The Quantum World

This page is to celebrate the Open University's Module SM358 "The Quantum World". This is one of the reasons why no additions were made to myWebPages between October 2013 and July 2014 while studying this module. 249 people shared this course. 24% earned a Grade 1 Pass . Very well done to them. 25% achieved a Grade 2 Pass, my result. Well done to us. There were no practical experiments carried out as part of the course but the internet identified two that could be carried out at home. These are:-

Dirac Three Polarizers Experiment

Dirac Three Polarizers Experiment demonstrates quantum states, the projection of a given state vector in another basis set of vectors, the preparation of quantum systems in states with known properties, and the measurement of properties.

Two 10cm by 5 cm polarising were purchased from Greenweld at a total cost of £ 15.93 including postage. These and were cut into four 5cm by 5cm squares. The technical specification of the film was

The 5cm by 5cm polarising sheets were arranged and interleaved on a bright window and photographed. The resulting JPEG was loaded into Windows Paint, it was then cropped and the 4 shades of gray were labelled one to four.

Four shades of gray

One

Half (38%) of the incoming unpolarized photons were transmitted through the first polarizing sheet which was oriented vertically. These transmitted photons have then been prepared in a vertically oriented polarized state, along the z axis. The ket for this state |V〉.

Two

A second polarizing sheet, oriented horizontally is placed over the first sheet. Vertical and horizontal polarized state give a complete orthonormal basis so the probability of a photon in state |V〉 being transmitted through this arrangement to state |H〉 is shown by

〈 H|V〉 = 0, i.e. no (0.005%) photons are transmitted through the overlap.

Three

A third polarizing sheet is inserted at 45 degrees to the z axis between the first and second sheets. Now some light is transmitted through the horizontal polarizing sheet. The Quantum World explanation for this can be found in equation 6.22 from Chapter 6 of Book 2 "Quantum mechanics and its interpretation". Any polarization state can be described as a linear combination of polarisation states that are vertical |V〉 along the z axis and horizontal |H〉 along the x axis where a and b are the probability amplitudes, such that |a|2 +|b|2 = 1.

|general>=a|H> + b|V>

A photon that is vertically polarised along an axis at angle θ to the z axis is represented by

|Vθ 〉 = Cosθ |V〉 + Sinθ |H〉

The probability amplitude for a photon in state |V〉 to be transmitted through a polarizer oriented at at θ to the z axis is

〈Vθ |V〉 = 〈Cosθ V|V〉 + 〈Sinθ H|V〉
              = Cosθ 〈V|V〉 + Sinθ 〈 H|V〉

Since |V> and | H〉 give a complete orthonormal basis 〈V|V〉 =1 and 〈 H|V> = 0 so

〈Vθ |V〉 = Cosθ

and the probability of transmission is |〈Vθ |V〉|2 = Cos2 θ

The probability amplitude for a photon in state |Vθ 〉 to be transmitted through a polarizer oriented horizontally along the x axis is

〈H|Vθ〉 = 〈H|CosθV〉 + 〈H|SinθH〉
              = Cosθ 〈H|V〉 + Sinθ 〈 H|H〉

Since |V〉 and |H〉 give a complete orthonormal basis 〈H|H〉 =1 and 〈 H|V〉 = 0 so

〈H|Vθ〉 = Sinθ

and the probability of transmission is |〈H|Vθ〉|2 = Sin2 θ

The combined probability of an unpolarised photon to be transmitted through a polarizer orientated in the z axis to state |V〉, then to be transmitted through a polarizer at angle θ to state |Vθ 〉 and then through a polarizer horizontal axis to state |H〉 is 1/2* Cos2 θ * Sin2 θ. With θ = 45 degrees the combined probability is 1/8. So an eighth of the incoming unpolarized photons are transmitted through the three layers of polarizing sheets.

Four

Here, the incoming unpolarized photons miss the polarizing sheet that is oriented along the z axis and are firstly incident on the polarizing sheet which is at angle θ to the z axis. Half (38%) of these unpolarized photons are transmitted through the polarizing sheet into state |Vθ 〉. The probability for a photon in state |Vθ 〉 to be transmitted through a polarizer oriented horizontally along the x axis to state |H〉 is Sin2 θ. With θ equal to 45 degrees the combined probability of the unpolarized photons to be transmitted is 1/2 * 1/2. So a quarter of the incoming unpolarized photons are transmitted through the two layers of polarizing sheets.

Four Shades of Gray

Four shades of gray

Quantum mechanics is a mathematical description of the behaviour of subatomic particles.

The Quantum Eraser

This experiment demonstrates several fundamental aspects of the quantum world.

Appendix

A new thing on this page is the vertical bar |. See a table on HTML Special Characters and inserting 〈meta charset="UTF-8"> in the Head of the page.