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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Quantum Mechanics – Statistical Universe

Quantum mechanics is the name for the mathematics that can describe physical systems on extremely small scales.  When we deal with the macroscopic – i.e scales that we experience in our everyday physical world, then Newtonian mechanics works just fine.  However on the microscopic level of particles, Newtonian mechanics no longer works – hence the need for quantum mechanics.

Quantum mechanics is both very complicated and very weird – I’m going to try and give a very simplified (though not simple!) example of how probabilities are at the heart of quantum mechanics.  Rather than speaking with certainty about the property of an object as we can in classical mechanics, we need to take about the probability that it holds such a property.

For example, one property of particles is spin.  We can have create a particle with the property of either up spin or down spin.  We can visualise this as an arrow pointing up or down:

Screen Shot 2016-01-10 at 8.43.57 PM
Screen Shot 2016-01-10 at 8.44.02 PM

We can then create an apparatus (say the slit below parallel to the z axis) which measures whether the particle is in either up state or down state.  If the particle is in up spin then it will return a value of +1 and if it is in down spin then it will return a value of -1.

Screen Shot 2016-01-10 at 8.46.36 PM

So far so normal.  But here is where things get weird.  If we then rotate the slit 90 degrees clockwise so that it is parallel to the x axis, we would expect  from classical mechanics to get a reading of 0.  i.e the “arrow” will not fit through the slit.  However that is not what happens.  Instead we will still get readings of -1 or +1.  However if we run the experiment a large number of times we find that the mean average reading will indeed be 0!

What has happened is that the act of measuring the particle with the slit has changed the state of the particle.  Say it was previously +1, i.e in up spin, by measuring it with the newly rotated slit we have forced the particle into a new state of either pointing right (right spin) or pointing left (left spin).  Our rotated slit will then return a value of +1 if the particle is in right spin, and will return a value of -1 if the particle in in left spin.

In this case the probability that the apparatus will return a value of +1 is 50% and the probability that the apparatus will return a value of -1 is also 50%.  Therefore when we run this experiment many times we get the average value of 0.  Therefore classical mechanics is achieved as an probabilistic approximation of repeated particle interactions

We can look at a slightly more complicated example – say we don’t rotate the slit 90 degrees, but instead rotate it an arbitrary number of degrees from the z axis as pictured below:

Screen Shot 2016-01-10 at 8.25.19 PM

Here the slit was initially parallel to the z axis in the x,y plane (i.e y=0), and has been rotated Θ degrees.  So the question is what is the probability that our previously up spin particle will return a value of +1 when measured through this new slit?

Screen Shot 2016-01-10 at 9.38.47 PM

The equations above give the probabilities of returning a +1 spin or a -1 spin depending on the angle of orientation.  So in the case of a 90 degree orientation we have both P(+1) and P(-1) = 1/2 as we stated earlier.  An orientation of 45 degrees would have P(+1) = 0.85 and P(-1) = 0.15.  An orientation of 10 degrees would have P(+1) = 0.99 and P(-1) = 0.01.

Screen Shot 2016-01-10 at 8.37.22 PMThe statistical average meanwhile is given by the above formula.  If we rotate the slit by Θ degrees from the z axis in the x,z plane, then run the experiment many times, we will get a long term average of cosΘ.  As we have seen before, when Θ = 90 this means we get an average value of 0.  if Θ = 45 degrees we would get an average reading of √2/2.

This gives a very small snapshot into the ideas of quantum mechanics and the crucial role that probability plays in understanding quantum states.  If you found that difficult, then don’t worry you’re in good company.  As Richard Feynman the legendary physicist once said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.”

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

Screen Shot 2018-03-19 at 4.42.05 PM.png

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2021-05-19 at 10.05.18 AM

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Essential Resources for IB Teachers

1) Intermathematics.com

Screen Shot 2021-08-21 at 1.07.49 PM

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

Screen Shot 2021-12-01 at 1.19.14 PM

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Quantum Mechanics – Statistical Universe

Quantum mechanics is the name for the mathematics that can describe physical systems on extremely small scales.  When we deal with the macroscopic – i.e scales that we experience in our everyday physical world, then Newtonian mechanics works just fine.  However on the microscopic level of particles, Newtonian mechanics no longer works – hence the need for quantum mechanics.

Quantum mechanics is both very complicated and very weird – I’m going to try and give a very simplified (though not simple!) example of how probabilities are at the heart of quantum mechanics.  Rather than speaking with certainty about the property of an object as we can in classical mechanics, we need to take about the probability that it holds such a property.

For example, one property of particles is spin.  We can have create a particle with the property of either up spin or down spin.  We can visualise this as an arrow pointing up or down:

Screen Shot 2016-01-10 at 8.43.57 PM
Screen Shot 2016-01-10 at 8.44.02 PM

We can then create an apparatus (say the slit below parallel to the z axis) which measures whether the particle is in either up state or down state.  If the particle is in up spin then it will return a value of +1 and if it is in down spin then it will return a value of -1.

Screen Shot 2016-01-10 at 8.46.36 PM

So far so normal.  But here is where things get weird.  If we then rotate the slit 90 degrees clockwise so that it is parallel to the x axis, we would expect  from classical mechanics to get a reading of 0.  i.e the “arrow” will not fit through the slit.  However that is not what happens.  Instead we will still get readings of -1 or +1.  However if we run the experiment a large number of times we find that the mean average reading will indeed be 0!

What has happened is that the act of measuring the particle with the slit has changed the state of the particle.  Say it was previously +1, i.e in up spin, by measuring it with the newly rotated slit we have forced the particle into a new state of either pointing right (right spin) or pointing left (left spin).  Our rotated slit will then return a value of +1 if the particle is in right spin, and will return a value of -1 if the particle in in left spin.

In this case the probability that the apparatus will return a value of +1 is 50% and the probability that the apparatus will return a value of -1 is also 50%.  Therefore when we run this experiment many times we get the average value of 0.  Therefore classical mechanics is achieved as an probabilistic approximation of repeated particle interactions

We can look at a slightly more complicated example – say we don’t rotate the slit 90 degrees, but instead rotate it an arbitrary number of degrees from the z axis as pictured below:

Screen Shot 2016-01-10 at 8.25.19 PM

Here the slit was initially parallel to the z axis in the x,y plane (i.e y=0), and has been rotated Θ degrees.  So the question is what is the probability that our previously up spin particle will return a value of +1 when measured through this new slit?

Screen Shot 2016-01-10 at 9.38.47 PM

The equations above give the probabilities of returning a +1 spin or a -1 spin depending on the angle of orientation.  So in the case of a 90 degree orientation we have both P(+1) and P(-1) = 1/2 as we stated earlier.  An orientation of 45 degrees would have P(+1) = 0.85 and P(-1) = 0.15.  An orientation of 10 degrees would have P(+1) = 0.99 and P(-1) = 0.01.

Screen Shot 2016-01-10 at 8.37.22 PMThe statistical average meanwhile is given by the above formula.  If we rotate the slit by Θ degrees from the z axis in the x,z plane, then run the experiment many times, we will get a long term average of cosΘ.  As we have seen before, when Θ = 90 this means we get an average value of 0.  if Θ = 45 degrees we would get an average reading of √2/2.

This gives a very small snapshot into the ideas of quantum mechanics and the crucial role that probability plays in understanding quantum states.  If you found that difficult, then don’t worry you’re in good company.  As Richard Feynman the legendary physicist once said, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.”

Essential Resources for IB Teachers

1) Intermathematics.com

Screen Shot 2021-08-21 at 1.07.49 PM

If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

Screen Shot 2021-12-01 at 1.19.14 PM

I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

Available to download here

IB Maths Super Exploration Guide

A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.

Available to download here.

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