mixtrue of
other states are called pure quantum states, all other
states are called mixed quantum state.
Mathematically, a pure quantum state can be represent-
ed by a ray in a Hilbert space over the complex numbers.
The ray is a set of nonzero vecctor differing by just a
complex factor; any of them can be chosen as a state vector
to represent the ray and thus the state. A unit vector is
usually picked, but its phase factor can be chosen freely
anyway. Nevertheless, such factors are important when
state vectors are added together to form a superposition.
A mixed quantum state corresponds to a probabilistic
mixture of pure state; however, different distribution of
pure states can generate equavalent mixed states. Mixed
states are describe by density matrices. A pure state can
also be recast as a density matrix; in this way, pure statee
can be represented as a subset of more general mixed
states.
II. BRA -KE T NOT ATI ON
Consider a microscopic system composed of particles or
bodies, with specific properties(mass, moment of inerti-
a,etc.) interacting according to specific laws of force. There
will be various possible motions of the particles, or bodies,
consistant with these laws of force. Let us term each such
motion a state of the system. According to the principle
of superposition of states. Thus, states must be related
to mathematical quantities of a kind that can be added
together to give other quantities of the same kind. The
most obvious examples of such quantities are vector.[1]
Consider a particular microscopic system in a particular
state, which we label A. We can represent this state as a
particular vector in some vector space. Dirac name it ket
space. The state vector is often written as |A〉. Suppose
state A is the superposition of two different states, B and
C. This interrelation is represented in ket space by writing
|A〉=|B〉 + |C〉.
Note that we cannot form a new state by superposing a
state with itself. This implies that the ket vector c1|A〉 +
c2|A〉 = (c1+c2)|A〉.
The absence of a state is represented by the null vector
|0〉in ket space. The null vector has the fairly obvious
property that. |A〉 + |0〉=|A〉
Now we know that any plane polarized state of a
photon can be represented as a linear superposition of two
orthogonal polarization states in wichi the weights are real
numbers. Suppose we want to construct a circular polar-
ized photon state. Remind that a circularly polarized wave
is a superposition of two waves of equal amplitude, plane
polarized in the X-direction and a photon polarized in the
y-direction, with equal weights given to the two states. To
descrive a circularly polarized photon, we can use complex
numbers to simultaneously represent the weighting and
relative phase phase in a linear superposition. Suppose
state B represents the photon polarizing in the x-direction,
state C represents the photon polarizing in the y-direction,
written as |B〉 + i|C〉. Then we can conclude: a general
elliptically polarized photon is represeted by c1|B〉 + c2|C〉
where c1and c2are complex numbers. It implies that a ket
space must be a complex vector space.
Any ket vector that is expressible linearly in terms of
cortain orthers is said to be dependent on them. Likewise,
a set of ket vecotrs are termed independent if none of them
are expressible linearly interms of them. The dimension of
a ket space is equivalent to the number of independent ket
vectors it contains. If there are N independent states then
the possible states of the system are represtned as an N-
dimensional ket space. Remind some microscopic sysyems
