Download 1 Polarization of Light

January 7, 2008
January 16, 2008
J. Rothberg
minor correction
Physics 225/315
Outline: Introduction to Quantum Mechanics
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1.1
Polarization of Light
Classical Description
~ = E0 x̂ cos(kz − ωt)
Light polarized in the x direction has an electric field vector E
~ = E0 ŷ cos(kz − ωt)
Light polarized in the y direction has an electric field vector E
Light polarized at 45 degrees (call this direction x0 ) has an electric field vector
Ã
!
~ 0 x̂0 cos(kz − ωt) = E0 √1 x̂ cos(kz − ωt) + √1 ŷ cos(kz − ωt)
E
2
2
Light which is right circularly polarized has an electric field vector
Ã
!
~ R = E0 √1 x̂ cos(kz − ωt) + √1 ŷ cos(kz − ωt + π )
E
2
2
2
There are other sign conventions for circularly polarized light; see French and Taylor[1]
Chapter 6 and Le Bellac[2] p. 65 and Sakurai[3] p. 9,10.
Ex and Ey are 90 degrees out of phase and the electric field vector in right circularly polarized
light rotates.
The fields may be represented using complex notation; the actual electric field is the Real
part of the complex electric field. The real part of the exponential is the cosine function
π
cos φ = Re(eiφ ) and recall i = ei 2 .
Ã
~R
E
1.2
Ã
1
= Re E0 √ x̂ ei(kz−ωt) +
2
Ã
Ã
1
= Re E0 √ x̂ ei(kz−ωt) +
2
!!
π
1
√ ŷ ei(kz−ωt+ 2 )
2
!!
i
√ ŷ ei(kz−ωt)
2
Polaroid Filters
The component of the electric field along the transmission direction of the filter is transmitted. After the filter the light is polarized along the transmission direction. Polaroid filters
“prepare” light in a state of polarization.
The intensity of light is proportional to the square of the electric field; therefore after the
filter the intensity is proportional to the square of the cosine of the angle between the
incoming electric field and the Polaroid transmission direction.
See French and Taylor[1] p. 234
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1.3
State Vector Description
We can represent the polarization states by “state” vectors in a two dimensional vector space.
Light polarized in the x direction can be represented by |xi and in the y direction by |yi.
These vectors like |xi are called “kets”, a name proposed by Dirac; it is part of the word
“bracket”. This way of writing vectors in Quantum Mechanics is called Dirac Notation.
Polarization at 45 degrees can be represented by the state |θ = 45o i = √12 (|xi + |yi)
Polarization at an angle θ in the x, y plane, the direction n̂θ , can be described by
|θi = cos θ |xi + sin θ |yi
Right and Left circular polarization can be described by
1
|Ri = √ (|xi + i |yi)
2
and
1
|Li = √ (|xi − i |yi)
2
Notice that the state vectors have complex coefficients in this case.
In the state notation the x, y, θ, R, and L in the brackets are labels for the states.
Basis states can be chosen to be |xi and |yi or |Ri and |Li or |θi and |θ⊥ i and all states can
be written as linear combinations of two orthogonal basis states.
What does the State Vector mean?
“A state vector is not a property of a physical system, but rather represents an
experimental procedure for preparing or testing one or more physical systems.”
F. Laloë[4] quoting A. Peres.[5]
There are other interpretations of the state vector: it may refer to an ensemble of identically
prepared systems or even to a single system. See Le Bellac[2] page 97 footnote.
1.4
Inner Product of State Vectors
With ordinary vectors the inner product can be written as A · B or B · A but with state
vectors we define for each vector a “dual” vector.
There is a one-to-one correspondence between a vector and its dual vector. From a more
mathematical perspective the dual vector provides a mapping from the ket vector onto complex numbers (the inner product). see Cohen-Tannoudji[6] page 110.
The dual vector corresponding to c|θi is c∗ hθ|. Notice the complex conjugate of the constant
c.
The inner products of the basis states are hx|xi = 1 and hy|yi = 1 and hy|xi = 0.
Suppose
|ψi = λ|xi + µ|yi
hψ| = λ∗ hx| + µ∗ hy|
|φi = α|xi + β|yi
hφ| = α∗ hx| + β ∗ hy|
2
where |xi and |yi are orthogonal and normalized.
Show that
hψ|φi = hφ|ψi∗
and from this we see that hψ|ψi is real.
1.4.1
Polarization Analyzers
An ideal Polaroid sheet has these properties:
a) When light goes through a Polaroid sheet (analyzer) the outgoing light is polarized in the
transmission direction of the analyzer.
b) The Polaroid sheet transmits, selects (or projects) the component of the incoming polarization state which is along its transmission direction.
The outgoing state is the component of the state vector of the incoming light projected along
the Polaroid transmission direction.
The amplitude of the outgoing state is the inner product of the state representing the incoming
light and the state representing the analyzer selection.
~
Recall the analogy with ordinary vectors, Ex = x̂ · E
For example if the incoming light is polarized at angle θ and the analyzer transmission
direction is in the y direction then the outgoing amplitude is
hy|θi = hy| (cos θ |xi + sin θ |yi) = cos θhy|xi + sin θhy|yi = sin θ
After the analyzer the polarization direction is y and the amplitude is sin θ.
Classically the intensity of light is proportional to the square of the electric field so the
outgoing intensity is proportional to the magnitude squared of the amplitude.
In this example the intensity I = I0 sin2 θ.
Question: One light beam is linearly polarized at 45 degrees; a second beam is Right Circularly polarized.
Each of these beams is sent through a Polaroid whose transmission direction is y.
How do the intensities compare? How can you distinguish between those two beams of light
using Polaroid filters?
If we multiply a state vector by eiα does the intensity change?
1.5
Photons
In the photon description of light we have to associate a polarization state with a beam of
photons which has passed a Polaroid filter or has been prepared in some way.
We can assign a polarization state |xi, |yi, |θi, |Ri, etc. to a beam of photons.
We prepare a beam of photons in a state |θi by sending the light through a Polaroid filter
whose transmission axis is at angle θ. Any photon observed after the Polaroid filter will be
in the state |θi. The angle θ is measured with respect to the x axis.
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We expect to find the same intensity relationships using the photon picture as with the wave
picture. Recall that each photon carries a quantum of energy (E = hν) and that the intensity
will be proportional to the number of photons.
The probability that a photon in the state |θi will be transmitted by a Polaroid sheet
whose transmission axis is in the x direction is cos2 θ. This is expected since the result must
agree with the classical picture. (The energy is proportional to the square of the Electric
field.)
Suppose we have a beam prepared in the state |θi. The Amplitude for finding a photon
after a Polaroid analyzer is given by the above inner product rules for finding amplitudes. For
example for a Polaroid whose transmission axis is y the amplitude is hy|θi. The “amplitude”
is sometimes called the “probability amplitude”.
The Probability for finding a photon after the analyzer is the magnitude squared of the
amplitude; the probability is |hy|θi|2 . If the states are normalized then the probability will
be ≤ 1.
Of course after the analyzer any photon that we observe will be in state |yi but we know
(almost) nothing about the state of the photon before the analyzer.
We must not conclude that the photon was in state |yi before the analyzer.
“quantum theory is incompatible with the proposition that measuremements are
processes by which we discover some unknown and pre-existing property.”
F. Laloë[4] quoting A. Peres.[5]
1.6
Polarization States
The most general polarization state is |Φi = λ|xi+µ|yi where λ and µ are in general complex
with |λ|2 + |µ|2 = 1. Then |Φi is normalized.
In the case of linear polarization λ and µ are real and can be written as cos θ and sin θ.
The basis states, |xi and |yi, form an orthonormal basis.
The state of a system can be expanded in a linear combination of basis states. The basis is
said to be “complete” if any state can be written as a linear combination of basis states.
Another orthonormal basis can be written as
|θi = cos θ |xi + sin θ |yi
|θ⊥ i = − sin θ |xi + cos θ |yi.
In the case of elliptical polarization (circular is a special case) the coefficients are complex
and can be written as λ = cos θ and µ = sin θ eiδ .
For circular polarization θ = π4 and δ = π2 .
The overall phase is not significant; a state can be multiplied by eiα with no physical consequence. However if we form a superposition of states the relative phase is certainly important.
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2
Principles of Quantum Mechanics
2.1
Postulates - Preliminary Version
1. The state of a system is represented by a vector |Φi. This is chosen to be normalized and
|hΦ|Φi|2 = 1. This is called the state vector of the system. hΦ| is the dual vector.
2. If |Φi and |Ψi represent two physical states the probability amplitude for finding the
system prepared in state Φ to be observed in the state Ψ is given by the inner product hΨ|Φi.
The probability is the magnitude squared of the amplitude, |hΨ|Φi|2 .
After testing for state |Ψi the system is in the state |Ψi with the probability given above.
2.2
Measurement
A system has been prepared in the state |Φi. The state of the system after the testing or
“measurement” is |Ψi and we can take the point of view that the original state has been
projected along this “direction” in the vector space.
This projection of the state after a measurement is sometimes called ”state-vector collapse”
or wave function collapse or reduction.
This state-collapse is sometimes considered a postulate of Quantum Mechanics. In this
perspective we do not discuss the details of the measurement process itself.
Suppose we prepare a state |Φi and expand it in a series of basis states |xi.
|Φi =
X
ax |xi
x
The ax are the amplitudes for each basis state. The amplitude is ax = hx|Φi.
Mermin[7] says
“...the link between |Φi and the value of x revealed by the measurement is this:
the probability of getting the output x is just px = |ax |2 , where ax is the amplitude of |xi in the expansion of |Φi. This connection between amplitudes and
the probabilities of measurement outcomes is known as the Born rule, after the
physicist Max Born.”
Paraphrasing Mermin
“ The postmeasurement state |xi contains no trace of the information present in
the premeasurement state |Φi. (beyond revealing that the amplitude ax 6= 0) ....
the state |Φi collapses or is reduced to the state |xi by the measurement.
The amplitude is ax = hx|Φi.
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2.3
Operators, Projections
An operator in Quantum Mechanics transforms a state into another state. An analogy is a
rotation operator for ordinary vectors in space.
The original state is |Φi and we can write the state after the measurement of |Ψi
|ΨihΨ|Φi = (|ΨihΨ|i) |Φi = PΨ |Φi
and this suggests the definition of a projection operator which transforms |Φi into |Ψi with
a coefficient.
PΨ = |ΨihΨ|
.
Notice that projection operators on basis states have the property that they sum to the
identity operator.
Px + Py = |xihx| + |yihy| = I
An analogy for ordinary two dimensional vectors would be
P = îî· + ĵĵ·
~ ?
What happenes if you apply P to an ordinary two dimensional vector E
2.4
~ =?
PE
Expansion in basis states
A state |Φi can be expanded in a series of basis states:
|Φi = |+ih+|Φi + |−ih−|Φi
In general
|Φi =
X
|ui ihui |Φi
i
where |ui i are the basis states and hui |Φi are the expansion coefficient or amplitudes.
A linear combination of state vectors is (usually) a valid state vector.
The probability of |Φi being found in state |ui i is |hui |Φi|2 , the magnitude squared of the
expansion coefficient (amplitude).
2.5
Mixture vs. Superposition
We must distinguish between a linear combination or superposition of states (sometimes called
a pure state) and a mixture of states.
a) A beam of light polarized at 45 degrees (between x and y) can be described as an equal
weighted superposition of x and y polarization states. √12 (|xi + |yi).
b) A beam of light formed by combining equal intensities of light polarized in the y direction
and light polarized in the x direction (using mirrors, for example) is completely different and
will give different results if you send it through a Polaroid analyzer in certain directions. This
is called a mixture.
What is such a light beam called? What Polaroid directions will enble you to distinguish
these beams?
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2.6
Matrix Representation
States in the two dimensional vector space and operators can be “represented” by two dimensional matrices. “Represented” means that the matrices will obey the same algebraic
relationships as the abstract states and operators.
When we form a matrix representation we do this in a particular basis.
The states |+i and |−i can be represented as column matrices in the +, − basis this way:
Ã
|+i is represented by
1
0
!
Ã
and
|−i is represented by
Ã
In general a state |ψi is represented by:
h+|ψi
h−|ψi
!
The dual vectors are represented by row matrices.
h+| is represented by
Ã
The dual of
a
b
!
is
³
³
´
1 0
a∗ b∗
and
h−| is represented by
!
0
1
³
´
0 1
´
Notice that Ã
the two
and orthogonal: Ã !
Ã
!
! basis states are normalized
³
´ 1
³
´ 0
³
´ 1
0 1
0 1
1 0
=0
=1
=1
0
1
0
Operators acting on such states will be represented by 2 × 2 matrices. for example:
h̄
Sx = 2
Ã
0 1
1 0
!
h̄
Sy = 2
Ã
0 −i
i 0
!
h̄
Sz = 2
Ã
Ã
The matrix representation of the projection operator |+ih+| is
!
1
0
0 −1
1
0
Ã
The matrix representation of the projection operator |−ih−| is
!
0
1
³
!
1 0
³
´
0 1
Notice the rules for multiplying row and column matrices.
Ã
The sum of the projection operators in matrix form is the identity matrix.
Ã
=
´
1 0
0 0
Ã
=
1 0
0 1
!
0 0
0 1
!
!
The most general normalized state is represented by
Ã
|χi =
λ
µ
!
where |λ|2 + |µ|2 = 1
λ and µ are complex and can be parametrized by λ = cos θ and µ = sin θ eiδ .
Still expressing the matrices in the x, y basis we can write the projection operators for Righthanded and Left-handed circular polarization; see Le Bellac [2] page 72.
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PR = |RihR| and in matrix form this is
1
√
2
Ã
1
i
!
Ã
!
´
1 ³
1 1 −i
√
1 −i =
2 i 1
2
We had to take the complex conjugate to form the dual (row) matrix
1
PR =
2
Ã
1 −i
i 1
!
1
and PL =
2
Ã
1 i
−i 1
³
´
1 −i .
!
Ã
!
a
Describe what happens if you operate with PR on an arbitrary state
.
b
We are sometimes casual about equating an operator to its matrix representation. They are
different objects and more correctly a special symbol rather than an “=” should be used.
2.7
Eigenstates and Measurements
Following Le Bellac let’s construct an operator S = PR − PL for circularly polarized light
Ã
S=
0 −i
i 0
!
A state |Ψi is an eigenstate or eigenvector of an operator A provided A|Ψi = a|Ψi where a
is a number called the eigenvalue.
From the definition of S we can see that |Ri and |Li are both eigenvectors of S with eigenvalues +1 and −1. That is S|Ri = |Ri and S|Li = −|Li.
We can associate the operator S with the measurement of circular polarization. Following a
measurement of circular polarization the photon will have collapsed into either state |Ri or
state |Li and the measured value will be either +1 or −1.
Sakurai[3] quotes P. A. M. Dirac, one of the founders of modern quantum mechanics and
comments:
“A measurement always causes the system to jump into an eigenstate of the
dynamical variable that is being measured”
What does all this mean? We interpret Dirac’s words as follows: Before a measurement of observable A is made the system is assumed to be represented by
some linear combination...
|Φi =
X
ci |ui i =
i
X
|ui ihui |Φi
i
where the |ui i are the eigenstates of an operator A corresponding to what is being measured.
“When the measurement is performed the system is ‘thrown into’ one of the
eigenstates, say |u0 i of the observable A. ...
When the measurement causes |Φi to change into |u0 i it is said that A is measured
to be u0 . It is in this sense that the result of a measurement yields one of the
eigenvalues of the observable being measured.”
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2.8
Averages and Expectation Values
Recall that if A|u0 i = u0 |u0 i we say that |u0 i is an eigenstate or eigenvector of the operator A
and u0 , a number, is the eigenvalue.
The probability for finding the system in the state |u0 i after the measurement or equivalently
for measuring u0 is |hu0 |Φi|2 as we have seen before.
If we make measurements on a collection of identically prepared systems (an ensemble) what
will be the average of the measured values?
The average of a collection of values (even in everyday situations) is
Ave =
X
Valuei × Probabilityi
i
The average is
2.8.1
P
i
ui |hui |Φi|2 which can be written
P
i
ui hΦ|ui ihui |Φi =
P
i
ui hΦ|Pi |Φi
An example based on Polarized light
Suppose we have an ensemble of photons prepared in the state |xi and we measure the circular
polarization. What is the average value of the circular polarization?
Recall that Right circular polarization would yield +1 and Left, −1.
The average value of the polarization is then the difference of the probabilities pR − pL , of
the two states.
The probability of finding a Right circularly polarized photon is pR = |hR|xi|2 = hx|RihR|xi
and the probability of finding a Left handed photon is pL = |hL|xi|2 = hx|LihL|xi.
Recall that the projection PR = |RihR| so the average value is
Ave = pR − pL = hx|PR − PL |xi = hx|S|xi
This expression hSi = hx|S|xi is the expectation value of S in the state |xi.
The expectation value of an operator A in the state |Φi is hAi = hΦ|A|Φi.
This is the average value of the results of measurements of A on an ensemble of identically
prepared systems.
2.9
Conjugate Operators
An operator A transforms a state |ψi into a new state |ψ 0 i. We write |ψ 0 i = A|ψi.
The operator which transforms the corresponding dual state (the “bra” in Dirac’s language)
is written as A† .
Then hψ 0 | = hψ|A† .
0
0 ∗
From hψ |φi = hφ|ψ i we obtain the important result
hψ|A† |φi = hφ|A|ψi∗
The Hermitian conjugate of A is A† . see Cohen-Tannoudji[6] p. 118.
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References
[1] A. P. French and Edwin F. Taylor, An Introduction to Quantum Physics, W. W. Norton,
1978.
[2] M. Le Bellac, Quantum Physics. Cambridge University Press, 2006.
[3] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1985.
[4] F. Laloë, “Do we really understand quantum mechanics?”, Am. J. Phys. 69 (6), 2001
[5] A. Peres, “What is a state vector?”, Am J. Phys. 52 (7) 1984.
[6] C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, John Wiley,1977.
[7] N. David Mermin, “From Cbits to Qbits”, Am. J. Phys. 71 (1), 2003.
[8] John S. Townsend, A Modern Approach to Quantum Mechanics, McGraw-Hill, 1992.
[9] M. Le Bellac, Quantum Information and Quantum Computing. Cambridge University
Press, 2006.
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