Quantization of the electromagnetic field
Encyclopedia
After quantization of the electromagnetic field, the EM (electromagnetic) field consists of discrete energy parcels, photon
s. Photons are massless particles of definite energy
, definite momentum
, and definite spin
.
In order to explain the photoelectric effect
, Einstein assumed heuristically in 1905 that an electromagnetic field consists of parcels of energy hν, where h is Planck's constant. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabrics of the new quantum mechanics
and to describe the interaction of photons with matter. He applied a technique which is now generally called second quantization, although this term is somewhat of a misnomer for EM fields, because they are, after all, solutions of the classical Maxwell equations. In Dirac's theory the fields are quantized for the first time and it is also the first time that Planck's constant enters the expressions. In its original work, Dirac took the phases of the different EM modes (Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as operator
s and postulated commutation relations between them). At present it is more common to quantize the Fourier components of the vector potential
. This is what will be done below.
A quantum mechanical photon state |k,μ⟩ belonging to mode (k,μ) will be introduced. It will be shown that it has the following properties
These equations say respectively: a photon has zero rest mass; the photon energy is hν=hc|k| (k is the wave vector
, c is speed of light); its electromagnetic momentum is ℏk [ℏ=h/(2π)]; the polarization μ=±1 is the eigenvalue of the z-component of the photon spin.
s and (anti)commutation relations between these new operators are imposed, commutation relations for boson
s and anticommutation relations
for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function.
In the case of EM fields the required expansion of the field is the Fourier expansion.
As the term suggests, an EM field consists of two vector fields, an electric field
E(r,t) and a magnetic field
B(r,t). Both are time-dependent vector field
s that in vacuum depend on a third vector field A(r,t) (the vector potential), under the Coulomb gauge condition, through
here ∇×A is the curl
of A.
The Fourier expansion of the vector potential enclosed in a finite cubic box of volume V = L3 is (bar on top indicates complex conjugate):
where the wave vector k gives the propagation direction of the corresponding Fourier component (a polarized monochromatic wave) of A(r,t); the length of the wave vector is |k| = 2πν/c = ω/c, with ν the frequency of the mode. The components of the vector k have discrete values (a consequence of the boundary condition that A has the same value on opposite walls of the box),
The two unit vectors e(μ) ("polarization vectors") are perpendicular to k. They are related to the orthonormal Cartesian vectors ex and ey through a unitary transformation,
The k-th Fourier component of A is a vector perpendicular to k and hence is a linear combination of e(1) and e(−1). The superscript μ indicates a component along e(μ). The Coulomb gauge is imposed,
which makes A into a transverse field
.
Clearly, the (discrete infinite) set of Fourier coefficients
and 
are variables defining the vector potential. In the following they will be promoted to operators.
of a particle by the rule
.
Note that Planck's constant is introduced here and that the time-dependence of the classical expression is not taken over in the quantum mechanical operator (this is true in the so-called Schrödinger picture
).
For the EM field we do something similar. The quantity ε0 is the electric constant
, which appears here because of the use of electromagnetic SI
units. The quantization rules are:
subject to the boson commutation relations
The square brackets indicate a commutator, defined by
for any two quantum mechanical operators A and B. The introduction of Planck's constant is essential in the transition from a classical to a quantum theory. The factor (2ωVε0)−½ is introduced to give the Hamiltonian (energy operator) a simple form, see below.
The quantized fields (operator fields) are the following
where ω = c |k| = ck.
Substitution of the field operators into the classical Hamiltonian gives the Hamilton operator of the EM field,
By the use of the commutation relations the second line follows from the first. Note again that
ℏω = hν = ℏc|k| and remember that ω depends on k, even though it is not explicit in the notation. The notation ω(k) could have been introduced, but is not common as it clutters the equations.
where ω ≡ 2πν is the fundamental frequency of the oscillator. The ground state of the oscillator is designated by | 0 ⟩ and is referred to as vacuum state. It can be shown that
is an excitation operator, it excites from an n fold excited state to an n+1 fold excited state:
Since harmonic oscillator energies are equidistant, the n-fold excited state | n⟩ can be looked upon as a single state containing n particles (sometimes called vibrons) all of energy hν. These particles are bosons. For obvious reason the excitation operator
is called a creation operator.
From the commutation relation follows that the Hermitian adjoint
de-excites:
so that
For obvious reason the de-excitation operator
is called an annihilation operator.
By mathematical induction the following "differentiation rule", that will be needed later, is easily proved,
Suppose now we have a number of non-interacting (independent) one-dimensional harmonic oscillators, each with its own fundamental frequency ωi. Because the oscillators are independent, the Hamiltonian is a simple sum:
Making the substitution
we see that the Hamiltonian of the EM field can be looked upon as a Hamiltonian of independent oscillators of energy ω = |k| c and oscillating along direction e(μ) with μ=1,−1.
gives a quantum state of m photons in mode (k,μ) and n photons in mode (k', μ'). The proportionality symbol is used because the state on the left-hand is not normalized to unity, whereas the state on the right-hand may be normalized.
The operator
is the number operator. When acting on a quantum mechanical photon state, it returns the number of photons in mode (k,μ). This also holds when the number of photons in this mode is zero, then the number operator returns zero. To show the action of the number operator on a one-photon ket, we consider
i.e., a number operator of mode (k,μ) returns zero if the mode is unoccupied and returns unity if the mode is singly occupied. To consider the action of the number operator of mode (k, μ) on a n-photon ket of the same mode, we drop the indices k and μ and consider
Use the "differentiation rule" introduced earlier and it follows that
A photon state is an eigenstate of the number operator. This is why the formalism described here, is often referred to as the occupation number representation.
was introduced. The zero of energy can be shifted, which leads to an expression in terms of the number operator,
The effect of H on a single-photon state is
Apparently, the single-photon state is an eigenstate of H and ℏω = hν is the corresponding energy. In the very same way
with
The classical approximation to EM radiation is good when the number of photons is much larger than unity in the volume
where λ is the length of the radio waves. In that case quantum fluctuations are negligible and cannot be heard.
Suppose the radio station broadcasts at ν = 100 MHz, then it is sending out photons with an energy content of νh = 1·108× 6.6·10−34 = 6.6·10−26 J, where h is Planck's constant. The wavelength of the station is λ = c/ν = 3 m, so that λ/(2π) = 48 cm and the volume is 0.111 m3. The energy content of this volume element is 2.1·10−10 × 0.111 = 2.3 ·10−11 J, which amounts to
Obviously, 3.5 ·1012 is much larger than one and hence quantum effects do not play a role; the waves emitted by this station are well into the classical limit.
yields
Quantization gives
The term 1/2 could be dropped, because when one sums over the allowed k, k cancels with −k. The effect of PEM on a single-photon state is
Apparently, the single-photon state is an eigenstate of the momentum operator, and ℏk is the eigenvalue (the momentum of a single photon).
Since the photon propagates with the speed of light
, special relativity
is called for. The relativistic expressions for energy and momentum squared are,
From p2/E2,
Use
and it follows that
so that m0 = 0.
with spin quantum number S = 1. This is similar to, say, the nuclear spin of the 14N isotope
, but with the important difference that the state with MS = 0 is zero, only the states with MS = ±1 are non-zero.
Define spin operators:
The products between the two orthogonal unit vectors are dyadic products. The unit vectors are perpendicular to the propagation direction k (the direction of the z axis, which is the spin quantization axis).
The spin operators satisfy the usual angular momentum commutation relations
Indeed, use the dyadic product property
because ez is of unit length. In this manner,
By inspection it follows that
and therefore μ labels the photon spin,
Because the vector potential A is a transverse field, the photon has no forward (μ = 0) spin component.
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s. Photons are massless particles of definite energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
, definite momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
, and definite spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
.
In order to explain the photoelectric effect
Photoelectric effect
In the photoelectric effect, electrons are emitted from matter as a consequence of their absorption of energy from electromagnetic radiation of very short wavelength, such as visible or ultraviolet light. Electrons emitted in this manner may be referred to as photoelectrons...
, Einstein assumed heuristically in 1905 that an electromagnetic field consists of parcels of energy hν, where h is Planck's constant. In 1927 Paul A. M. Dirac was able to weave the photon concept into the fabrics of the new quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and to describe the interaction of photons with matter. He applied a technique which is now generally called second quantization, although this term is somewhat of a misnomer for EM fields, because they are, after all, solutions of the classical Maxwell equations. In Dirac's theory the fields are quantized for the first time and it is also the first time that Planck's constant enters the expressions. In its original work, Dirac took the phases of the different EM modes (Fourier components of the field) and the mode energies as dynamic variables to quantize (i.e., he reinterpreted them as operator
Operator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
s and postulated commutation relations between them). At present it is more common to quantize the Fourier components of the vector potential
Vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
. This is what will be done below.
A quantum mechanical photon state |k,μ⟩ belonging to mode (k,μ) will be introduced. It will be shown that it has the following properties
These equations say respectively: a photon has zero rest mass; the photon energy is hν=hc|k| (k is the wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
, c is speed of light); its electromagnetic momentum is ℏk [ℏ=h/(2π)]; the polarization μ=±1 is the eigenvalue of the z-component of the photon spin.
Second quantization
Second quantization starts with an expansion of a scalar of vector field (or wave functions) in a basis consisting of a complete set of functions. These expansion functions depend on the coordinates of a single particle. The coefficients multiplying the basis functions are interpreted as operatorOperator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
s and (anti)commutation relations between these new operators are imposed, commutation relations for boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
s and anticommutation relations
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
for fermions (nothing happens to the basis functions themselves). By doing this, the expanded field is converted into a fermion or boson operator field. The expansion coefficients have been promoted from ordinary numbers to operators, creation and annihilation operators. A creation operator creates a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function.
In the case of EM fields the required expansion of the field is the Fourier expansion.
Electromagnetic field and vector potential
- See Fourier expansion electromagnetic field for more details.
As the term suggests, an EM field consists of two vector fields, an electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
E(r,t) and a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
B(r,t). Both are time-dependent vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s that in vacuum depend on a third vector field A(r,t) (the vector potential), under the Coulomb gauge condition, through
here ∇×A is the curl
Curl
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector...
of A.
The Fourier expansion of the vector potential enclosed in a finite cubic box of volume V = L3 is (bar on top indicates complex conjugate):
where the wave vector k gives the propagation direction of the corresponding Fourier component (a polarized monochromatic wave) of A(r,t); the length of the wave vector is |k| = 2πν/c = ω/c, with ν the frequency of the mode. The components of the vector k have discrete values (a consequence of the boundary condition that A has the same value on opposite walls of the box),
The two unit vectors e(μ) ("polarization vectors") are perpendicular to k. They are related to the orthonormal Cartesian vectors ex and ey through a unitary transformation,
The k-th Fourier component of A is a vector perpendicular to k and hence is a linear combination of e(1) and e(−1). The superscript μ indicates a component along e(μ). The Coulomb gauge is imposed,
which makes A into a transverse field
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational vector field and a...
.
Clearly, the (discrete infinite) set of Fourier coefficients
and are variables defining the vector potential. In the following they will be promoted to operators.Quantization of EM field
The best known example of quantization is the replacement of the time-dependent linear momentumMomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
of a particle by the rule
.Note that Planck's constant is introduced here and that the time-dependence of the classical expression is not taken over in the quantum mechanical operator (this is true in the so-called Schrödinger picture
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
).
For the EM field we do something similar. The quantity ε0 is the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...
, which appears here because of the use of electromagnetic SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
units. The quantization rules are:
subject to the boson commutation relations
The square brackets indicate a commutator, defined by
for any two quantum mechanical operators A and B. The introduction of Planck's constant is essential in the transition from a classical to a quantum theory. The factor (2ωVε0)−½ is introduced to give the Hamiltonian (energy operator) a simple form, see below.
The quantized fields (operator fields) are the following
where ω = c |k| = ck.
Hamiltonian of the field
The classical Hamiltonian has the formSubstitution of the field operators into the classical Hamiltonian gives the Hamilton operator of the EM field,
By the use of the commutation relations the second line follows from the first. Note again that
ℏω = hν = ℏc|k| and remember that ω depends on k, even though it is not explicit in the notation. The notation ω(k) could have been introduced, but is not common as it clutters the equations.
Digression: harmonic oscillator
The second quantized treatment of the one-dimensional quantum harmonic oscillator is a well-known topic in quantum mechanical courses. We digress and say a few words about it. The harmonic oscillator Hamiltonian has the formwhere ω ≡ 2πν is the fundamental frequency of the oscillator. The ground state of the oscillator is designated by | 0 ⟩ and is referred to as vacuum state. It can be shown that
is an excitation operator, it excites from an n fold excited state to an n+1 fold excited state:Since harmonic oscillator energies are equidistant, the n-fold excited state | n⟩ can be looked upon as a single state containing n particles (sometimes called vibrons) all of energy hν. These particles are bosons. For obvious reason the excitation operator
is called a creation operator.From the commutation relation follows that the Hermitian adjoint
Hermitian adjoint
In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator.Adjoints of operators generalize conjugate transposes of square matrices to infinite-dimensional situations...
de-excites:so that
For obvious reason the de-excitation operator
is called an annihilation operator.By mathematical induction the following "differentiation rule", that will be needed later, is easily proved,
Suppose now we have a number of non-interacting (independent) one-dimensional harmonic oscillators, each with its own fundamental frequency ωi. Because the oscillators are independent, the Hamiltonian is a simple sum:
Making the substitution
we see that the Hamiltonian of the EM field can be looked upon as a Hamiltonian of independent oscillators of energy ω = |k| c and oscillating along direction e(μ) with μ=1,−1.
Photon states
The quantized EM field has a vacuum (no photons) state | 0 ⟩. The application to it of, say,gives a quantum state of m photons in mode (k,μ) and n photons in mode (k', μ'). The proportionality symbol is used because the state on the left-hand is not normalized to unity, whereas the state on the right-hand may be normalized.
The operator
is the number operator. When acting on a quantum mechanical photon state, it returns the number of photons in mode (k,μ). This also holds when the number of photons in this mode is zero, then the number operator returns zero. To show the action of the number operator on a one-photon ket, we consider
i.e., a number operator of mode (k,μ) returns zero if the mode is unoccupied and returns unity if the mode is singly occupied. To consider the action of the number operator of mode (k, μ) on a n-photon ket of the same mode, we drop the indices k and μ and consider
Use the "differentiation rule" introduced earlier and it follows that
A photon state is an eigenstate of the number operator. This is why the formalism described here, is often referred to as the occupation number representation.
Photon energy
Earlier the Hamiltonian,was introduced. The zero of energy can be shifted, which leads to an expression in terms of the number operator,
The effect of H on a single-photon state is
Apparently, the single-photon state is an eigenstate of H and ℏω = hν is the corresponding energy. In the very same way
with
Example photon density
In this article the electromagnetic energy density was computed that a 100kW radio station creates in its environment; at 5 km from the station it was estimated to be 2.1·10−10 J/m3. Is quantum mechanics needed to describe the broadcasting of this station?The classical approximation to EM radiation is good when the number of photons is much larger than unity in the volume
where λ is the length of the radio waves. In that case quantum fluctuations are negligible and cannot be heard.
Suppose the radio station broadcasts at ν = 100 MHz, then it is sending out photons with an energy content of νh = 1·108× 6.6·10−34 = 6.6·10−26 J, where h is Planck's constant. The wavelength of the station is λ = c/ν = 3 m, so that λ/(2π) = 48 cm and the volume is 0.111 m3. The energy content of this volume element is 2.1·10−10 × 0.111 = 2.3 ·10−11 J, which amounts to
- 3.5 ·1012 photons per
Obviously, 3.5 ·1012 is much larger than one and hence quantum effects do not play a role; the waves emitted by this station are well into the classical limit.
Photon momentum
Introducing the Fourier expansion of the electromagnetic field into the classical formyields
Quantization gives
The term 1/2 could be dropped, because when one sums over the allowed k, k cancels with −k. The effect of PEM on a single-photon state is
Apparently, the single-photon state is an eigenstate of the momentum operator, and ℏk is the eigenvalue (the momentum of a single photon).
Photon mass
The photon having non-zero linear momentum, one could imagine that it has a non-vanishing rest mass m0, which is its mass at zero speed. However, we will now show that this is not the case: m0 = 0.Since the photon propagates with the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
, special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
is called for. The relativistic expressions for energy and momentum squared are,
From p2/E2,
Use
and it follows that
so that m0 = 0.
Photon spin
The photon can be assigned a triplet spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
with spin quantum number S = 1. This is similar to, say, the nuclear spin of the 14N isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
, but with the important difference that the state with MS = 0 is zero, only the states with MS = ±1 are non-zero.
Define spin operators:
The products between the two orthogonal unit vectors are dyadic products. The unit vectors are perpendicular to the propagation direction k (the direction of the z axis, which is the spin quantization axis).
The spin operators satisfy the usual angular momentum commutation relations
Indeed, use the dyadic product property
because ez is of unit length. In this manner,
By inspection it follows that
and therefore μ labels the photon spin,
Because the vector potential A is a transverse field, the photon has no forward (μ = 0) spin component.