©Fernando Caracena 2013
Resolving light into its fundamental components
Results of experiments that explore the emission and absorption spectra of light show that white light can be resolved into color components (see, Photons and the Electromagnetic field). A simple prism or grating will do the trick. It is possible to prepare a beam of practically monochromatic light which cannot be further resolved in this way into other colors. Even when resolved into a monochromatic component, light still has another degree of freedom that can be analysed.
There are filters, through which monochromatic light passes in a way that reveals still another degree of freedom. Light transmitted through an ideal polarizing filter becomes polarized, but its color does not change, although the light may be dimmed a bit. There are several types of polarizations: (1) plane polarization, (2) circular polarization, and (3) elliptical polarization. The discussion here deals mainly with plane polarized light. The reader is no doubt quite familiar with plane polarizing filters: polarizing glasses remove glare from reflecting surfaces; they are also used in creating 3-D movies.
The degree of freedom of light light, called polarization is an effect related to the spin of the photon, and is a quantum mechanical effect as is shown below. The best way to do an optical experiment on photon polarization is to fix all of the rest of degrees of freedom of light so that only its polarization changes.
Fig. 1a. Two p-filters are aligned for maximum transmission of light polarized by p-filter 1, which subsequently passes through p-filter 2. Note that the p-filter planes are perpendicular to the optical axis, which passes through their centers, and about which axis they can be turned independently through various angles. The arrows on the p-filters depict their polarization alignments, which in this figure show identical allignments.
The optical experiment that we present here may be simple conceptually, but it can be performed in a standard university laboratory. Monochromatic light, perhaps generated by a laser, is sent in a fixed direction (the optical axis) along an optical bench, on which are mounted two plane polarizing filters (p-filters in Fig. 1a and b), which are round, flat, and mounted perpendicular to and concentric with the light beam (the optical axis). Note that the p-filters are perpendicular to the beam because they filter the light in accordance with the orientation of its vibrating electric field, which is transverse to the direction of propagation.
The portion of the light beam that continues beyond p-filter 2 goes through the aperture of a photo detector. Fixed in a plane perpendicular to the beam, the filters can be rotated about the optical axis allowing them to effect polarized light passing through them. The first polarizing filter (p-filter 1 in Figs. 1) sets up a beam of polarized light that gives the zero angle reference of polarization. The orientation of the p-filter 1 is held fast through out the experiment to maintain a beam of fixed polarization, the reference beam which is incident on p-filter 2.
The color of the transmitted light is not altered by an ideal polarizer. The reference beam now plane polarized, passing through the second plane polarizer (p-filter 2), transmits with an intensity that is observed to vary with the angle that p-filter 2 makes with the refence beam prepared with p-filter 1. For convenience let us mark the polarization alignment of p-filter 2(Fig. 1a), which transmits the maximum intensity [I(0)] of light, with a line drawn through its center (short arrows in Fig. 1a) parallel to the refence line drwan in p-filter 1. This establishes the direction of polarization of the reference beam incident on p-filter 2, as the zero polarization direction. When p-filter 2 is rotated at some angle (θ) with respect to the marked reference polarization, the intensity of transmitted light is observed to decrease (Fig 1b) from maximum at to 0°, to zero at 90°. Call the intensity of light at a given angle I(θ), where intensity is defined as the energy flux of light per unit cross section per unit time. A photoelectric detector farther along the beam from p-filter 2 detects the intensity of light passing through the filter with a signal that is proportional to the intensity of the light beam.
Fig. 1b. As in Fig. 1a, but having p-filter 2 rotated through an angle θ relative to the reference polarization.
An empirical relation exists between the maximum transmitted intensity of light through aligned, p-filters and the transmitted intensity when the second filter is turned through an angle, θ, relative to the first p-filter (See Fig. 1b), called Malus' s Law, which runs as follows:
I(θ)/I(0) = cos2(θ). (4)
The following discussion makes use of Malus's Law combined with the concept of a smallest unit of light energy or photon to introduce the mathematical structure of quantum mechanics.
The discrete character of light can be detected directly through the use of a device called a photo-multiplier tube. By hitting a plate a photon entering a photomultiplier tube knocks out a photo-electron that initiates a cascade of electrons, which in a strong, internal electric field grows into an avalanche of electrons that is strong enough to register on electronic circuitry.
Note, never try to detect laser light directly with you eye, because the intensity of laser light is strong enough to blind you.
The initial electron, accelerated by an electric field inside the tube, strikes a second plate, which releases a cascade of a few electrons. In turn these electrons, also accelerated by the electric field, striking another electric plate release a still larger cascade of electrons. After the processes is repeated several times, an avalanche of electrons grows into a pulse of electric current that is strong enough enough to register on electronic equipment and to make a click on a loudspeaker. Ordinary monochromatic light, which is a continuous stream unless otherwise specified, presents such a high density of photons impacts that a photomultiplier produces a hissing sound. Note that light must be attenuated considerably before it reaches the appropriate level of dimness to properly register on the photomultiplier as discrete clicks that are characteristic of photons.
Malus's Law is a property of light that is satisfied down to the single quantum scale of polarized light. The state vectors |θ> and |0> are associated with polarization states of light to give the polarization of photons a quantum description. Suppose that N(0) represents the number of photons entering the photomultiplier tube per unit time when p-filter 2 is aligned to pass the maximum amount of polarized light. When p-filter 2 is turned through an angle (θ) the light intensity emerging from the filter is decreased, not by changing the energy of the photons in the stream, but by decreasing the number of photons that are allowed to pass through p-filter 2, N(θ).
Since nothing has been changed other than rotating p-filter 2, the numbers N(θ) and N(0) are proportional to the intensities of light I((θ) and I(0), respectively, which appear in (4). Therefore by applying Malus' Law (4), we get the following relation
N(θ)/N(0) = cos2(θ), (5)
even if the photomultiplier had registered the counts at a small enough rate that only one photon passed through the apparatus at any given time.
The above ideas set us up to be able to interpret the relationship between bra and ket vectors regarding the plane polarization of a photon. First of all, the ket |0> indicates that a photon spin state is aligned with the reference polarization, and the ket |θ> that a photon spin state is aligned at a polarization angle θ to that of the reference beam. The scaler product of a vector with itself is the amplitude of the vector squared. The rule for taking the scaler product of two vectors in Hilbert space, is to form the dual of one and make a bracket product of it with the other. The dual of the ket |0> is defined as follows
<0| = (|0>)†, (6a)
and of the bra <0|
|0> =(<0|)†, (6b)
where the symbol † stands for the Hermitian adjoint of a ket or a bra. The above relations appear as rather abstract. One way to picture bras and kets is in terms of row and column vectors, which are normally used in vectors over the real numbers, but with some weirdness as is explained below,
, (6c)
, (6d)
.
The blue arrow shows the direction that the index of the vector runs, and it is a solid color because it may be continuous as well as discrete. The vector elements are also complex in general, instead of being limited to just real numbers.
The scalar product is carried out as a sum over the product of the absolute squares of the components, either over a discrete index as in the following
<A|A> =∑ i=0N a*i ai , (6e)
or a continuous index
<A|A> = ∫-∞∞ a*(ξ) a(ξ) dξ , (6f)
in which case the coefficients are functions of a dummy variable ξ, which serves as a continuous index. Henceforth, unless otherwise specified, the bra ket scalar product is taken as in (6f).
In quantum mechanics, it is customary to use normalized bra and ket vectors,
which in this case, applies to the reference polarization ket and the polarization ket for a photon polarized at an angle θ to it
<0|0> = 1 (7a)
and
<θ|θ> =1 . (7b)
Not the natural question to ask is what is the scalar product <θ|0>? In analogy with vectors over the real numbers, we would expect something like
<θ|0> =eiα cos(θ), (7c)
where eiα is an arbitrary complex number of unit magnitude. It is the absolute square of an amplitude such as in(7c), which has physical meaning
|<0|θ>|2 = cos2(θ). (8a)
Comparing (8a) and (5), we see that this amplitude square represents the fraction of photons, which approaching p-filter 2 at polarization |0> are transmitted with the new polarization |θ>
|<0|θ>|2 = N(θ)/N(0). (8b)
The fraction of photons that make it through the p-filter (8b) is always positive definite and ranges between 0 and 1. Because of this property, Niels Bohr and Werner Heisenberg interpreted equations like (8b) as representing a probability, and <0|θ> as a probability amplitude; this interpretation is called the Copenhagen interpretation. In this interpretation, (7c) applies to a single photon approaching a p-filter, which is oriented at an angle θ to the photon's original polarization. On the other side of the p-filter, a photon either randomly emerges polarized in the θ-direction, or it does not. Therefore, the Copenhagen interpretation is that <0|θ> represents the probability amplitude that a photon will pass through the p-filter with polarization direction altered to that of the orientation of the filter.
Dirac's bra and ket notation have an elegant simplicity, yet accurately and powerfully portray quantum states. Because of their simplicity, they may seen too abstract, for which reason, some people may prefer to work with the Shörodinger equation and wave functions. The elegance of Dirac's notation is demonstrated below using plane polarization states, holding all other properties fixed, which form a two dimensional vector space. In this case, we can write out the orthonormal base vectors explicitly.
First, let us look at the operator, P(θ), which describes the action of a p-filter
P(θ) = |θ><θ|. (9a)
A photon in the state |0> approaches a p-filter described by (9) resulting in a new state (|F>)
|F> = P(θ) |0> , (9b)
which reduces to
|F> = <θ|0> |θ> . (9c)
Now, we evaluate the probability amplitude that an incident photon with orientation, |0> has passed through the p-filter
<θ|F> = <θ|0><θ|θ> . (9d)
Substituting for |F> using (9b) we have the expectation value for such a probability amplitude
<θ|P(θ) |0> =cos θ. (9e)
In what follows, we repeat the above steps, but using a more cumbersome row and column vector notation for the bras and kets. First, define the state of the incident photon in terms of a column vector
. (10a)
The orthogonal vector is apparent by inspection
. (10b)
Using the following equation, we can represent a θ-polarized photon state
|θ> = cos θ |0> + sin θ |90> , (10c)
which written out as a column vector is
. (10d)
multiplying the column vector with its adjoint in reverse order results in a square matrix
, (10e)
which is a square matrix representation of the operator P(θ) in (9a). Note that showing these steps is left to the reader as an exercise. Finally, show that the square matrix operating on the column vector in (10a) , results in the equation
P(θ) |0> = cos θ |0>, (10f)
where |0> is written here as a short hand for the column vector in (10a). The continued multiplication by the row vector
<θ| = (cos θ , sin θ) (10g)
results in
<θ|P(θ) |0> = cos θ (cos2 θ + sin2 θ), (10h)
which reduces to
<θ|P(θ) |0> = cos θ, (10i)
the same value as in (9e). Another take on (10i) is that it represents a matrix element in the quantum transition between plane polarization state 0 and final state θ.
Note that Dr. Susskind has an interesting lecture online in which he discusses the mathematics of complex numbers and state vectors.
This ends the discussion on polarization states. The next blog will solve the harmonic oscillator quantum mechanically using the Dirac notation.