A Pedastrian Approach to Radiation-Matter Interaction-Part I
June 28, 2020
TL;DR
This blog introduces the formalism of quantum mechanics in Hilbert space.
Motivation
While working on my side project of ab initio polaron theory, I realize that none of my notes is explicitly related to the radiation-matter interaction (RMI). It’s a bit weird at this point that RMI never occur to me as an important thing to learn even though the polaron is a result of interactions between electrons and radiation-excited phonons in crystalline lattice. After a search for proper sources to learn RMI, I laid my eyes on Edward Harris’ A Pedestrian Approach to Quantum Field Theory. For a guy who just finished Dr. Klauber’s excellent Student Friendly Quantum Field Theory, I was kinda reluctant to start reading another introductory book on QFT with concerns of wasting too much time. But Edward’s book turned out to be a huge page-turner after I scanned through the first chapter. This blog is dedicated to reproduce some of the important conclusions in the first chapter of Edward’s book. All the derivation in this blog is prepared for those who had an introductory non-relativistic quantum mechanics class, but a brief review of axioms of QM is also given somewhere along the derivation.
Quantum Mechanics in Hilbert Space
Properties and definitions in Hilbert space
Hilbert space is a vector space spanned by a complete set of base vectors. The vector space is similar to the scalar space except that each coordiante in the space corresponds to a vector instead of a scalar. The vectors live in Hilbert space have the following properties:
- they follow commutative and associative rules,
- there is a null vector,
- and for an arbitrary vector
there is a so that
Except the vector properties the scalar product in Hilbert space is defined as
where
Other important properties of Hilbert space include “completeness” and “separable”:
- separable: for any vector
and an arbitrary positive value in Hilbert space there is a set of vectors that satisfy
- completeness: any Cauchy vector sequence that has the property of
defines a unique limit of
Enough for the properties, let us now define some other useful things in Hilbert space which will be used in our new formalism for quantum mechanics. First, a vector set
The set is complete if any vector
where
Operators in Hilbert space
Mathematically the operators in Hilbert space are mappings to the space itself. We use hatted symbols (e.g.
The properties of Hermitian adjoint include:
An adjoint is said to be Hermitian when
In physics Hermitian operators usually represent real-word detectable observables (e.g. energy, momentum, and total number of particles).
Eigenvalues and eigenvectors
For each operator
with
- the eigenvalues are real numbers,
- if
and are two eigenvectors of we have - The eigenvectors of a bounded Hermitian operator after normalization form a denumerably complete orthonormal system. Consequently, its eigenvalues form a discrete set (discrete spectrum).
Based on the property #3 we now can represent arbitrary vector
If we assume that
and
with
where
We use a matrix instead to present an operator
Note that the matrix
and
Further we can prove the trace of an operator is independent of representation using base transition, i.e.
where the last equivalence is due to the invariance of matrix trace.
So far we only considered the discrete base sets in our formalism but we must also consider continuous base sets. A good example of such base set is the spatial coordinate set
with
In the real-world physical problems, the operator functions are also defined as functions that take operators as their arguments. By means of eigenvalues the operator function
We conclude this section by defining the inverse of an operator as
Unitary transformations
An operator
for states and
for operators. Notice that the unitary transformations conserve:
- the eigenvalues of operators,i.e.
- and the inner products of states,i.e.
Direct product space
Due to the variety of particle states (e.g. spin-up and spin-down states of electrons) sometimes we need to expand our Hilbert space through the direct product method. Here we use a nucleon as an example. A nucleon can be a proton or a neutron. A proton is positively charged but a neutron is neutral. They can be treated as two states of the same particle in the charge space (a specific example of Hilbert space):
with
then a spin-up proton state can be represented by a 4-vector as
Thus, whenever one wish to accommondate more particle attributes in a physical system, the Hilbert space must be expanded.
Axioms of quantum mechanics
The axioms used in QM are postulations that assume the correspondences between mathematical objects and physical quantities. This section simply serves as a reminder of basic axioms that we will use later. For comprehensive explanations you might need to consult introductory QM textbooks.
Axiom I: The result of any measurement of an observable can only be one of the eigenvalues of the corresponding operator, and the system state is represented by the corresponding eigenvector.
Axiom II: If a system is known to be at a state of
, then the probability of a measurement of yields is . If measures a continuous quantity then we have the probability of finding the measurement results within a range of is .
Axiom III: The operators
and that correspond classical dynamic variables satisfy the following commutation rules:
where the curly bracket is the Poisson bracket of dynamic variables
and :
and
and are classical momenta and coordinates of the particles in a system. One could easily find that
We shall now derive a consequence of Axiom III before we proceed. If we define the expectation value of observable
Substitute
Axiom IV: let
and be the system state at time and repectively, then they are related to each other through the unitary transformation:
Using Axiom IV we can recast QM in the Heisenberg picture (before this point we derived everything in Schrodinger’s Picture,SP). To accomplish this we let
and the state vector in Heisenberg picture is derived from SP as
The consequence of Eq.(18) is obviously that the system states are time-invariant in Heisenberg picture(HP). The operators in HP is then
and it is now time-dependent. We can get the Heisenberg equation of motion by time-differentiating (19):
It can be seen immediately from (20) that any operator commutates with Hamiltonian is a constant of the motion in HP.
Simultaneous eigenstates
The concept of simultaneous eigenstate arises due to the commutative operators. For operators
Thus
If the state
then we can only conclude that
A brief summary before we proceed. We have at this point laid out the general formalism of QM using mathematical objects in Hilbert space. We start with basic properties and definitions of eigenvectors and operators to prepare our review of axioms of QM. The axiom IV, in particular, help us to recast Schrodinger picture into Heisenberg picture where operators become time-dependent and eigenstates are time-invariant. In the following section we use this Hilbert-space formalism to solve some simple problems in QM.
Solved QM Problems
Spin particle in magnetic field
Spin is an intrinsic form of angular momentum carried by elementary particles. In this section we use Hilbert-space formalism to derive the expression of general spin operator around an arbitrary axis. For a
where
are called Pauli metrices. Now the task is to find the Hamiltonian for a particle in magnetic field. If we only focus on magnetic momentum energy, and let the magnetic field strength be
where
Note that if we relax the direction of
and use Eq.(21), we can obtain a more general representation for angular momentum operator about an axis in the direction of
which has its two eigenvectors being
and
As in the case of
Next we discuss the dynamics of our system by first defining a particle state as
Use (27) and (23) in time-dependent Schrodinger’s equation, we obtain the equation of motion for the dynamic state,
from which
At
then (29) becomes
we can get
from by setting and .
By axiom II of QM the probability of finding the up-spin at x-direction at time
Propagator of free particle
In this section we introduce the concept of propagator, the probability amplitude of a free particle trasporting between two coordinates in spacetime. The particle propagator is the building block of quantum perturbation theory, the most important theoretical framework that allows people doing ab initio calculations of fermion-boson interactions. We shall appreciate the convenience that our Hilbert-space formalism brings to our derivation of particle propagator.
For free particles we only care about their dynamic variables:coordinates
In free space a particle can occupy any coordinate or momentum, so the normalization of eigenvectors is:
From Axiom III of QM we have the commutation rule for
which determines the matrix element of
We also know that
and the famous property of Dirac
so we obtain from (32)
By similar steps we can also prove that
By taking a matrix product, we can also find that
In general we have
With the coordinate representation of momentum operator matrix, we now turn our attention to the eigenvalue problem of
Since
which indicates that
Thus, we can choose
The results in (34),(35) and (36) can be easily extended to 3-dimensional space and Eq.(38) in 3D space becomes
We shall now see that the particle propagator arises naturally when we consider time-dependent eigenvectors of the Hamiltonian. For a free particle the Hamiltonian is just the particle’s kinetic energy,
from the results in (36). Using Axiom IV of QM we find that the eigenstate of free particle at time
The eigenvector in coordinate
where
and
After the substitution of (38) in (42), we find
This integration can be carried out with the result
Fermi’s golden rule
We should now be comfortable with the Hilbert-space formalism. Using the first-order approximation in the quantum perturbation theory, we procced to derive the Fermi’s Golden rule in this section. Since mordern research of quantum field theory mostly stems from perturbative theory, getting some exposure to it could really help us to appreciate the motives behind modern studies that use quantum field theory to understand the properties of condensed matters.
In a perturbed system, we are often interested to solve the following equation:
where we have for non-pertubative Hamiltonian
If we let
and apply a
If we integrate (47) from time
Eq.(48) is too bulky to handle, so we need to make some approximations to make it easier the calculation of the probability of finding our perturbed system at state
The first approximation is that the system is at state
Then the probability of
where
Now, regarded as a function of
the function becomes very sharply peaked about when becomes large. Most of the area under a graph of the function is under the central peak. Also Therefore, we can say that
If we use (51) in (50) memtioned above, we have
Eq.(52) may be interpreted as the transition probability per unit time for a transition from an initial state
Conclusion
There are some typos in Edward’s book and I’m glad I worked all of them out. In the part II I’ll develop a quantum theory of free electromagnetic field.