Passing SM358 (Quantum Mechanics) at the OU

I received a note today from Dylan H. who works in defence R&D. She's studying quantum theory with the Open University and wrote:

"I read your blog from 2009 on QM SM358 having had my interest prompted on a Google search. I’m currently doing this same course and am already freaking out about the exam in October – do you have any study/revision tips you could pass on?"
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Here are the blog references Dylan cited. Always keen to offer advice here's my reply.

"Hope you're finding the course interesting. I've subsequently been looking at Quantum Field Theory and found the OU course an excellent preparation. The exam is straightforward if you know the material but it DOES reward time assigned to revision.

My own approach was to read through the entire course again, all three volumes, doing the inline questions where I felt any puzzlement. It's amazing how much clearer it seems second time round. There was also a web-based series of revision tests which I faithfully did and although I found them both stressful and irritating they were worth it.

I assume you attended the summer school. If so, you will have been given a number of question papers together with the answers. I did these systematically as part of my revision.

Finally, one of the key issues in a beginning QM course is to figure out the many confusing concepts. SM358 doesn't do a particularly bad job at conceptual clarity [e.g. distinguishing between operators, eigenvectors, eigenvalues, amplitudes, superpositions, representations, observables, ...] but it sometimes helps to have a complementary book. My choice would be Essential Quantum Mechanics by Gary Bowman which focuses relentlessly on the key concepts and you might find it helpful too.

About a week before the exam I worked through three past papers under semi-exam conditions (I made sure I attempted ALL the questions where there were options). I did get a distinction, so I guess the answer is (i) revise ALL the material; (ii) do lots of questions, don't just read stuff."

To my embarrassment I then had to write her a follow-up email:

"In my haste to reply to you before dinner, I omitted to notice that you're just starting out on the course. Let me therefore add:

1. The course is very conceptually demanding in the first few months. If you feel you will never understand it, don't worry, I felt just the same. Stick at it and immerse yourself and it WILL become clearer by June!

2. Go to the Summer School.

3. The book I recommended is great to read in September when you’ve completed Book 2 of the course. It won't make a lot of sense at this stage and it's a waste of your time and money to look at it now. It's use is in revision for a 'second opinion' on the conceptual foundations.

Best,

Nigel."

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We were in Taunton today, about an hour's ride from Wells. The expected torrential rain did not materialise but on the way back we saw a wind turbine spinning at a fantastic rate. It might, however, have been the result of the low-flying Chinook which was powering over the Somerset Levels dressed in its best camo. I'm glad they've finally got the software bugs out of that machine ...

Film Review: Source Code

Following a publishing glitch, my review of the hard-SF short-story collection Engineering Infinity has now been posted at sciencefiction.com.

The Old Bristol Road from Wells to Bristol is charmingly rustic, running alongside Chew Valley Lake and entering Bristol via a steep descent at Dundry. It took us less than an hour to get to the Cribbs Causeway Vue where we enjoyed 'Source Code'. Here's how Wikipedia describes the start of the story.

"Captain Colter Stevens (Jake Gyllenhaal) is a decorated army helicopter pilot whose last memory is of his recent mission in Afghanistan, flying with his team while taking on enemy gunfire. He awakens on a train headed for Chicago with no memory of how he arrived there. His reflection is of a different man and his wallet says his name is Sean Fentress. Sitting across from him is a woman named Christina Warren (Michelle Monaghan) who seems to know him as history teacher Sean Fentress. Before he can understand what is happening, a bomb goes off and destroys the train."

It turns out that Colter is within a simulation based on the memories of the last eight minutes of Sean Fentress's life. He is sent back again and again into those memories with a mission to find the bomb and the bomber to avert an even greater threat to Chicago. To Colter's surprise, the simulation is malleable: as he relives the train experience he learns the sequence of events and can act to change it: recurrences are different every time. Is this just a simulation, as the Source Code mission team seems to believe, or is he occupying alternate histories of the quantum multiverse? (What do you think?)

Although a couple of people (obviously of little imagination) walked out before the end, we thought the film was great. It's thought-provoking and exciting in equal measure and it even manages to end happily - amazing for a movie where everything is based on damage and death.

'Engineering Infinity' and Daffodils

Yesterday morning the sun beautifully illuminated the daffodils in our front hall. When Clare saw it she was impelled to turn it into a big image. After much arithmetic, we were able to print out six blown-up jigsaw pieces and she set to work.


Here is the artist trimming and arranging and glueing onto a cardboard backing.


And here is the result in its chosen location, waiting for some final trimming tomorrow.


A bonus snap of Shadow doing that "can I have some more?" thing.

In other news, my latest review, of the hard-SF short-story collection 'Engineering Infinity' is now at sciencefiction.com.

A taste of summer

We're blessed by some wonderful walks here in the Mendips. This afternoon we parked at the top end of the Cheddar Gorge and walked up the path through Black Rock and into Long Wood. We came back via Velvet Bottom: a couple of pictures of myself and Clare on the way back down.



When I bought my ASUS Netbook last Saturday the salesperson at the Cribbs Causeway PC World store looked at me sharply and said, "You do have a proper computer don't you, as well?" At the time I thought this was a curious point but now I understand what he meant.

The Netbook, running Windows 7 starter, is just fine. Configuring it has taken a couple of days (see previous post) but really presented few problems. The trouble is inherent in a design which costs just a couple of hundred pounds, uses a low-power processor and utilises just 1 GB of RAM.

The machine is midway between a serious device and a toy. It can do everything but it's just so slow! And just a bit brittle as compared to the robust HP laptop which I'm using to write this post. Still, it's just 1 kg and runs for ages on its battery; ideal for travelling around.

Lighting the gas fire

When we first moved into our new house in the winter of last year, we were unable to light the gas fire (the pilot had been switched off). I made frantic phone calls to the previous owners and eventually got instructions. Ever since I have been haunted by the thought that come next winter we will have forgotten again how to light it.



I therefore coerced Clare this afternoon into showing me how to do it (video above). Naturally we messed it up.

Congratulations Adrian on passing your Level-3 Snowboarding exam yesterday at Sun Peaks, Canada!


Here's Clare discussing the good news this afternoon with him. Adrian is due back here from Canada some time next week.


We had the electrician in today to replace our tentacled mess of nested power bars with proper office-like sockets (below). They did some other work too including installing the lighting set above in the upstairs and downstairs halls.


We seem to be overrun by laptops according to the picture above. On the left is my new ASUS Netbook, bought for client presentations and on-site working to replace the Advent Notebook on the right whose fan died some months ago.

I generally budget one day to configure a new machine.

1. Power up, configure Windows 7 and remove the pre-installed clutter.
2. Install AVG free, Skype, TruCrypt and DropBox.
3. Install MS Office 2007, MS FrontPage, Maple 13.
4. Copy all my data files across (around 50 GB).
5. Manage interminable downloads and restarts from MS Update.

All done without a hitch although I have a residual issue with downloading a free email client and configuring it to connect with Pro4's IMAP mail server. The above has taken two calendar days so far ...

Oh, and this post done from the new Netbook.

Noether's Theorem

One of the most foundational concepts in physics is Noether's theorem, which states that symmetries of the laws of nature mathematically imply conserved quantities. Thus conservation of energy is revealed as a logical consequence of the fact that the laws of physics are timeshift-invariant; conservation of momentum is similarly understood to be a consequence of the spatial-translation invariance of the laws of physics. More abstract symmetries (gauge symmetries) imply such things as the conservation of electrical charge.

John Baez in 2002 gave the 'simplest possible proof' of Noether's theorem which I reproduce below. It repays attention as the concepts are quite subtle even though the maths is sixth-form level.

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Note that some of the derivatives below are partial but for simplicity they are all shown with 'd'.

Suppose we have a particle able to move on a line with position x and velocity v = dx/dt under the influence of a potential V(x). Note that both x(t) and v(t) are functions of time t.

The Lagrangian is L = KE - PE = (1/2) mv² - V(x).

The momentum of the particle is p = mv = dL/dv.

The force (caused by the potential) on the particle is F = -dV/dx = dL/dx.

The Euler-Lagrange equation (derived from the principle of least action) says that dL/dx = d/dt(dL/dv), so working it out:

F = dL/dx = d/dt(dL/dv) = d(mv)/dt = dp/dt = mf which is Newton's second law. Note in particular that dp/dt = dL/dx.

So far this is just how Lagrangians work.

A symmetry transformation

Now we suppose that the Lagrangian has a symmetry defined by a parameter s, so that if there is a transformation which sends x (at some t) to some new position x(s) (at time t) then the Lagrangian is unchanged. Note that this makes both x and v now functions of t and s: x(t, s) and v(t, s). I will often suppress one or both of these arguments for notational declutter.

As the Lagrangian is unchanged by the application of s, this means:

dL(x(t, s), v(t, s))/ds = 0.

With less clutter we can write dL(x, v)/ds = 0.

The symmetry might, for example, be: x_t(s) -> x_t + s which simply shifts the particle at x_t spatially by amount s (at every time point). This corresponds to doing the experiment again in a different part of the lab.

Or x_t(s) -> x_t+s which shifts the particle at (x, t) through time by time-increment s. This corresponds to doing the experiment in the afternoon rather than the morning.

So here is the proposed conserved quantity: pdx(s)/ds.

Examples

We assume that the potential V(x) = 0 so there are no applied forces (which makes the conservation law simpler). In the first example, dx(s)/ds = d(x_t + s)/ds = 1 (as t is held constant). So pdx(s)/ds = p. This says that momentum is the conserved quantity in space translation.

In the second example, we hold t constant and let s do the job of incrementing time. Define a new t-coordinate t' = t + s, noting that dt'/ds = 1.

Then dx(s)/ds = (dx/dt')(dt'/ds) = dx/dt' = v so

pdx(s)/ds = pv = 2 * (1/2)mv² which shows that the particle's energy is what will be conserved in time translation.

Proof

Differentiate the allegedly conserved quantity pdx(s)/ds with respect to t. [Conservation of anything means conservation as time passes].

(dp/dt)(dx(s)/ds) + pd[dx(s)/ds]/dt =

(dp/dt)(dx(s)/ds) + pdv(s)/ds ... assuming s is time-independent.

Now we replace p and its derivatives by their Lagrangian equivalents.

(dL/dx)(dx/ds) + (dL/dv)(dv/ds) = dL(x, v)/ds by the chain rule.

But dL/ds = 0 as transformations via s are a symmetry of L, so the conserved quantity pdx(s)/ds is indeed conserved, as Emmy Noether indicated.

Nigel the teenager

As Elaine and myself were clearing my mother's front bedroom over the weekend prior to painting it, I came across this.

This picture of a very gawky teenage-me with my mother in our back garden in Henbury, Bristol is probably from the mid-sixties. It testifies to a lifelong addiction to jeans, cardigans and rolled-up sleeves; I'm probably wearing sandals too. You just can't suppress the essential geek.

A switch 360

Adrian called us yesterday. Due to visa delays he'll be back in the UK in a week or two. Here's an eight second video of him doing something in the snow. I'll let him explain in his own words:

"The clip by the way is a switch 360, normally people ride either left or right foot forward (right in my case). That's what made this shot hard, you can see I'm pretty unbalanced on the way in..."

Pretty in Pink

I was drafted by my sister, Elaine, into helping redecorate my mother's front bedroom today.

Normally I never do interior decoration: my job is to hunt woolly mammoths while Clare sorts out things like painting and decorating. Still, one has to rise to these challenges.


I discovered that painting walls and ceilings is paintbrush stuff around the edges and rollering in the middle. If, like me, you basically want to chuck as much paint as possible onto the wall in the shortest possible time, you absolutely soak the roller in paint. Then, as it spins, it throws off a fine pink mist which covers your arm. There, isn't that interesting? Don't you think that the result (above) is actually rather purple?


After more than a year, building work on our house in Wells is coming to an end. Except, that is, for the area outside the back door (pictured above) which was the interior of a 'utility lean-to' for the previous owner. It desperately needs resurfacing and proper drainage, a project for the warmer weather.


This note of odds-and-ends would not be complete without an update on the Amaryllis, which has beautifully flowered while not failing to exhibit a darker, more suggestive side.